Riemann hypothesis

Riemann hypothesis   (Redirected from Critical line theorem) Jump to navigation Jump to search For the musical term, see Riemannian theory. The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first nontrivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011. 0:23 Animation showing in 3D the Riemann zeta function critical strip (blue), critical line (red) and zeroes (cross between red and orange): [x,y,z] = [Re(ζ(r + it), Im(ζ(r + it), t] with 0.1 ≤ r ≤ 0.9 and 1 ≤ t ≤ 51 Riemann zeta function along critical line Re(s) = 1/2 (real values are on the horizontal axis and imaginary values are on the vertical axis): Re(ζ(1/2 + it), Im(ζ(1/2 + it) with t ranging between −30 and 30 Millennium Prize Problems Birch and Swinnerton-Dyer conjecture Hodge conjecture Navier–Stokes existence and smoothness P versus NP problem Poincaré conjecture (solved) Riemann hypothesis Yang–Mills existence and mass gap vte In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1 / 2 . Many consider it to be the most important unsolved problem in pure mathematics.[1] It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.

The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems, which offers a million dollars to anyone who solves any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.

The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that: The real part of every nontrivial zero of the Riemann zeta function is  1 / 2 .

Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers 1 / 2 + i t, where t is a real number and i is the imaginary unit.

Contents 1 Riemann zeta function 2 Origin 3 Consequences 3.1 Distribution of prime numbers 3.2 Growth of arithmetic functions 3.3 Lindelöf hypothesis and growth of the zeta function 3.4 Large prime gap conjecture 3.5 Analytic criteria equivalent to the Riemann hypothesis 3.6 Consequences of the generalized Riemann hypothesis 3.7 Excluded middle 3.7.1 Littlewood's theorem 3.7.2 Gauss's class number conjecture 3.7.3 Growth of Euler's totient 4 Generalizations and analogs 4.1 Dirichlet L-series and other number fields 4.2 Function fields and zeta functions of varieties over finite fields 4.3 Arithmetic zeta functions of arithmetic schemes and their L-factors 4.4 Selberg zeta functions 4.5 Ihara zeta functions 4.6 Montgomery's pair correlation conjecture 4.7 Other zeta functions 5 Attempted proofs 5.1 Operator theory 5.2 Lee–Yang theorem 5.3 Turán's result 5.4 Noncommutative geometry 5.5 Hilbert spaces of entire functions 5.6 Quasicrystals 5.7 Arithmetic zeta functions of models of elliptic curves over number fields 5.8 Multiple zeta functions 6 Location of the zeros 6.1 Number of zeros 6.2 Theorem of Hadamard and de la Vallée-Poussin 6.3 Zero-free regions 7 Zeros on the critical line 7.1 Hardy–Littlewood conjectures 7.2 Selberg's zeta function conjecture 7.3 Numerical calculations 7.4 Gram points 8 Arguments for and against the Riemann hypothesis 9 Notes 10 References 10.1 Popular expositions 11 External links Riemann zeta function The Riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergent infinite series {displaystyle zeta (s)=sum _{n=1}^{infty }{frac {1}{n^{s}}}={frac {1}{1^{s}}}+{frac {1}{2^{s}}}+{frac {1}{3^{s}}}+cdots } Leonhard Euler already considered this series in the 1730s for real values of s, in conjunction with his solution to the Basel problem. He also proved that it equals the Euler product {displaystyle zeta (s)=prod _{p{text{ prime}}}{frac {1}{1-p^{-s}}}={frac {1}{1-2^{-s}}}cdot {frac {1}{1-3^{-s}}}cdot {frac {1}{1-5^{-s}}}cdot {frac {1}{1-7^{-s}}}cdot {frac {1}{1-11^{-s}}}cdots } where the infinite product extends over all prime numbers p.[2] The Riemann hypothesis discusses zeros outside the region of convergence of this series and Euler product. To make sense of the hypothesis, it is necessary to analytically continue the function to obtain a form that is valid for all complex s. Because the zeta function is meromorphic, all choices of how to perform this analytic continuation will lead to the same result, by the identity theorem. A first step in this continuation observes that the series for the zeta function and the Dirichlet eta function satisfy the relation {displaystyle left(1-{frac {2}{2^{s}}}right)zeta (s)=eta (s)=sum _{n=1}^{infty }{frac {(-1)^{n+1}}{n^{s}}}={frac {1}{1^{s}}}-{frac {1}{2^{s}}}+{frac {1}{3^{s}}}-cdots ,} within the region of convergence for both series. However, the eta function series on the right converges not just when the real part of s is greater than one, but more generally whenever s has positive real part. Thus, the zeta function can be redefined as {displaystyle eta (s)/(1-2/2^{s})} , extending it from Re(s) > 1 to a larger domain: Re(s) > 0, except for the points where {displaystyle 1-2/2^{s}} is zero. These are the points {displaystyle s=1+2pi in/log 2} where {displaystyle n} can be any nonzero integer; the zeta function can be extended to these values too by taking limits (see Dirichlet eta function § Landau's problem with ζ(s) = η(s)/0 and solutions), giving a finite value for all values of s with positive real part except for the simple pole at s = 1.

In the strip 0 < Re(s) < 1 this extension of the zeta function satisfies the functional equation {displaystyle zeta (s)=2^{s}pi ^{s-1} sin left({frac {pi s}{2}}right) Gamma (1-s) zeta (1-s).} One may then define ζ(s) for all remaining nonzero complex numbers s (Re(s) ≤ 0 and s ≠ 0) by applying this equation outside the strip, and letting ζ(s) equal the right-hand side of the equation whenever s has non-positive real part (and s ≠ 0). If s is a negative even integer then ζ(s) = 0 because the factor sin(πs/2) vanishes; these are the trivial zeros of the zeta function. (If s is a positive even integer this argument does not apply because the zeros of the sine function are cancelled by the poles of the gamma function as it takes negative integer arguments.) The value ζ(0) = −1/2 is not determined by the functional equation, but is the limiting value of ζ(s) as s approaches zero. The functional equation also implies that the zeta function has no zeros with negative real part other than the trivial zeros, so all non-trivial zeros lie in the critical strip where s has real part between 0 and 1. Origin ...es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habe indess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für den nächsten Zweck meiner Untersuchung entbehrlich schien. ...it is very probable that all roots are real. Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the immediate objective of my investigation. — Riemann's statement of the Riemann hypothesis, from (Riemann 1859). (He was discussing a version of the zeta function, modified so that its roots (zeros) are real rather than on the critical line.) Riemann's original motivation for studying the zeta function and its zeros was their occurrence in his explicit formula for the number of primes π(x) less than or equal to a given number x, which he published in his 1859 paper "On the Number of Primes Less Than a Given Magnitude". His formula was given in terms of the related function {displaystyle Pi (x)=pi (x)+{tfrac {1}{2}}pi (x^{frac {1}{2}})+{tfrac {1}{3}}pi (x^{frac {1}{3}})+{tfrac {1}{4}}pi (x^{frac {1}{4}})+{tfrac {1}{5}}pi (x^{frac {1}{5}})+{tfrac {1}{6}}pi (x^{frac {1}{6}})+cdots } which counts the primes and prime powers up to x, counting a prime power pn as 1⁄n. The number of primes can be recovered from this function by using the Möbius inversion formula, {displaystyle {begin{aligned}pi (x)&=sum _{n=1}^{infty }{frac {mu (n)}{n}}Pi (x^{frac {1}{n}})\&=Pi (x)-{frac {1}{2}}Pi (x^{frac {1}{2}})-{frac {1}{3}}Pi (x^{frac {1}{3}})-{frac {1}{5}}Pi (x^{frac {1}{5}})+{frac {1}{6}}Pi (x^{frac {1}{6}})-cdots ,end{aligned}}} where μ is the Möbius function. Riemann's formula is then {displaystyle Pi _{0}(x)=operatorname {li} (x)-sum _{rho }operatorname {li} (x^{rho })-log 2+int _{x}^{infty }{frac {dt}{t(t^{2}-1)log t}}} where the sum is over the nontrivial zeros of the zeta function and where Π0 is a slightly modified version of Π that replaces its value at its points of discontinuity by the average of its upper and lower limits: {displaystyle Pi _{0}(x)=lim _{varepsilon to 0}{frac {Pi (x-varepsilon )+Pi (x+varepsilon )}{2}}.} The summation in Riemann's formula is not absolutely convergent, but may be evaluated by taking the zeros ρ in order of the absolute value of their imaginary part. The function li occurring in the first term is the (unoffset) logarithmic integral function given by the Cauchy principal value of the divergent integral {displaystyle operatorname {li} (x)=int _{0}^{x}{frac {dt}{log t}}.} The terms li(xρ) involving the zeros of the zeta function need some care in their definition as li has branch points at 0 and 1, and are defined (for x > 1) by analytic continuation in the complex variable ρ in the region Re(ρ) > 0, i.e. they should be considered as Ei(ρ log x). The other terms also correspond to zeros: the dominant term li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. For some graphs of the sums of the first few terms of this series see Riesel & Göhl (1970) or Zagier (1977).

This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions. Riemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line s = 1/2 + it, and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(s) ≤ 1. He checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis.

The result has caught the imagination of most mathematicians because it is so unexpected, connecting two seemingly unrelated areas in mathematics; namely, number theory, which is the study of the discrete, and complex analysis, which deals with continuous processes. (Burton 2006, p. 376) Consequences The practical uses of the Riemann hypothesis include many propositions known to be true under the Riemann hypothesis, and some that can be shown to be equivalent to the Riemann hypothesis.

Distribution of prime numbers Riemann's explicit formula for the number of primes less than a given number in terms of a sum over the zeros of the Riemann zeta function says that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function. In particular the error term in the prime number theorem is closely related to the position of the zeros. For example, if β is the upper bound of the real parts of the zeros, then [3] {displaystyle pi (x)-operatorname {li} (x)=Oleft(x^{beta }log xright).} It is already known that 1/2 ≤ β ≤ 1.[4] Von Koch (1901) proved that the Riemann hypothesis implies the "best possible" bound for the error of the prime number theorem. A precise version of Koch's result, due to Schoenfeld (1976), says that the Riemann hypothesis implies {displaystyle |pi (x)-operatorname {li} (x)|<{frac {1}{8pi }}{sqrt {x}}log(x),qquad {text{for all }}xgeq 2657,} where {displaystyle pi (x)} is the prime-counting function, {displaystyle operatorname {li} (x)} is the logarithmic integral function, {displaystyle log(x)} is the natural logarithm of x. Schoenfeld (1976) also showed that the Riemann hypothesis implies {displaystyle |psi (x)-x|<{frac {1}{8pi }}{sqrt {x}}log ^{2}x,qquad {text{for all }}xgeq 73.2,} where {displaystyle psi (x)} is Chebyshev's second function. Dudek (2014) proved that the Riemann hypothesis implies that for all {displaystyle xgeq 2} there is a prime {displaystyle p} satisfying {displaystyle x-{frac {4}{pi }}{sqrt {x}}log x 5040 if and only if the Riemann hypothesis is true, where γ is the Euler–Mascheroni constant.

A related bound was given by Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that: {displaystyle sigma (n) 1, where {displaystyle H_{n}} is the nth harmonic number.[6] The Riemann hypothesis is also true if and only if the inequality {displaystyle {frac {n}{varphi (n)}} 0 {displaystyle sum _{i=1}^{m}|F_{n}(i)-{tfrac {i}{m}}|=Oleft(n^{{frac {1}{2}}+epsilon }right)} is equivalent to the Riemann hypothesis. Here {displaystyle m=sum _{i=1}^{n}phi (i)} is the number of terms in the Farey sequence of order n.

For an example from group theory, if g(n) is Landau's function given by the maximal order of elements of the symmetric group Sn of degree n, then Massias, Nicolas & Robin (1988) showed that the Riemann hypothesis is equivalent to the bound {displaystyle log g(n)<{sqrt {operatorname {Li} ^{-1}(n)}}} for all sufficiently large n. Lindelöf hypothesis and growth of the zeta function The Riemann hypothesis has various weaker consequences as well; one is the Lindelöf hypothesis on the rate of growth of the zeta function on the critical line, which says that, for any ε > 0, {displaystyle zeta left({frac {1}{2}}+itright)=O(t^{varepsilon }),} as {displaystyle tto infty } .

The Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip. For example, it implies that {displaystyle e^{gamma }leq limsup _{trightarrow +infty }{frac {|zeta (1+it)|}{log log t}}leq 2e^{gamma }} {displaystyle {frac {6}{pi ^{2}}}e^{gamma }leq limsup _{trightarrow +infty }{frac {1/|zeta (1+it)|}{log log t}}leq {frac {12}{pi ^{2}}}e^{gamma }} so the growth rate of ζ(1+it) and its inverse would be known up to a factor of 2.[8] Large prime gap conjecture The prime number theorem implies that on average, the gap between the prime p and its successor is log p. However, some gaps between primes may be much larger than the average. Cramér proved that, assuming the Riemann hypothesis, every gap is O(√p log p). This is a case in which even the best bound that can be proved using the Riemann hypothesis is far weaker than what seems true: Cramér's conjecture implies that every gap is O((log p)2), which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis. Numerical evidence supports Cramér's conjecture.[9] Analytic criteria equivalent to the Riemann hypothesis Many statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in proving (or disproving) it. Some typical examples are as follows. (Others involve the divisor function σ(n).) The Riesz criterion was given by Riesz (1916), to the effect that the bound {displaystyle -sum _{k=1}^{infty }{frac {(-x)^{k}}{(k-1)!zeta (2k)}}=Oleft(x^{{frac {1}{4}}+epsilon }right)} holds for all ε > 0 if and only if the Riemann hypothesis holds.

Nyman (1950) proved that the Riemann hypothesis is true if and only if the space of functions of the form {displaystyle f(x)=sum _{nu =1}^{n}c_{nu }rho left({frac {theta _{nu }}{x}}right)} where ρ(z) is the fractional part of z, 0 ≤ θν ≤ 1, and {displaystyle sum _{nu =1}^{n}c_{nu }theta _{nu }=0} , is dense in the Hilbert space L2(0,1) of square-integrable functions on the unit interval. Beurling (1955) extended this by showing that the zeta function has no zeros with real part greater than 1/p if and only if this function space is dense in Lp(0,1). This Nyman-Beurling criterion was strengthened by Baez-Duarte [10] to the case where {displaystyle theta _{nu }in {1/k}_{kgeq 1}} .

Salem (1953) showed that the Riemann hypothesis is true if and only if the integral equation {displaystyle int _{0}^{infty }{frac {z^{-sigma -1}phi (z)}{{e^{x/z}}+1}},dz=0} has no non-trivial bounded solutions {displaystyle phi } for {displaystyle 1/22}(-1)^{(p+1)/2}x^{p}=+infty ,} which says that primes 3 mod 4 are more common than primes 1 mod 4 in some sense. (For related results, see Prime number theorem § Prime number race.) In 1923 Hardy and Littlewood showed that the generalized Riemann hypothesis implies a weak form of the Goldbach conjecture for odd numbers: that every sufficiently large odd number is the sum of three primes, though in 1937 Vinogradov gave an unconditional proof. In 1997 Deshouillers, Effinger, te Riele, and Zinoviev showed that the generalized Riemann hypothesis implies that every odd number greater than 5 is the sum of three primes. In 2013 Harald Helfgott proved the ternary Goldbach conjecture without the GRH dependence, subject to some extensive calculations completed with the help of David J. Platt. In 1934, Chowla showed that the generalized Riemann hypothesis implies that the first prime in the arithmetic progression a mod m is at most Km2log(m)2 for some fixed constant K. In 1967, Hooley showed that the generalized Riemann hypothesis implies Artin's conjecture on primitive roots. In 1973, Weinberger showed that the generalized Riemann hypothesis implies that Euler's list of idoneal numbers is complete. Weinberger (1973) showed that the generalized Riemann hypothesis for the zeta functions of all algebraic number fields implies that any number field with class number 1 is either Euclidean or an imaginary quadratic number field of discriminant −19, −43, −67, or −163. In 1976, G. Miller showed that the generalized Riemann hypothesis implies that one can test if a number is prime in polynomial time via the Miller test. In 2002, Manindra Agrawal, Neeraj Kayal and Nitin Saxena proved this result unconditionally using the AKS primality test. Odlyzko (1990) discussed how the generalized Riemann hypothesis can be used to give sharper estimates for discriminants and class numbers of number fields. Ono & Soundararajan (1997) showed that the generalized Riemann hypothesis implies that Ramanujan's integral quadratic form x2 + y2 + 10z2 represents all integers that it represents locally, with exactly 18 exceptions. Excluded middle Some consequences of the RH are also consequences of its negation, and are thus theorems. In their discussion of the Hecke, Deuring, Mordell, Heilbronn theorem, Ireland & Rosen (1990, p. 359) say The method of proof here is truly amazing. If the generalized Riemann hypothesis is true, then the theorem is true. If the generalized Riemann hypothesis is false, then the theorem is true. Thus, the theorem is true!!     (punctuation in original) Care should be taken to understand what is meant by saying the generalized Riemann hypothesis is false: one should specify exactly which class of Dirichlet series has a counterexample.

Littlewood's theorem This concerns the sign of the error in the prime number theorem. It has been computed that π(x) < li(x) for all x ≤ 1025 (see this table), and no value of x is known for which π(x) > li(x).

In 1914 Littlewood proved that there are arbitrarily large values of x for which {displaystyle pi (x)>operatorname {li} (x)+{frac {1}{3}}{frac {sqrt {x}}{log x}}log log log x,} and that there are also arbitrarily large values of x for which {displaystyle pi (x)C{frac {sqrt {|D|}}{log |D|}}.} Theorem (Deuring; 1933) — If the RH is false then h(D) > 1 if |D| is sufficiently large.

Theorem (Mordell; 1934) — If the RH is false then h(D) → ∞ as D → −∞.

Theorem (Heilbronn; 1934) — If the generalized RH is false for the L-function of some imaginary quadratic Dirichlet character then h(D) → ∞ as D → −∞.

(In the work of Hecke and Heilbronn, the only L-functions that occur are those attached to imaginary quadratic characters, and it is only for those L-functions that GRH is true or GRH is false is intended; a failure of GRH for the L-function of a cubic Dirichlet character would, strictly speaking, mean GRH is false, but that was not the kind of failure of GRH that Heilbronn had in mind, so his assumption was more restricted than simply GRH is false.) In 1935, Carl Siegel later strengthened the result without using RH or GRH in any way.

Growth of Euler's totient In 1983 J. L. Nicolas proved that {displaystyle varphi (n)0,} where λ(n) is the Liouville function given by (−1)r if n has r prime factors. He showed that this in turn would imply that the Riemann hypothesis is true. But Haselgrove (1958) proved that T(x) is negative for infinitely many x (and also disproved the closely related Pólya conjecture), and Borwein, Ferguson & Mossinghoff (2008) showed that the smallest such x is 72185376951205. Spira (1968) showed by numerical calculation that the finite Dirichlet series above for N=19 has a zero with real part greater than 1. Turán also showed that a somewhat weaker assumption, the nonexistence of zeros with real part greater than 1+N−1/2+ε for large N in the finite Dirichlet series above, would also imply the Riemann hypothesis, but Montgomery (1983) showed that for all sufficiently large N these series have zeros with real part greater than 1 + (log log N)/(4 log N). Therefore, Turán's result is vacuously true and cannot help prove the Riemann hypothesis.

Noncommutative geometry Connes (1999, 2000) has described a relationship between the Riemann hypothesis and noncommutative geometry, and showed that a suitable analog of the Selberg trace formula for the action of the idèle class group on the adèle class space would imply the Riemann hypothesis. Some of these ideas are elaborated in Lapidus (2008).

Hilbert spaces of entire functions Louis de Branges (1992) showed that the Riemann hypothesis would follow from a positivity condition on a certain Hilbert space of entire functions. However Conrey & Li (2000) showed that the necessary positivity conditions are not satisfied. Despite this obstacle, de Branges has continued to work on an attempted proof of the Riemann hypothesis along the same lines, but this has not been widely accepted by other mathematicians.[19] Quasicrystals The Riemann hypothesis implies that the zeros of the zeta function form a quasicrystal, a distribution with discrete support whose Fourier transform also has discrete support. Dyson (2009) suggested trying to prove the Riemann hypothesis by classifying, or at least studying, 1-dimensional quasicrystals.

Arithmetic zeta functions of models of elliptic curves over number fields When one goes from geometric dimension one, e.g. an algebraic number field, to geometric dimension two, e.g. a regular model of an elliptic curve over a number field, the two-dimensional part of the generalized Riemann hypothesis for the arithmetic zeta function of the model deals with the poles of the zeta function. In dimension one the study of the zeta integral in Tate's thesis does not lead to new important information on the Riemann hypothesis. Contrary to this, in dimension two work of Ivan Fesenko on two-dimensional generalisation of Tate's thesis includes an integral representation of a zeta integral closely related to the zeta function. In this new situation, not possible in dimension one, the poles of the zeta function can be studied via the zeta integral and associated adele groups. Related conjecture of Fesenko (2010) on the positivity of the fourth derivative of a boundary function associated to the zeta integral essentially implies the pole part of the generalized Riemann hypothesis. Suzuki (2011) proved that the latter, together with some technical assumptions, implies Fesenko's conjecture.

Multiple zeta functions Deligne's proof of the Riemann hypothesis over finite fields used the zeta functions of product varieties, whose zeros and poles correspond to sums of zeros and poles of the original zeta function, in order to bound the real parts of the zeros of the original zeta function. By analogy, Kurokawa (1992) introduced multiple zeta functions whose zeros and poles correspond to sums of zeros and poles of the Riemann zeta function. To make the series converge he restricted to sums of zeros or poles all with non-negative imaginary part. So far, the known bounds on the zeros and poles of the multiple zeta functions are not strong enough to give useful estimates for the zeros of the Riemann zeta function.

Location of the zeros Number of zeros The functional equation combined with the argument principle implies that the number of zeros of the zeta function with imaginary part between 0 and T is given by {displaystyle N(T)={frac {1}{pi }}mathop {mathrm {Arg} } (xi (s))={frac {1}{pi }}mathop {mathrm {Arg} } (Gamma ({tfrac {s}{2}})pi ^{-{frac {s}{2}}}zeta (s)s(s-1)/2)} for s=1/2+iT, where the argument is defined by varying it continuously along the line with Im(s)=T, starting with argument 0 at ∞+iT. This is the sum of a large but well understood term {displaystyle {frac {1}{pi }}mathop {mathrm {Arg} } (Gamma ({tfrac {s}{2}})pi ^{-s/2}s(s-1)/2)={frac {T}{2pi }}log {frac {T}{2pi }}-{frac {T}{2pi }}+7/8+O(1/T)} and a small but rather mysterious term {displaystyle S(T)={frac {1}{pi }}mathop {mathrm {Arg} } (zeta (1/2+iT))=O(log T).} So the density of zeros with imaginary part near T is about log(T)/2π, and the function S describes the small deviations from this. The function S(t) jumps by 1 at each zero of the zeta function, and for t ≥ 8 it decreases monotonically between zeros with derivative close to −log t.

Trudgian (2014) proved that, if {displaystyle T>e} , then {displaystyle |N(T)-{frac {T}{2pi }}log {frac {T}{2pi e}}|leq 0.112log T+0.278log log T+3.385+{frac {0.2}{T}}} .

Karatsuba (1996) proved that every interval (T, T+H] for {displaystyle Hgeq T^{{frac {27}{82}}+varepsilon }} contains at least {displaystyle H(log T)^{frac {1}{3}}e^{-c{sqrt {log log T}}}} points where the function S(t) changes sign.

Selberg (1946) showed that the average moments of even powers of S are given by {displaystyle int _{0}^{T}|S(t)|^{2k}dt={frac {(2k)!}{k!(2pi )^{2k}}}T(log log T)^{k}+O(T(log log T)^{k-1/2}).} This suggests that S(T)/(log log T)1/2 resembles a Gaussian random variable with mean 0 and variance 2π2 (Ghosh (1983) proved this fact). In particular |S(T)| is usually somewhere around (log log T)1/2, but occasionally much larger. The exact order of growth of S(T) is not known. There has been no unconditional improvement to Riemann's original bound S(T)=O(log T), though the Riemann hypothesis implies the slightly smaller bound S(T)=O(log T/log log T).[8] The true order of magnitude may be somewhat less than this, as random functions with the same distribution as S(T) tend to have growth of order about log(T)1/2. In the other direction it cannot be too small: Selberg (1946) showed that S(T) ≠ o((log T)1/3/(log log T)7/3), and assuming the Riemann hypothesis Montgomery showed that S(T) ≠ o((log T)1/2/(log log T)1/2).

Numerical calculations confirm that S grows very slowly: |S(T)| < 1 for T < 280, |S(T)| < 2 for T < 6800000, and the largest value of |S(T)| found so far is not much larger than 3.[20] Riemann's estimate S(T) = O(log T) implies that the gaps between zeros are bounded, and Littlewood improved this slightly, showing that the gaps between their imaginary parts tends to 0. Theorem of Hadamard and de la Vallée-Poussin Hadamard (1896) and de la Vallée-Poussin (1896) independently proved that no zeros could lie on the line Re(s) = 1. Together with the functional equation and the fact that there are no zeros with real part greater than 1, this showed that all non-trivial zeros must lie in the interior of the critical strip 0 < Re(s) < 1. This was a key step in their first proofs of the prime number theorem. Both the original proofs that the zeta function has no zeros with real part 1 are similar, and depend on showing that if ζ(1+it) vanishes, then ζ(1+2it) is singular, which is not possible. One way of doing this is by using the inequality {displaystyle |zeta (sigma )^{3}zeta (sigma +it)^{4}zeta (sigma +2it)|geq 1} for σ > 1, t real, and looking at the limit as σ → 1. This inequality follows by taking the real part of the log of the Euler product to see that {displaystyle |zeta (sigma +it)|=exp Re sum _{p^{n}}{frac {p^{-n(sigma +it)}}{n}}=exp sum _{p^{n}}{frac {p^{-nsigma }cos(tlog p^{n})}{n}},} where the sum is over all prime powers pn, so that {displaystyle |zeta (sigma )^{3}zeta (sigma +it)^{4}zeta (sigma +2it)|=exp sum _{p^{n}}p^{-nsigma }{frac {3+4cos(tlog p^{n})+cos(2tlog p^{n})}{n}}} which is at least 1 because all the terms in the sum are positive, due to the inequality {displaystyle 3+4cos(theta )+cos(2theta )=2(1+cos(theta ))^{2}geq 0.} Zero-free regions De la Vallée-Poussin (1899–1900) proved that if σ + i t is a zero of the Riemann zeta function, then 1 − σ ≥ C / log(t) for some positive constant C. In other words, zeros cannot be too close to the line σ = 1: there is a zero-free region close to this line. This zero-free region has been enlarged by several authors using methods such as Vinogradov's mean-value theorem. Ford (2002) gave a version with explicit numerical constants: ζ(σ + i t ) ≠ 0 whenever |t | ≥ 3 and {displaystyle sigma geq 1-{frac {1}{57.54(log {|t|})^{2/3}(log {log {|t|}})^{1/3}}}.} In 2015, Mossinghoff and Trudgian proved[21] that zeta has no zeros in the region {displaystyle sigma geq 1-{frac {1}{5.573412log |t|}}} for |t| ≥ 2. This is the largest known zero-free region in the critical strip for {displaystyle 3.06cdot 10^{10}<|t|0} , and {displaystyle H=T^{a+varepsilon }} and with as small as possible value of {displaystyle a>0} , where {displaystyle varepsilon >0} is an arbitrarily small number, open two new directions in the investigation of the Riemann zeta function: For any {displaystyle varepsilon >0} there exists a lower bound {displaystyle T_{0}=T_{0}(varepsilon )>0} such that for {displaystyle Tgeq T_{0}} and {displaystyle H=T^{{tfrac {1}{4}}+varepsilon }} the interval {displaystyle (T,T+H]} contains a zero of odd order of the function {displaystyle zeta {bigl (}{tfrac {1}{2}}+it{bigr )}} .

Let {displaystyle N(T)} be the total number of real zeros, and {displaystyle N_{0}(T)} be the total number of zeros of odd order of the function {displaystyle ~zeta left({tfrac {1}{2}}+itright)~} lying on the interval {displaystyle (0,T]~} .

For any {displaystyle varepsilon >0} there exists {displaystyle T_{0}=T_{0}(varepsilon )>0} and some {displaystyle c=c(varepsilon )>0} , such that for {displaystyle Tgeq T_{0}} and {displaystyle H=T^{{tfrac {1}{2}}+varepsilon }} the inequality {displaystyle N_{0}(T+H)-N_{0}(T)geq cH} is true. Selberg's zeta function conjecture Main article: Selberg's zeta function conjecture Atle Selberg (1942) investigated the problem of Hardy–Littlewood 2 and proved that for any ε > 0 there exists such {displaystyle T_{0}=T_{0}(varepsilon )>0} and c = c(ε) > 0, such that for {displaystyle Tgeq T_{0}} and {displaystyle H=T^{0.5+varepsilon }} the inequality {displaystyle N(T+H)-N(T)geq cHlog T} is true. Selberg conjectured that this could be tightened to {displaystyle H=T^{0.5}} . A. A. Karatsuba (1984a, 1984b, 1985) proved that for a fixed ε satisfying the condition 0 < ε < 0.001, a sufficiently large T and {displaystyle H=T^{a+varepsilon }} , {displaystyle a={tfrac {27}{82}}={tfrac {1}{3}}-{tfrac {1}{246}}} , the interval (T, T+H) contains at least cH log(T) real zeros of the Riemann zeta function {displaystyle zeta left({tfrac {1}{2}}+itright)} and therefore confirmed the Selberg conjecture. The estimates of Selberg and Karatsuba can not be improved in respect of the order of growth as T → ∞. Karatsuba (1992) proved that an analog of the Selberg conjecture holds for almost all intervals (T, T+H], {displaystyle H=T^{varepsilon }} , where ε is an arbitrarily small fixed positive number. The Karatsuba method permits to investigate zeros of the Riemann zeta function on "supershort" intervals of the critical line, that is, on the intervals (T, T+H], the length H of which grows slower than any, even arbitrarily small degree T. In particular, he proved that for any given numbers ε, {displaystyle varepsilon _{1}} satisfying the conditions {displaystyle 0 0, the partial sums {displaystyle M(x)=sum _{nleq x}mu (n)} (the values of which are positions in a simple random walk) satisfy the bound {displaystyle M(x)=O(x^{1/2+varepsilon })} with probability 1. The Riemann hypothesis is equivalent to this bound for the Möbius function μ and the Mertens function M derived in the same way from it. In other words, the Riemann hypothesis is in some sense equivalent to saying that μ(x) behaves like a random sequence of coin tosses. When μ(x) is nonzero its sign gives the parity of the number of prime factors of x, so informally the Riemann hypothesis says that the parity of the number of prime factors of an integer behaves randomly. Such probabilistic arguments in number theory often give the right answer, but tend to be very hard to make rigorous, and occasionally give the wrong answer for some results, such as Maier's theorem. The calculations in Odlyzko (1987) show that the zeros of the zeta function behave very much like the eigenvalues of a random Hermitian matrix, suggesting that they are the eigenvalues of some self-adjoint operator, which would imply the Riemann hypothesis. All attempts to find such an operator have failed. There are several theorems, such as Goldbach's weak conjecture for sufficiently large odd numbers, that were first proved using the generalized Riemann hypothesis, and later shown to be true unconditionally. This could be considered as weak evidence for the generalized Riemann hypothesis, as several of its "predictions" are true. Lehmer's phenomenon,[25] where two zeros are sometimes very close, is sometimes given as a reason to disbelieve the Riemann hypothesis. But one would expect this to happen occasionally by chance even if the Riemann hypothesis is true, and Odlyzko's calculations suggest that nearby pairs of zeros occur just as often as predicted by Montgomery's conjecture. Patterson (1988) suggests that the most compelling reason for the Riemann hypothesis for most mathematicians is the hope that primes are distributed as regularly as possible.[26] Notes ^ Bombieri (2000). ^ Leonhard Euler. Variae observationes circa series infinitas. Commentarii academiae scientiarum Petropolitanae 9, 1744, pp. 160–188, Theorems 7 and 8. In Theorem 7 Euler proves the formula in the special case {displaystyle s=1} , and in Theorem 8 he proves it more generally. In the first corollary to his Theorem 7 he notes that {displaystyle zeta (1)=log infty } , and makes use of this latter result in his Theorem 19, in order to show that the sum of the inverses of the prime numbers is {displaystyle log log infty } . ^ Ingham (1932), Theorem 30, p. 83; Montgomery & Vaughan (2007), p. 430 ^ Ingham (1932), p. 82. ^ Robin (1984). ^ Lagarias, Jeffrey C. (2002), "An elementary problem equivalent to the Riemann hypothesis", The American Mathematical Monthly, 109 (6): 534–543, arXiv:math/0008177, doi:10.2307/2695443, ISSN 0002-9890, JSTOR 2695443, MR 1908008, S2CID 15884740 ^ Broughan 2017, Corollary 5.35. ^ Jump up to: a b c Titchmarsh (1986). ^ Nicely (1999). ^ Baez-Duarte, Luis (2005). "A general strong Nyman-Beurling criterion for the Riemann hypothesis". Publications de l'Institut Mathématique. Nouvelle Série. 78 (92): 117–125. doi:10.2298/PIM0578117B. S2CID 17406178. ^ Rodgers & Tao (2020). ^ Platt & Trudgian (2021). ^ Ribenboim (1996), p. 320. ^ Radziejewski (2007). ^ Wiles (2000). ^ Connes (1999). ^ Leichtnam (2005). ^ Knauf (1999). ^ Sarnak (2005). ^ Odlyzko (2002). ^ Mossinghoff, Michael J.; Trudgian, Timothy S. (2015). "Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function". J. Number Theory. 157: 329–349. arXiv:1410.3926. doi:10.1016/J.JNT.2015.05.010. S2CID 117968965. ^ Pratt, Kyle; Robles, Nicolas; Zaharescu, Alexandru; Zeindler, Dirk (2020). "More than five-twelfths of the zeros of ζ are on the critical line". Res Math Sci. 7. arXiv:1802.10521. doi:10.1007/s40687-019-0199-8. S2CID 202542332. ^ Weisstein, Eric W., "Riemann Zeta Function Zeros", MathWorld: "ZetaGrid is a distributed computing project attempting to calculate as many zeros as possible. It had reached 1029.9 billion zeros as of Feb. 18, 2005." ^ Edwards (1974). ^ Lehmer (1956). ^ p. 75: "One should probably add to this list the 'Platonic' reason that one expects the natural numbers to be the most perfect idea conceivable, and that this is only compatible with the primes being distributed in the most regular fashion possible..." References There are several nontechnical books on the Riemann hypothesis, such as Derbyshire (2003), Rockmore (2005), Sabbagh (2003a, 2003b), du Sautoy (2003), and Watkins (2015). The books Edwards (1974), Patterson (1988), Borwein et al. (2008), Mazur & Stein (2015) and Broughan (2017) give mathematical introductions, while Titchmarsh (1986), Ivić (1985) and Karatsuba & Voronin (1992) are advanced monographs.

Artin, Emil (1924), "Quadratische Körper im Gebiete der höheren Kongruenzen. II. Analytischer Teil", Mathematische Zeitschrift, 19 (1): 207–246, doi:10.1007/BF01181075, S2CID 117936362 Backlund, R. J. (1914), "Sur les Zéros de la Fonction ζ(s) de Riemann", C. R. Acad. Sci. Paris, 158: 1979–1981 Beurling, Arne (1955), "A closure problem related to the Riemann zeta-function", Proceedings of the National Academy of Sciences of the United States of America, 41 (5): 312–314, Bibcode:1955PNAS...41..312B, doi:10.1073/pnas.41.5.312, MR 0070655, PMC 528084, PMID 16589670 Bohr, H.; Landau, E. (1914), "Ein Satz über Dirichletsche Reihen mit Anwendung auf die ζ-Funktion und die L-Funktionen", Rendiconti del Circolo Matematico di Palermo, 37 (1): 269–272, doi:10.1007/BF03014823, S2CID 121145912 Bombieri, Enrico (2000), The Riemann Hypothesis – official problem description (PDF), Clay Mathematics Institute, retrieved 2008-10-25 Reprinted in (Borwein et al. 2008). Borwein, Peter; Choi, Stephen; Rooney, Brendan; Weirathmueller, Andrea, eds. (2008), The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, CMS Books in Mathematics, New York: Springer, doi:10.1007/978-0-387-72126-2, ISBN 978-0-387-72125-5 Borwein, Peter; Ferguson, Ron; Mossinghoff, Michael J. (2008), "Sign changes in sums of the Liouville function", Mathematics of Computation, 77 (263): 1681–1694, Bibcode:2008MaCom..77.1681B, doi:10.1090/S0025-5718-08-02036-X, MR 2398787 de Branges, Louis (1992), "The convergence of Euler products", Journal of Functional Analysis, 107 (1): 122–210, doi:10.1016/0022-1236(92)90103-P, MR 1165869 Broughan, Kevin (2017), Equivalents of the Riemann Hypothesis, Cambridge University Press, ISBN 978-1108290784 Burton, David M. (2006), Elementary Number Theory, Tata McGraw-Hill Publishing Company Limited, ISBN 978-0-07-061607-3 Cartier, P. (1982), "Comment l'hypothèse de Riemann ne fut pas prouvée", Seminar on Number Theory, Paris 1980–81 (Paris, 1980/1981), Progr. Math., vol. 22, Boston, MA: Birkhäuser Boston, pp. 35–48, MR 0693308 Connes, Alain (1999), "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function", Selecta Mathematica, New Series, 5 (1): 29–106, arXiv:math/9811068, doi:10.1007/s000290050042, MR 1694895, S2CID 55820659 Connes, Alain (2000), "Noncommutative geometry and the Riemann zeta function", Mathematics: frontiers and perspectives, Providence, R.I.: American Mathematical Society, pp. 35–54, MR 1754766 Connes, Alain (2016), "An Essay on the Riemann Hypothesis", in Nash, J. F.; Rassias, Michael (eds.), Open Problems in Mathematics, New York: Springer, pp. 225–257, arXiv:1509.05576, doi:10.1007/978-3-319-32162-2_5 Conrey, J. B. (1989), "More than two fifths of the zeros of the Riemann zeta function are on the critical line", J. Reine Angew. Math., 1989 (399): 1–26, doi:10.1515/crll.1989.399.1, MR 1004130, S2CID 115910600 Conrey, J. Brian (2003), "The Riemann Hypothesis" (PDF), Notices of the American Mathematical Society: 341–353 Reprinted in (Borwein et al. 2008). Conrey, J. B.; Li, Xian-Jin (2000), "A note on some positivity conditions related to zeta and L-functions", International Mathematics Research Notices, 2000 (18): 929–940, arXiv:math/9812166, doi:10.1155/S1073792800000489, MR 1792282, S2CID 14678312 Deligne, Pierre (1974), "La conjecture de Weil. I", Publications Mathématiques de l'IHÉS, 43: 273–307, doi:10.1007/BF02684373, MR 0340258, S2CID 123139343 Deligne, Pierre (1980), "La conjecture de Weil : II", Publications Mathématiques de l'IHÉS, 52: 137–252, doi:10.1007/BF02684780, S2CID 189769469 Deninger, Christopher (1998), "Some analogies between number theory and dynamical systems on foliated spaces", Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), Documenta Mathematica, pp. 163–186, MR 1648030 Dudek, Adrian W. (2014-08-21), "On the Riemann hypothesis and the difference between primes", International Journal of Number Theory, 11 (3): 771–778, arXiv:1402.6417, Bibcode:2014arXiv1402.6417D, doi:10.1142/S1793042115500426, ISSN 1793-0421, S2CID 119321107 Dyson, Freeman (2009), "Birds and frogs" (PDF), Notices of the American Mathematical Society, 56 (2): 212–223, MR 2483565 Edwards, H. M. (1974), Riemann's Zeta Function, New York: Dover Publications, ISBN 978-0-486-41740-0, MR 0466039 Fesenko, Ivan (2010), "Analysis on arithmetic schemes. II", Journal of K-theory, 5 (3): 437–557, doi:10.1017/is010004028jkt103 Ford, Kevin (2002), "Vinogradov's integral and bounds for the Riemann zeta function", Proceedings of the London Mathematical Society, Third Series, 85 (3): 565–633, arXiv:1910.08209, doi:10.1112/S0024611502013655, MR 1936814, S2CID 121144007 Franel, J.; Landau, E. (1924), "Les suites de Farey et le problème des nombres premiers" (Franel, 198–201); "Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel (Landau, 202–206)", Göttinger Nachrichten: 198–206 Ghosh, Amit (1983), "On the Riemann zeta function—mean value theorems and the distribution of |S(T)|", J. Number Theory, 17: 93–102, doi:10.1016/0022-314X(83)90010-0 Gourdon, Xavier (2004), The 1013 first zeros of the Riemann Zeta function, and zeros computation at very large height (PDF) Gram, J. P. (1903), "Note sur les zéros de la fonction ζ(s) de Riemann", Acta Mathematica, 27: 289–304, doi:10.1007/BF02421310, S2CID 115327214 Hadamard, Jacques (1896), "Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques", Bulletin de la Société Mathématique de France, 14: 199–220, doi:10.24033/bsmf.545 Reprinted in (Borwein et al. 2008). Hardy, G. H. (1914), "Sur les Zéros de la Fonction ζ(s) de Riemann", C. R. Acad. Sci. Paris, 158: 1012–1014, JFM 45.0716.04 Reprinted in (Borwein et al. 2008). Hardy, G. H.; Littlewood, J. E. (1921), "The zeros of Riemann's zeta-function on the critical line", Math. Z., 10 (3–4): 283–317, doi:10.1007/BF01211614, S2CID 126338046 Haselgrove, C. B. (1958), "A disproof of a conjecture of Pólya", Mathematika, 5 (2): 141–145, doi:10.1112/S0025579300001480, ISSN 0025-5793, MR 0104638, Zbl 0085.27102 Reprinted in (Borwein et al. 2008). Haselgrove, C. B.; Miller, J. C. P. (1960), Tables of the Riemann zeta function, Royal Society Mathematical Tables, Vol. 6, Cambridge University Press, ISBN 978-0-521-06152-0, MR 0117905 Review Hutchinson, J. I. (1925), "On the Roots of the Riemann Zeta-Function", Transactions of the American Mathematical Society, 27 (1): 49–60, doi:10.2307/1989163, JSTOR 1989163 Ingham, A.E. (1932), The Distribution of Prime Numbers, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 30, Cambridge University Press. Reprinted 1990, ISBN 978-0-521-39789-6, MR1074573 Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, ISBN 0-387-97329-X Ivić, A. (1985), The Riemann Zeta Function, New York: John Wiley & Sons, ISBN 978-0-471-80634-9, MR 0792089 (Reprinted by Dover 2003) Ivić, Aleksandar (2008), "On some reasons for doubting the Riemann hypothesis", in Borwein, Peter; Choi, Stephen; Rooney, Brendan; Weirathmueller, Andrea (eds.), The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, CMS Books in Mathematics, New York: Springer, pp. 131–160, arXiv:math.NT/0311162, ISBN 978-0-387-72125-5 Karatsuba, A. A. (1984a), "Zeros of the function ζ(s) on short intervals of the critical line", Izv. Akad. Nauk SSSR, Ser. Mat. (in Russian), 48 (3): 569–584, MR 0747251 Karatsuba, A. A. (1984b), "Distribution of zeros of the function ζ(1/2 + it)", Izv. Akad. Nauk SSSR, Ser. Mat. (in Russian), 48 (6): 1214–1224, MR 0772113 Karatsuba, A. A. (1985), "Zeros of the Riemann zeta-function on the critical line", Trudy Mat. Inst. Steklov. (in Russian) (167): 167–178, MR 0804073 Karatsuba, A. A. (1992), "On the number of zeros of the Riemann zeta-function lying in almost all short intervals of the critical line", Izv. Ross. Akad. Nauk, Ser. Mat. (in Russian), 56 (2): 372–397, Bibcode:1993IzMat..40..353K, doi:10.1070/IM1993v040n02ABEH002168, MR 1180378 Karatsuba, A. A.; Voronin, S. M. (1992), The Riemann zeta-function, de Gruyter Expositions in Mathematics, vol. 5, Berlin: Walter de Gruyter & Co., doi:10.1515/9783110886146, ISBN 978-3-11-013170-3, MR 1183467 Keating, Jonathan P.; Snaith, N. C. (2000), "Random matrix theory and ζ(1/2 + it)", Communications in Mathematical Physics, 214 (1): 57–89, Bibcode:2000CMaPh.214...57K, doi:10.1007/s002200000261, MR 1794265, S2CID 11095649 Knapowski, S. (1962), "On sign-changes of the difference {displaystyle pi (x)-operatorname {li} x} ", Acta Arithmetica, 7: 107–119, doi:10.4064/aa-7-2-107-119, MR 0133308 Knauf, Andreas (1999), "Number theory, dynamical systems and statistical mechanics", Reviews in Mathematical Physics, 11 (8): 1027–1060, Bibcode:1999RvMaP..11.1027K, doi:10.1142/S0129055X99000325, MR 1714352 von Koch, Niels Helge (1901), "Sur la distribution des nombres premiers", Acta Mathematica, 24: 159–182, doi:10.1007/BF02403071, S2CID 119914826 Kurokawa, Nobushige (1992), "Multiple zeta functions: an example", Zeta functions in geometry (Tokyo, 1990), Adv. Stud. Pure Math., vol. 21, Tokyo: Kinokuniya, pp. 219–226, MR 1210791 Lapidus, Michel L. (2008), In search of the Riemann zeros, Providence, R.I.: American Mathematical Society, doi:10.1090/mbk/051, ISBN 978-0-8218-4222-5, MR 2375028 Lavrik, A. F. (2001) [1994], "Zeta-function", Encyclopedia of Mathematics, EMS Press Lehmer, D. H. (1956), "Extended computation of the Riemann zeta-function", Mathematika, 3 (2): 102–108, doi:10.1112/S0025579300001753, MR 0086083 Leichtnam, Eric (2005), "An invitation to Deninger's work on arithmetic zeta functions", Geometry, spectral theory, groups, and dynamics, Contemp. Math., vol. 387, Providence, RI: Amer. Math. Soc., pp. 201–236, doi:10.1090/conm/387/07243, MR 2180209. Levinson, N. (1974), "More than one-third of the zeros of Riemann's zeta function are on σ = 1/2", Advances in Mathematics, 13 (4): 383–436, doi:10.1016/0001-8708(74)90074-7, MR 0564081 Littlewood, J. E. (1962), "The Riemann hypothesis", The scientist speculates: an anthology of partly baked idea, New York: Basic books van de Lune, J.; te Riele, H. J. J.; Winter, D. T. (1986), "On the zeros of the Riemann zeta function in the critical strip. IV", Mathematics of Computation, 46 (174): 667–681, doi:10.2307/2008005, JSTOR 2008005, MR 0829637 Massias, J.-P.; Nicolas, Jean-Louis; Robin, G. (1988), "Évaluation asymptotique de l'ordre maximum d'un élément du groupe symétrique", Acta Arithmetica, 50 (3): 221–242, doi:10.4064/aa-50-3-221-242, MR 0960551 Mazur, Barry; Stein, William (2015), Prime Numbers and the Riemann Hypothesis Montgomery, Hugh L. (1973), "The pair correlation of zeros of the zeta function", Analytic number theory, Proc. Sympos. Pure Math., vol. XXIV, Providence, R.I.: American Mathematical Society, pp. 181–193, MR 0337821 Reprinted in (Borwein et al. 2008). Montgomery, Hugh L. (1983), "Zeros of approximations to the zeta function", in Erdős, Paul (ed.), Studies in pure mathematics. To the memory of Paul Turán, Basel, Boston, Berlin: Birkhäuser, pp. 497–506, ISBN 978-3-7643-1288-6, MR 0820245 Montgomery, Hugh L.; Vaughan, Robert C. (2007), Multiplicative Number Theory I. Classical Theory, Cambridge studies in advanced mathematics, vol. 97, Cambridge University Press.ISBN 978-0-521-84903-6 Nicely, Thomas R. (1999), "New maximal prime gaps and first occurrences", Mathematics of Computation, 68 (227): 1311–1315, Bibcode:1999MaCom..68.1311N, doi:10.1090/S0025-5718-99-01065-0, MR 1627813. Nyman, Bertil (1950), On the One-Dimensional Translation Group and Semi-Group in Certain Function Spaces, PhD Thesis, University of Uppsala: University of Uppsala, MR 0036444 Odlyzko, A. M.; te Riele, H. J. J. (1985), "Disproof of the Mertens conjecture", Journal für die reine und angewandte Mathematik, 1985 (357): 138–160, doi:10.1515/crll.1985.357.138, MR 0783538, S2CID 13016831, archived from the original on 2012-07-11 Odlyzko, A. M. (1987), "On the distribution of spacings between zeros of the zeta function", Mathematics of Computation, 48 (177): 273–308, doi:10.2307/2007890, JSTOR 2007890, MR 0866115 Odlyzko, A. M. (1990), "Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results", Séminaire de Théorie des Nombres de Bordeaux, Série 2, 2 (1): 119–141, doi:10.5802/jtnb.22, MR 1061762 Odlyzko, A. M. (1992), The 1020-th zero of the Riemann zeta function and 175 million of its neighbors (PDF) This unpublished book describes the implementation of the algorithm and discusses the results in detail. Odlyzko, A. M. (1998), The 1021st zero of the Riemann zeta function (PDF) Ono, Ken; Soundararajan, K. (1997), "Ramanujan's ternary quadratic form", Inventiones Mathematicae, 130 (3): 415–454, Bibcode:1997InMat.130..415O, doi:10.1007/s002220050191, S2CID 122314044 Patterson, S. J. (1988), An introduction to the theory of the Riemann zeta-function, Cambridge Studies in Advanced Mathematics, vol. 14, Cambridge University Press, doi:10.1017/CBO9780511623707, ISBN 978-0-521-33535-5, MR 0933558 Platt, Dave; Trudgian, Tim (January 2021), "The Riemann hypothesis is true up to {displaystyle 3cdot 10^{12}} ", Bulletin of the London Mathematical Society, Wiley, arXiv:2004.09765, doi:10.1112/blms.12460, S2CID 234355998 Radziejewski, Maciej (2007), "Independence of Hecke zeta functions of finite order over normal fields", Transactions of the American Mathematical Society, 359 (5): 2383–2394, doi:10.1090/S0002-9947-06-04078-5, MR 2276625, There are infinitely many nonisomorphic algebraic number fields whose Dedekind zeta functions have infinitely many nontrivial multiple zeros. Ribenboim, Paulo (1996), The New Book of Prime Number Records, New York: Springer, ISBN 0-387-94457-5 Riemann, Bernhard (1859), "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse", Monatsberichte der Berliner Akademie. In Gesammelte Werke, Teubner, Leipzig (1892), Reprinted by Dover, New York (1953). Original manuscript (with English translation). Reprinted in (Borwein et al. 2008) and (Edwards 1974) Riesel, Hans; Göhl, Gunnar (1970), "Some calculations related to Riemann's prime number formula", Mathematics of Computation, 24 (112): 969–983, doi:10.2307/2004630, JSTOR 2004630, MR 0277489 Riesz, M. (1916), "Sur l'hypothèse de Riemann", Acta Mathematica, 40: 185–190, doi:10.1007/BF02418544 Robin, G. (1984), "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", Journal de Mathématiques Pures et Appliquées, Neuvième Série, 63 (2): 187–213, MR 0774171 Rodgers, Brad; Tao, Terence (2020), "The de Bruijn–Newman constant is non-negative", Forum of Mathematics, 8: e6, 62, doi:10.1017/fmp.2020.6, MR 4089393; see also announcement on Tao's blog, January 19, 2018 Rosser, J. Barkley; Yohe, J. M.; Schoenfeld, Lowell (1969), "Rigorous computation and the zeros of the Riemann zeta-function. (With discussion)", Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968), Vol. 1: Mathematics, Software, Amsterdam: North-Holland, pp. 70–76, MR 0258245 Rudin, Walter (1973), Functional Analysis, 1st edition (January 1973), New York: McGraw-Hill, ISBN 0-070-54225-2 Salem, Raphaël (1953), "Sur une proposition équivalente à l'hypothèse de Riemann", Les Comptes rendus de l'Académie des sciences, 236: 1127–1128, MR 0053148 Sarnak, Peter (2005), Problems of the Millennium: The Riemann Hypothesis (2004) (PDF), Clay Mathematics Institute, retrieved 2015-07-28 Reprinted in (Borwein et al. 2008). Schoenfeld, Lowell (1976), "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II", Mathematics of Computation, 30 (134): 337–360, doi:10.2307/2005976, JSTOR 2005976, MR 0457374 Schumayer, Daniel; Hutchinson, David A. W. (2011), "Physics of the Riemann Hypothesis", Reviews of Modern Physics, 83 (2): 307–330, arXiv:1101.3116, Bibcode:2011RvMP...83..307S, doi:10.1103/RevModPhys.83.307, S2CID 119290777 Selberg, Atle (1942), "On the zeros of Riemann's zeta-function", SKR. Norske Vid. Akad. Oslo I., 10: 59 pp, MR 0010712 Selberg, Atle (1946), "Contributions to the theory of the Riemann zeta-function", Arch. Math. Naturvid., 48 (5): 89–155, MR 0020594 Selberg, Atle (1956), "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series", J. Indian Math. Soc., New Series, 20: 47–87, MR 0088511 Serre, Jean-Pierre (1969–1970), "Facteurs locaux des fonctions zeta des varietés algébriques (définitions et conjectures)", Séminaire Delange-Pisot-Poitou, 19 Sheats, Jeffrey T. (1998), "The Riemann hypothesis for the Goss zeta function for Fq[T]", Journal of Number Theory, 71 (1): 121–157, arXiv:math/9801158, doi:10.1006/jnth.1998.2232, MR 1630979, S2CID 119703557 Siegel, C. L. (1932), "Über Riemanns Nachlaß zur analytischen Zahlentheorie", Quellen Studien zur Geschichte der Math. Astron. Und Phys. Abt. B: Studien 2: 45–80 Reprinted in Gesammelte Abhandlungen, Vol. 1. Berlin: Springer-Verlag, 1966. Speiser, Andreas (1934), "Geometrisches zur Riemannschen Zetafunktion", Mathematische Annalen, 110: 514–521, doi:10.1007/BF01448042, JFM 60.0272.04, S2CID 119413347, archived from the original on 2015-06-27 Spira, Robert (1968), "Zeros of sections of the zeta function. II", Mathematics of Computation, 22 (101): 163–173, doi:10.2307/2004774, JSTOR 2004774, MR 0228456 Stein, William; Mazur, Barry (2007), What is Riemann's Hypothesis? (PDF), archived from the original (PDF) on 2009-03-27 Suzuki, Masatoshi (2011), "Positivity of certain functions associated with analysis on elliptic surfaces", Journal of Number Theory, 131 (10): 1770–1796, doi:10.1016/j.jnt.2011.03.007 Titchmarsh, Edward Charles (1935), "The Zeros of the Riemann Zeta-Function", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, The Royal Society, 151 (873): 234–255, Bibcode:1935RSPSA.151..234T, doi:10.1098/rspa.1935.0146, JSTOR 96545 Titchmarsh, Edward Charles (1936), "The Zeros of the Riemann Zeta-Function", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, The Royal Society, 157 (891): 261–263, Bibcode:1936RSPSA.157..261T, doi:10.1098/rspa.1936.0192, JSTOR 96692 Titchmarsh, Edward Charles (1986), The theory of the Riemann zeta-function (2nd ed.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853369-6, MR 0882550 Trudgian, Timothy S. (2014), "An improved upper bound for the argument of the Riemann zeta function on the critical line II", J. Number Theory, 134: 280–292, arXiv:1208.5846, doi:10.1016/j.jnt.2013.07.017 Trudgian, Timothy (2011), "On the success and failure of Gram's Law and the Rosser Rule", Acta Arithmetica, 125 (3): 225–256, doi:10.4064/aa148-3-2 Turán, Paul (1948), "On some approximative Dirichlet-polynomials in the theory of the zeta-function of Riemann", Danske Vid. Selsk. Mat.-Fys. Medd., 24 (17): 36, MR 0027305 Reprinted in (Borwein et al. 2008). Turing, Alan M. (1953), "Some calculations of the Riemann zeta-function", Proceedings of the London Mathematical Society, Third Series, 3: 99–117, doi:10.1112/plms/s3-3.1.99, MR 0055785 de la Vallée-Poussin, Ch.J. (1896), "Recherches analytiques sur la théorie des nombers premiers", Ann. Soc. Sci. Bruxelles, 20: 183–256 de la Vallée-Poussin, Ch.J. (1899–1900), "Sur la fonction ζ(s) de Riemann et la nombre des nombres premiers inférieurs à une limite donnée", Mem. Couronnes Acad. Sci. Belg., 59 (1) Reprinted in (Borwein et al. 2008). Weil, André (1948), Sur les courbes algébriques et les variétés qui s'en déduisent, Actualités Sci. Ind., no. 1041 = Publ. Inst. Math. Univ. Strasbourg 7 (1945), Hermann et Cie., Paris, MR 0027151 Weil, André (1949), "Numbers of solutions of equations in finite fields", Bulletin of the American Mathematical Society, 55 (5): 497–508, doi:10.1090/S0002-9904-1949-09219-4, MR 0029393 Reprinted in Oeuvres Scientifiques/Collected Papers by Andre Weil ISBN 0-387-90330-5 Weinberger, Peter J. (1973), "On Euclidean rings of algebraic integers", Analytic number theory ( St. Louis Univ., 1972), Proc. Sympos. Pure Math., vol. 24, Providence, R.I.: Amer. Math. Soc., pp. 321–332, MR 0337902 Wiles, Andrew (2000), "Twenty years of number theory", Mathematics: frontiers and perspectives, Providence, R.I.: American Mathematical Society, pp. 329–342, ISBN 978-0-8218-2697-3, MR 1754786 Zagier, Don (1977), "The first 50 million prime numbers" (PDF), Math. Intelligencer, Springer, 1: 7–19, doi:10.1007/BF03039306, MR 0643810, S2CID 189886510, archived from the original (PDF) on 2009-03-27 Zagier, Don (1981), "Eisenstein series and the Riemann zeta function", Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., vol. 10, Tata Inst. Fundamental Res., Bombay, pp. 275–301, MR 0633666 Popular expositions Sabbagh, Karl (2003a), The greatest unsolved problem in mathematics, Farrar, Straus and Giroux, New York, ISBN 978-0-374-25007-2, MR 1979664 Sabbagh, Karl (2003b), Dr. Riemann's zeros, Atlantic Books, London, ISBN 978-1-843-54101-1 du Sautoy, Marcus (2003), The music of the primes, HarperCollins Publishers, ISBN 978-0-06-621070-4, MR 2060134 Rockmore, Dan (2005), Stalking the Riemann hypothesis, Pantheon Books, ISBN 978-0-375-42136-5, MR 2269393 Derbyshire, John (2003), Prime Obsession, Joseph Henry Press, Washington, DC, ISBN 978-0-309-08549-6, MR 1968857 Watkins, Matthew (2015), Mystery of the Prime Numbers, Liberalis Books, ISBN 978-1782797814, MR 0000000 Frenkel, Edward (2014), The Riemann Hypothesis Numberphile, Mar 11, 2014 (video) External links Media related to Riemann hypothesis at Wikimedia Commons Wikiquote has quotations related to Riemann hypothesis Mathematics portal American institute of mathematics, Riemann hypothesis Zeroes database, 103 800 788 359 zeroes The Key to the Riemann Hypothesis - Numberphile, a YouTube video about the Riemann hypothesis by Numberphile Apostol, Tom, Where are the zeros of zeta of s? Poem about the Riemann hypothesis, sung by John Derbyshire. Borwein, Peter, The Riemann Hypothesis (PDF), archived from the original (PDF) on 2009-03-27 (Slides for a lecture) Conrad, K. (2010), Consequences of the Riemann hypothesis Conrey, J. Brian; Farmer, David W, Equivalences to the Riemann hypothesis, archived from the original on 2010-03-16 Gourdon, Xavier; Sebah, Pascal (2004), Computation of zeros of the Zeta function (Reviews the GUE hypothesis, provides an extensive bibliography as well). Odlyzko, Andrew, Home page including papers on the zeros of the zeta function and tables of the zeros of the zeta function Odlyzko, Andrew (2002), Zeros of the Riemann zeta function: Conjectures and computations (PDF) Slides of a talk Pegg, Ed (2004), Ten Trillion Zeta Zeros, Math Games website, archived from the original on 2004-11-02, retrieved 2004-10-20. A discussion of Xavier Gourdon's calculation of the first ten trillion non-trivial zeros Pugh, Glen, Java applet for plotting Z(t), archived from the original on 2015-06-30, retrieved 2009-03-13 Rubinstein, Michael, algorithm for generating the zeros, archived from the original on 2007-04-27. du Sautoy, Marcus (2006), Prime Numbers Get Hitched, Seed Magazine, archived from the original on 2017-09-22, retrieved 2006-03-27 Stein, William A., What is Riemann's hypothesis, archived from the original on 2009-01-04 de Vries, Andreas (2004), The Graph of the Riemann Zeta function ζ(s), a simple animated Java applet. Watkins, Matthew R. (2007-07-18), Proposed proofs of the Riemann Hypothesis Zetagrid (2002) A distributed computing project that attempted to disprove Riemann's hypothesis; closed in November 2005 show vte L-functions in number theory show vte Bernhard Riemann show Authority control Categories: 1859 introductionsAnalytic number theoryBernhard RiemannConjecturesHilbert's problemsHypothesesMillennium Prize ProblemsUnsolved problems in number theoryZeta and L-functions