# Riemann hypothesis

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]~} .

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