Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, denoted by the

defined as $${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots }$$ for ${\displaystyle \operatorname {Re} (s)>1}$, and its analytic continuation elsewhere.^[2]

The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.

The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ(2), provides a solution to the Basel problem. In 1979 Roger Apéry proved the irrationality of ζ(3). The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known.

The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it, where σ and t are real numbers. (The notation s, σ, and t is used traditionally in the study of the zeta function, following Riemann.) When Re(s) = σ > 1, the function can be written as a converging summation or as an integral:

${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\,\mathrm {d} x\,,}$

where

${\displaystyle \Gamma (s)=\int _{0}^{\infty }x^{s-1}\,e^{-x}\,\mathrm {d} x}$

is the gamma function. The Riemann zeta function is defined for other complex values via analytic continuation of the function defined for σ > 1.

extended the definition to ${\displaystyle \operatorname {Re} (s)>1.}$^[4]

The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for s such that σ > 1 and diverges for all other values of s. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠ 1. For s = 1, the series is the harmonic series which diverges to +∞, and $${\displaystyle \lim _{s\to 1}(s-1)\zeta (s)=1.}$$ Thus the Riemann zeta function is a

1.

Euler's product formula

In 1737, the connection between the zeta function and prime numbers was discovered by Euler, who proved the identity

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}},}$

where, by definition, the left hand side is ζ(s) and the infinite product on the right hand side extends over all prime numbers p (such expressions are called Euler products):

${\displaystyle \prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}={\frac {1}{1-2^{-s}}}\cdot {\frac {1}{1-3^{-s}}}\cdot {\frac {1}{1-5^{-s}}}\cdot {\frac {1}{1-7^{-s}}}\cdot {\frac {1}{1-11^{-s}}}\cdots {\frac {1}{1-p^{-s}}}\cdots }$

Both sides of the Euler product formula converge for Re(s) > 1. The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s = 1, diverges, Euler's formula (which becomes Π_p ⁠p/p − 1⁠) implies that there are infinitely many primes.^[5] Since the logarithm of ⁠p/p − 1⁠ is approximately ⁠1/p⁠, the formula can also be used to prove the stronger result that the sum of the reciprocals of the primes is infinite. On the other hand, combining that with the sieve of Eratosthenes shows that the density of the set of primes within the set of positive integers is zero.

The Euler product formula can be used to calculate the

it is divisible by nm, an event which occurs with probability ⁠1/nm⁠). Thus the asymptotic probability that s numbers are coprime is given by a product over all primes,

${\displaystyle \prod _{p{\text{ prime}}}\left(1-{\frac {1}{p^{s}}}\right)=\left(\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}\right)^{-1}={\frac {1}{\zeta (s)}}.}$

Riemann's functional equation

This zeta function satisfies the functional equation $${\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s)\ ,}$$ where Γ(s) is the

, which cancels the simple zero of the sine factor.

The functional equation was established by Riemann in his 1859 paper "On the Number of Primes Less Than a Given Magnitude" and used to construct the analytic continuation in the first place.

Riemann also found a symmetric version of the functional equation by setting $${\displaystyle \xi (s)={\frac {s(s-1)}{2}}\times \pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)\zeta (s)=(s-1)\pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}+1\right)\zeta (s)\ ,}$$ which satisfies: $${\displaystyle \xi (s)=\xi (1-s)~.}$$

Returning to the functional equation's derivation in the previous section, we have

$${\displaystyle \xi (s)={\frac {1}{2}}+{\frac {s(s-1)}{2}}\int _{1}^{\infty }\left(x^{-{\frac {s}{2}}-{\frac {1}{2}}}+x^{{\frac {s}{2}}-1}\right)\psi (x)dx}$$

Using integration by parts, $${\displaystyle \xi (s)={\frac {1}{2}}-\left[\left(sx^{\frac {1-s}{2}}+(1-s)x^{\frac {s}{2}}\right)\psi (x)\right]_{1}^{\infty }+\int _{1}^{\infty }\left(sx^{\frac {1-s}{2}}+(1-s)x^{\frac {s}{2}}\right)\psi '(x)dx}$$ $${\displaystyle \xi (s)={\frac {1}{2}}+\psi (1)+\int _{1}^{\infty }\left(sx^{\frac {1-s}{2}}+(1-s)x^{\frac {s}{2}}\right)\psi '(x)dx}$$

Using integration by parts again with a factorization of ${\displaystyle x^{\frac {3}{2}}}$, $${\displaystyle \xi (s)={\frac {1}{2}}+\psi (1)-2\left[x^{\frac {3}{2}}\psi '(x)\left(x^{\frac {s-1}{2}}+x^{-{\frac {s}{2}}}\right)\right]_{1}^{\infty }+2\int _{1}^{\infty }\left(x^{\frac {s-1}{2}}+x^{-{\frac {s}{2}}}\right){\frac {d}{dx}}\left[x^{\frac {3}{2}}\psi '(x)\right]dx}$$ $${\displaystyle \xi (s)={\frac {1}{2}}+\psi (1)+4\psi '(1)+2\int _{1}^{\infty }{\frac {d}{dx}}\left[x^{\frac {3}{2}}\psi '(x)\right]\left(x^{\frac {s-1}{2}}+x^{-{\frac {s}{2}}}\right)dx}$$

As ${\displaystyle {\frac {1}{2}}+\psi (1)+4\psi '(1)=0}$,

$${\displaystyle \xi (s)=2\int _{1}^{\infty }{\frac {d}{dx}}\left[x^{\frac {3}{2}}\psi '(x)\right]\left(x^{\frac {s-1}{2}}+x^{-{\frac {s}{2}}}\right)dx}$$

Remove a factor of ${\displaystyle x^{-{\frac {1}{4}}}}$ to make the exponents in the remainder opposites.

$${\displaystyle \xi (s)=2\int _{1}^{\infty }{\frac {d}{dx}}\left[x^{\frac {3}{2}}\psi '(x)\right]x^{-{\frac {1}{4}}}\left(x^{\frac {s-{\frac {1}{2}}}{2}}+x^{\frac {{\frac {1}{2}}-s}{2}}\right)dx}$$

Using the hyperbolic functions, namely ${\displaystyle \cos(x)=\cosh(ix)={\frac {e^{ix}+e^{-ix}}{2}}}$, and letting ${\displaystyle s={\frac {1}{2}}+it}$ gives $${\displaystyle \xi (s)=4\int _{1}^{\infty }{\frac {d}{dx}}\left[x^{\frac {3}{2}}\psi '(x)\right]x^{-{\frac {1}{4}}}\cos({\frac {t}{2}}\log x)dx}$$

and by separating the integral and using the power series for ${\displaystyle \cos }$, $${\displaystyle \xi (s)=\sum _{n=0}^{\infty }a_{2n}t^{2n}}$$ which led Riemann to his famous hypothesis.

The functional equation shows that the Riemann zeta function has zeros at −2, −4,.... These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin ⁠πs/2⁠ being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields important results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip ${\displaystyle \{s\in \mathbb {C} :0<\operatorname {Re} (s)<1\}}$, which is called the critical strip. The set ${\displaystyle \{s\in \mathbb {C} :\operatorname {Re} (s)=1/2\}}$ is called the critical line. The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that all non-trivial zeros are on the critical line. In 1989, Conrey proved that more than 40% of the non-trivial zeros of the Riemann zeta function are on the critical line.^[8] This has since been improved to 41.7%.^[9]

For the Riemann zeta function on the critical line, see Z-function.

First few nontrivial zeros^[10][11]

Zero
1/2 ± 14.134725... i
1/2 ± 21.022040... i
1/2 ± 25.010858... i
1/2 ± 30.424876... i
1/2 ± 32.935062... i
1/2 ± 37.586178... i
1/2 ± 40.918719... i

Let ${\displaystyle N(T)}$ be the number of zeros of ${\displaystyle \zeta (s)}$ in the critical strip ${\displaystyle 0<\operatorname {Re} (s)<1}$, whose imaginary parts are in the interval ${\displaystyle 0<\operatorname {Im} (s)<T}$. Timothy Trudgian proved that, if ${\displaystyle T>e}$, then^[12]

${\displaystyle \left|N(T)-{\frac {T}{2\pi }}\log {\frac {T}{2\pi e}}\right|\leq 0.112\log T+0.278\log \log T+3.385+{\frac {0.2}{T}}}$.

In 1914, G. H. Hardy proved that ζ (⁠1/2⁠ + it) has infinitely many real zeros.^[13][14]

Hardy and J. E. Littlewood formulated two conjectures on the density and distance between the zeros of ζ (⁠1/2⁠ + it) on intervals of large positive real numbers. In the following, N(T) is the total number of real zeros and N₀(T) the total number of zeros of odd order of the function ζ (⁠1/2⁠ + it) lying in the interval (0, T].

These two conjectures opened up new directions in the investigation of the Riemann zeta function.

The location of the Riemann zeta function's zeros is of great importance in number theory. The prime number theorem is equivalent to the fact that there are no zeros of the zeta function on the Re(s) = 1 line.^[15] It is also known that zeros do not exist in certain regions slightly to the left of the Re(s) = 1 line, known as zero-free regions. For instance, Korobov^[16] and Vinogradov^[17] independently showed via the Vinogradov's mean-value theorem that for sufficiently large ${\displaystyle |t|}$, ${\displaystyle \zeta (\sigma +it)\neq 0}$ for

${\displaystyle \sigma \geq 1-{\frac {c}{(\log |t|)^{2/3+\varepsilon }}}}$

for any ${\displaystyle \varepsilon >0}$ and a number ${\displaystyle c>0}$ depending on ${\displaystyle \varepsilon }$. Asymptotically, this is the largest known zero-free region for the zeta function.

Explicit zero-free regions are also known. Platt and Trudgian^[18] verified computationally that ${\displaystyle \zeta (\sigma +it)\neq 0}$ if ${\displaystyle \sigma \neq 1/2}$ and ${\displaystyle |t|\leq 3\cdot 10^{12}}$. Mossinghoff, Trudgian and Yang proved^[19] that zeta has no zeros in the region

${\displaystyle \sigma \geq 1-{\frac {1}{5.558691\log |t|}}}$

for |t| ≥ 2, which is the largest known zero-free region in the critical strip for ${\displaystyle 3\cdot 10^{12}<|t|<e^{64.1}\approx 7\cdot 10^{27}}$ (for previous results see^[20]). Yang^[21] showed that ${\displaystyle \zeta (\sigma +it)\neq 0}$ if

${\displaystyle \sigma \geq 1-{\frac {\log \log |t|}{21.233\log |t|}}}$ and ${\displaystyle |t|\geq 3}$

which is the largest known zero-free region for ${\displaystyle e^{170.2}<|t|<e^{4.8\cdot 10^{5}}}$. Bellotti proved^[22] (building on the work of Ford^[23]) the zero-free region

${\displaystyle \sigma \geq 1-{\frac {1}{53.989(\log |t|)^{2/3}(\log \log |t|)^{1/3}}}}$ and ${\displaystyle |t|\geq 3}$.

This is the largest known zero-free region for fixed ${\displaystyle |t|\geq \exp(4.8\cdot 10^{5}).}$ Bellotti also showed that for sufficiently large ${\displaystyle |t|}$, the following better result is known: ${\displaystyle \zeta (\sigma +it)\neq 0}$ for

${\displaystyle \sigma \geq 1-{\frac {1}{48.0718(\log |t|)^{2/3}(\log \log |t|)^{1/3}}}.}$

The strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound consequences in the theory of numbers.

It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence (γ_n) contains the imaginary parts of all zeros in the upper half-plane in ascending order, then

${\displaystyle \lim _{n\rightarrow \infty }\left(\gamma _{n+1}-\gamma _{n}\right)=0.}$

The

asserts that a positive proportion of the nontrivial zeros lies on the critical line. (The Riemann hypothesis would imply that this proportion is 1.)

In the critical strip, the zero with smallest non-negative imaginary part is ⁠1/2⁠ + 14.13472514...i (OEIS: A058303). The fact that

${\displaystyle \zeta (s)={\overline {\zeta ({\overline {s}})}}}$

for all complex s ≠ 1 implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line Re(s) = ⁠1/2⁠.

It is also known that no zeros lie on the line with real part 1.

For any positive even integer 2n, $${\displaystyle \zeta (2n)={\frac {|{B_{2n}}|(2\pi )^{2n}}{2(2n)!}},}$$ where B_2n is the 2n-th Bernoulli number. For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraic K-theory of the integers; see Special values of L-functions.

For nonpositive integers, one has $${\displaystyle \zeta (-n)=-{\frac {B_{n+1}}{n+1}}}$$ for n ≥ 0 (using the convention that B₁ = ⁠1/2⁠). In particular, ζ vanishes at the negative even integers because B_m = 0 for all odd m other than 1. These are the so-called "trivial zeros" of the zeta function.

Via analytic continuation, one can show that $${\displaystyle \zeta (-1)=-{\tfrac {1}{12}}}$$ This gives a pretext for assigning a finite value to the divergent series 1 + 2 + 3 + 4 + ⋯, which has been used in certain contexts (Ramanujan summation) such as string theory.^[24] Analogously, the particular value $${\displaystyle \zeta (0)=-{\tfrac {1}{2}}}$$ can be viewed as assigning a finite result to the divergent series 1 + 1 + 1 + 1 + ⋯.

The value $${\displaystyle \zeta {\bigl (}{\tfrac {1}{2}}{\bigr )}=-1.46035450880958681288\ldots }$$ is employed in calculating kinetic boundary layer problems of linear kinetic equations.^[25][26]

Although $${\displaystyle \zeta (1)=1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\cdots }$$ diverges, its Cauchy principal value $${\displaystyle \lim _{\varepsilon \to 0}{\frac {\zeta (1+\varepsilon )+\zeta (1-\varepsilon )}{2}}}$$ exists and is equal to the

The demonstration of the particular value $${\displaystyle \zeta (2)=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}}$$ is known as the

The value $${\displaystyle \zeta (3)=1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots =1.202056903159594285399...}$$ is Apéry's constant.

Taking the limit ${\displaystyle s\rightarrow +\infty }$ through the real numbers, one obtains ${\displaystyle \zeta (+\infty )=1}$. But at

Various properties

For sums involving the zeta function at integer and half-integer values, see rational zeta series.

The reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius function μ(n):

${\displaystyle {\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}}$

for every complex number s with real part greater than 1. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series.

The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than ⁠1/2⁠.

The critical strip of the Riemann zeta function has the remarkable property of universality. This zeta function universality states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable. The first proof of universality was provided by Sergei Mikhailovitch Voronin in 1975.^[29] More recent work has included effective versions of Voronin's theorem^[30] and extending it to Dirichlet L-functions.^[31][32]

Let the functions F(T;H) and G(s₀;Δ) be defined by the equalities

${\displaystyle F(T;H)=\max _{|t-T|\leq H}\left|\zeta \left({\tfrac {1}{2}}+it\right)\right|,\qquad G(s_{0};\Delta )=\max _{|s-s_{0}|\leq \Delta }|\zeta (s)|.}$

Here T is a sufficiently large positive number, 0 < H ≪ log log T, s₀ = σ₀ + iT, ⁠1/2⁠ ≤ σ₀ ≤ 1, 0 < Δ < ⁠1/3⁠. Estimating the values F and G from below shows, how large (in modulus) values ζ(s) can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip 0 ≤ Re(s) ≤ 1.

The case H ≫ log log T was studied by Kanakanahalli Ramachandra; the case Δ > c, where c is a sufficiently large constant, is trivial.

in particular, that if the values H and Δ exceed certain sufficiently small constants, then the estimates

${\displaystyle F(T;H)\geq T^{-c_{1}},\qquad G(s_{0};\Delta )\geq T^{-c_{2}},}$

hold, where c₁ and c₂ are certain absolute constants.

The argument of the Riemann zeta function

The function

${\displaystyle S(t)={\frac {1}{\pi }}\arg {\zeta \left({\tfrac {1}{2}}+it\right)}}$

is called the

argument

of the Riemann zeta function. Here arg ζ(⁠1/2⁠ + it) is the increment of an arbitrary continuous branch of arg ζ(s) along the broken line joining the points 2, 2 + it and ⁠1/2⁠ + it.

There are some theorems on properties of the function S(t). Among those results

mean value theorems

for S(t) and its first integral

${\displaystyle S_{1}(t)=\int _{0}^{t}S(u)\,\mathrm {d} u}$

on intervals of the real line, and also the theorem claiming that every interval (T, T + H] for

${\displaystyle H\geq T^{{\frac {27}{82}}+\varepsilon }}$

contains at least

${\displaystyle H{\sqrt[{3}]{\ln T}}e^{-c{\sqrt {\ln \ln T}}}}$

points where the function S(t) changes sign. Earlier similar results were obtained by Atle Selberg for the case

${\displaystyle H\geq T^{{\frac {1}{2}}+\varepsilon }.}$

An extension of the area of convergence can be obtained by rearranging the original series.^[37] The series

${\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=1}^{\infty }\left({\frac {n}{(n+1)^{s}}}-{\frac {n-s}{n^{s}}}\right)}$

converges for Re(s) > 0, while

${\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=1}^{\infty }{\frac {n(n+1)}{2}}\left({\frac {2n+3+s}{(n+1)^{s+2}}}-{\frac {2n-1-s}{n^{s+2}}}\right)}$

converge even for Re(s) > −1. In this way, the area of convergence can be extended to Re(s) > −k for any negative integer −k.

The recurrence connection is clearly visible from the expression valid for Re(s) > −2 enabling further expansion by integration by parts.

${\displaystyle {\begin{aligned}\zeta (s)=&1+{\frac {1}{s-1}}-{\frac {s}{2!}}[\zeta (s+1)-1]\\-&{\frac {s(s+1)}{3!}}[\zeta (s+2)-1]\\&-{\frac {s(s+1)(s+2)}{3!}}\sum _{n=1}^{\infty }\int _{0}^{1}{\frac {t^{3}dt}{(n+t)^{s+3}}}\end{aligned}}}$

The Mellin transform of a function f(x) is defined as^[38]

${\displaystyle \int _{0}^{\infty }f(x)x^{s}\,{\frac {\mathrm {d} x}{x}}}$

in the region where the integral is defined. There are various expressions for the zeta function as Mellin transform-like integrals. If the real part of s is greater than one, we have

${\displaystyle \Gamma (s)\zeta (s)=\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\,\mathrm {d} x\quad }$ and ${\displaystyle \quad \Gamma (s)\zeta (s)={\frac {1}{2s}}\int _{0}^{\infty }{\frac {x^{s}}{\cosh(x)-1}}\,\mathrm {d} x}$,

where Γ denotes the gamma function. By modifying the contour, Riemann showed that

${\displaystyle 2\sin(\pi s)\Gamma (s)\zeta (s)=i\oint _{H}{\frac {(-x)^{s-1}}{e^{x}-1}}\,\mathrm {d} x}$

for all s^[39] (where H denotes the Hankel contour).

We can also find expressions which relate to prime numbers and the prime number theorem. If π(x) is the prime-counting function, then

${\displaystyle \ln \zeta (s)=s\int _{0}^{\infty }{\frac {\pi (x)}{x(x^{s}-1)}}\,\mathrm {d} x,}$

for values with Re(s) > 1.

A similar Mellin transform involves the Riemann function J(x), which counts prime powers pⁿ with a weight of ⁠1/n⁠, so that

${\displaystyle J(x)=\sum {\frac {\pi \left(x^{\frac {1}{n}}\right)}{n}}.}$

Now

${\displaystyle \ln \zeta (s)=s\int _{0}^{\infty }J(x)x^{-s-1}\,\mathrm {d} x.}$

These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's prime-counting function is easier to work with, and π(x) can be recovered from it by Möbius inversion.

The Riemann zeta function can be given by a Mellin transform^[40]

${\displaystyle 2\pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)\zeta (s)=\int _{0}^{\infty }{\bigl (}\theta (it)-1{\bigr )}t^{{\frac {s}{2}}-1}\,\mathrm {d} t,}$

in terms of Jacobi's theta function

${\displaystyle \theta (\tau )=\sum _{n=-\infty }^{\infty }e^{\pi in^{2}\tau }.}$

However, this integral only converges if the real part of s is greater than 1, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all s except 0 and 1:

${\displaystyle \pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)\zeta (s)={\frac {1}{s-1}}-{\frac {1}{s}}+{\frac {1}{2}}\int _{0}^{1}\left(\theta (it)-t^{-{\frac {1}{2}}}\right)t^{{\frac {s}{2}}-1}\,\mathrm {d} t+{\frac {1}{2}}\int _{1}^{\infty }{\bigl (}\theta (it)-1{\bigr )}t^{{\frac {s}{2}}-1}\,\mathrm {d} t.}$

The Riemann zeta function is

${\displaystyle \zeta (s)={\frac {1}{s-1}}+\sum _{n=0}^{\infty }{\frac {\gamma _{n}}{n!}}(1-s)^{n}.}$

The constants γ_n here are called the Stieltjes constants and can be defined by the limit

${\displaystyle \gamma _{n}=\lim _{m\rightarrow \infty }{\left(\left(\sum _{k=1}^{m}{\frac {(\ln k)^{n}}{k}}\right)-{\frac {(\ln m)^{n+1}}{n+1}}\right)}.}$

The constant term γ₀ is the

.

For all s ∈ ℂ, s ≠ 1, the integral relation (cf. Abel–Plana formula)

${\displaystyle \ \zeta (s)\ =\ {\frac {1}{\ s-1\ }}+{\frac {\ 1\ }{2}}+2\int _{0}^{\infty }{\frac {\sin(\ s\ \arctan t\ )}{\ \left(1+t^{2}\right)^{s/2}\left(e^{2\pi t}-1\right)\ }}\ \operatorname {d} t\ }$

holds true, which may be used for a numerical evaluation of the zeta function.

Another series development using the

${\displaystyle \zeta (s)={\frac {s}{s-1}}-\sum _{n=1}^{\infty }{\bigl (}\zeta (s+n)-1{\bigr )}{\frac {s(s+1)\cdots (s+n-1)}{(n+1)!}}.}$

This can be used recursively to extend the Dirichlet series definition to all complex numbers.

The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the

Hadamard product

On the basis of

expansion

${\displaystyle \zeta (s)={\frac {e^{\left(\log(2\pi )-1-{\frac {\gamma }{2}}\right)s}}{2(s-1)\Gamma \left(1+{\frac {s}{2}}\right)}}\prod _{\rho }\left(1-{\frac {s}{\rho }}\right)e^{\frac {s}{\rho }},}$

where the product is over the non-trivial zeros ρ of ζ and the letter γ again denotes the

expansion is

${\displaystyle \zeta (s)=\pi ^{\frac {s}{2}}{\frac {\prod _{\rho }\left(1-{\frac {s}{\rho }}\right)}{2(s-1)\Gamma \left(1+{\frac {s}{2}}\right)}}.}$

This form clearly displays the simple pole at s = 1, the trivial zeros at −2, −4, ... due to the gamma function term in the denominator, and the non-trivial zeros at s = ρ. (To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i.e. the factors for a pair of zeros of the form ρ and 1 − ρ should be combined.)

Globally convergent series

A globally convergent series for the zeta function, valid for all complex numbers s except s = 1 + ⁠2πi/ln 2⁠n for some integer n, was conjectured by Konrad Knopp in 1926 ^[43] and proven by Helmut Hasse in 1930^[44] (cf. Euler summation):

${\displaystyle \zeta (s)={\frac {1}{1-2^{1-s}}}\sum _{n=0}^{\infty }{\frac {1}{2^{n+1}}}\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{(k+1)^{s}}}.}$

The series appeared in an appendix to Hasse's paper, and was published for the second time by Jonathan Sondow in 1994.^[45]

Hasse also proved the globally converging series

${\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{(k+1)^{s-1}}}}$

in the same publication.^[44] Research by Iaroslav Blagouchine^[46][43] has found that a similar, equivalent series was published by Joseph Ser in 1926.^[47]

In 1997 K. Maślanka gave another globally convergent (except s = 1) series for the Riemann zeta function:

${\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{k=0}^{\infty }{\biggl (}\prod _{i=1}^{k}(i-{\frac {s}{2}}){\biggl )}{\frac {A_{k}}{k!}}={\frac {1}{s-1}}\sum _{k=0}^{\infty }{\biggl (}1-{\frac {s}{2}}{\biggl )}_{k}{\frac {A_{k}}{k!}}}$

where real coefficients ${\displaystyle A_{k}}$ are given by:

${\displaystyle A_{k}=\sum _{j=0}^{k}(-1)^{j}{\binom {k}{j}}(2j+1)\zeta (2j+2)=\sum _{j=0}^{k}{\binom {k}{j}}{\frac {B_{2j+2}\pi ^{2j+2}}{\left(2\right)_{j}\left({\frac {1}{2}}\right)_{j}}}}$

Here ${\displaystyle B_{n}}$ are the Bernoulli numbers and ${\displaystyle (x)_{k}}$ denotes the Pochhammer symbol.^[48][49]

Note that this representation of the zeta function is essentially an interpolation with nodes, where the nodes are points ${\displaystyle s=2,4,6,\ldots }$, i.e. exactly those where the zeta values are precisely known, as Euler showed. An elegant and very short proof of this representation of the zeta function, based on Carlson's theorem, was presented by Philippe Flajolet in 2006.^[50]

The asymptotic behavior of the coefficients ${\displaystyle A_{k}}$ is rather curious: for growing ${\displaystyle k}$ values, we observe regular oscillations with a nearly exponentially decreasing amplitude and slowly decreasing frequency (roughly as ${\displaystyle k^{-2/3}}$). Using the saddle point method, we can show that

${\displaystyle A_{k}\sim {\frac {4\pi ^{3/2}}{\sqrt {3\kappa }}}\exp {\biggl (}-{\frac {3\kappa }{2}}+{\frac {\pi ^{2}}{4\kappa }}{\biggl )}\cos {\biggl (}{\frac {4\pi }{3}}-{\frac {3{\sqrt {3}}\kappa }{2}}+{\frac {{\sqrt {3}}\pi ^{2}}{4\kappa }}{\biggl )}}$

where ${\displaystyle \kappa }$ stands for:

${\displaystyle \kappa :={\sqrt[{3}]{\pi ^{2}k}}}$

(see ^[51] for details).

On the basis of this representation, in 2003 Luis Báez-Duarte provided a new criterion for the Riemann hypothesis.^[52][53][54] Namely, if we define the coefficients ${\displaystyle c_{k}}$ as

${\displaystyle c_{k}:=\sum _{j=0}^{k}(-1)^{j}{\binom {k}{j}}{\frac {1}{\zeta (2j+2)}}}$

then the Riemann hypothesis is equivalent to

${\displaystyle c_{k}={\mathcal {O}}{\biggl (}k^{-3/4+\varepsilon }{\biggl )}\qquad (\forall \varepsilon >0)}$

Series representation at positive integers via the primorial

${\displaystyle \zeta (k)={\frac {2^{k}}{2^{k}-1}}+\sum _{r=2}^{\infty }{\frac {(p_{r-1}\#)^{k}}{J_{k}(p_{r}\#)}}\qquad k=2,3,\ldots .}$

Here p_n# is the primorial sequence and J_k is Jordan's totient function.^[56]

The function ζ can be represented, for Re(s) > 1, by the infinite series

${\displaystyle \zeta (s)=\sum _{n=0}^{\infty }B_{n,\geq 2}^{(s)}{\frac {(W_{k}(-1))^{n}}{n!}},}$

where k ∈ {−1, 0}, W_k is the kth branch of the Lambert W-function, and B^(μ)
_{n, ≥2} is an incomplete poly-Bernoulli number.^[57]

The function ${\displaystyle g(x)=x\left(1+\left\lfloor x^{-1}\right\rfloor \right)-1}$ is iterated to find the coefficients appearing in Engel expansions.^[58]

The Mellin transform of the map ${\displaystyle g(x)}$ is related to the Riemann zeta function by the formula

${\displaystyle {\begin{aligned}\int _{0}^{1}g(x)x^{s-1}\,dx&=\sum _{n=1}^{\infty }\int _{\frac {1}{n+1}}^{\frac {1}{n}}(x(n+1)-1)x^{s-1}\,dx\\[6pt]&=\sum _{n=1}^{\infty }{\frac {n^{-s}(s-1)+(n+1)^{-s-1}(n^{2}+2n+1)+n^{-s-1}s-n^{1-s}}{(s+1)s(n+1)}}\\[6pt]&={\frac {\zeta (s+1)}{s+1}}-{\frac {1}{s(s+1)}}\end{aligned}}}$

Certain linear combinations of Dirichlet series whose coefficients are terms of the

For instance:

${\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {5t_{n-1}+3t_{n}}{n^{2}}}&=4\zeta (2)={\frac {2\pi ^{2}}{3}},\\\sum _{n\geq 1}{\frac {9t_{n-1}+7t_{n}}{n^{3}}}&=8\zeta (3),\end{aligned}}}$

where ${\displaystyle (t_{n})_{n\geq 0}}$ is the ${\displaystyle n^{\rm {th}}}$ term of the Thue-Morse sequence. In fact, for all ${\displaystyle s}$ with real part greater than ${\displaystyle 1}$, we have

${\displaystyle (2^{s}+1)\sum _{n\geq 1}{\frac {t_{n-1}}{n^{s}}}+(2^{s}-1)\sum _{n\geq 1}{\frac {t_{n}}{n^{s}}}=2^{s}\zeta (s).}$

Stochastic representations

The Brownian motion and Riemann zeta function are connected through the moment-generating functions of stochastic processes derived from the Brownian motion.

A classical algorithm, in use prior to about 1930, proceeds by applying the

to obtain, for n and m positive integers,

${\displaystyle \zeta (s)=\sum _{j=1}^{n-1}j^{-s}+{\tfrac {1}{2}}n^{-s}+{\frac {n^{1-s}}{s-1}}+\sum _{k=1}^{m}T_{k,n}(s)+E_{m,n}(s)}$

where, letting ${\displaystyle B_{2k}}$ denote the indicated Bernoulli number,

${\displaystyle T_{k,n}(s)={\frac {B_{2k}}{(2k)!}}n^{1-s-2k}\prod _{j=0}^{2k-2}(s+j)}$

and the error satisfies

${\displaystyle |E_{m,n}(s)|<\left|{\frac {s+2m+1}{\sigma +2m+1}}T_{m+1,n}(s)\right|,}$

with σ = Re(s).^[60]

A modern numerical algorithm is the Odlyzko–Schönhage algorithm.

The zeta function occurs in applied statistics including Zipf's law, Zipf–Mandelbrot law, and Lotka's law.

Musical tuning

In the theory of musical tunings, the zeta function can be used to find equal divisions of the octave (EDOs) that closely approximate the intervals of the harmonic series. For increasing values of ${\displaystyle t\in \mathbb {R} }$, the value of

${\displaystyle \left\vert \zeta \left({\frac {1}{2}}+{\frac {2\pi {i}}{\ln {(2)}}}t\right)\right\vert }$

peaks near integers that correspond to such EDOs.^[62] Examples include popular choices such as 12, 19, and 53.^[63]

The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.^[64]

• ${\displaystyle \sum _{n=2}^{\infty }{\bigl (}\zeta (n)-1{\bigr )}=1}$

In fact the even and odd terms give the two sums

• ${\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (2n)-1{\bigr )}={\frac {3}{4}}}$

and

• ${\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (2n+1)-1{\bigr )}={\frac {1}{4}}}$

Parametrized versions of the above sums are given by

• ${\displaystyle \sum _{n=1}^{\infty }(\zeta (2n)-1)\,t^{2n}={\frac {t^{2}}{t^{2}-1}}+{\frac {1}{2}}\left(1-\pi t\cot(t\pi )\right)}$

and

• ${\displaystyle \sum _{n=1}^{\infty }(\zeta (2n+1)-1)\,t^{2n}={\frac {t^{2}}{t^{2}-1}}-{\frac {1}{2}}\left(\psi ^{0}(t)+\psi ^{0}(-t)\right)-\gamma }$

with ${\displaystyle |t|<2}$ and where ${\displaystyle \psi }$ and ${\displaystyle \gamma }$ are the polygamma function and Euler's constant, respectively, as well as

• ${\displaystyle \sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n}}\,t^{2n}=\log \left({\dfrac {1-t^{2}}{\operatorname {sinc} (\pi \,t)}}\right)}$

all of which are continuous at ${\displaystyle t=1}$. Other sums include

• ${\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}=1-\gamma }$
• ${\displaystyle \sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n}}=\ln 2}$
• ${\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}\left(\left({\tfrac {3}{2}}\right)^{n-1}-1\right)={\frac {1}{3}}\ln \pi }$
• ${\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (4n)-1{\bigr )}={\frac {7}{8}}-{\frac {\pi }{4}}\left({\frac {e^{2\pi }+1}{e^{2\pi }-1}}\right).}$
• ${\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}\Im {\bigl (}(1+i)^{n}-1-i^{n}{\bigr )}={\frac {\pi }{4}}}$

where ${\displaystyle \Im }$ denotes the

of a complex number.

Another interesting series that relates to the natural logarithm of the lemniscate constant is the following

• ${\displaystyle \sum _{n=2}^{\infty }\left[{\frac {2(-1)^{n}\zeta (n)}{4^{n}n}}-{\frac {(-1)^{n}\zeta (n)}{2^{n}n}}\right]=\ln \left({\frac {\varpi }{2{\sqrt {2}}}}\right)}$

There are yet more formulas in the article Harmonic number.

There are a number of related

${\displaystyle \zeta (s,q)=\sum _{k=0}^{\infty }{\frac {1}{(k+q)^{s}}}}$

(the convergent series representation was given by

.

The polylogarithm is given by

${\displaystyle \operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{s}}}}$

which coincides with the Riemann zeta function when z = 1. The Clausen function Cl_s(θ) can be chosen as the real or imaginary part of Li_s(e^iθ).

The Lerch transcendent is given by

${\displaystyle \Phi (z,s,q)=\sum _{k=0}^{\infty }{\frac {z^{k}}{(k+q)^{s}}}}$

which coincides with the Riemann zeta function when z = 1 and q = 1 (the lower limit of summation in the Lerch transcendent is 0, not 1).

The

are defined by

${\displaystyle \zeta (s_{1},s_{2},\ldots ,s_{n})=\sum _{k_{1}>k_{2}>\cdots >k_{n}>0}{k_{1}}^{-s_{1}}{k_{2}}^{-s_{2}}\cdots {k_{n}}^{-s_{n}}.}$

One can analytically continue these functions to the n-dimensional complex space. The special values taken by these functions at positive integer arguments are called

by number theorists and have been connected to many different branches in mathematics and physics.

• 1 + 2 + 3 + 4 + ···
• Arithmetic zeta function
• Dirichlet eta function
• Generalized Riemann hypothesis
• Lehmer pair
• Particular values of the Riemann zeta function
• Prime zeta function
• Renormalization
• Riemann–Siegel theta function
• ZetaGrid

Sources

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  MR 2723248
  ..
• doi:10.1016/S0377-0427(00)00336-8
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• Cvijović, Djurdje; Klinowski, Jacek (2002). "Integral representations of the Riemann zeta function for odd-integer arguments". J. Comput. Appl. Math. 142 (2): 435–439.
  MR 1906742
  .
• Cvijović, Djurdje; Klinowski, Jacek (1997). "Continued-fraction expansions for the Riemann zeta function and polylogarithms". Proc. Amer. Math. Soc. 125 (9): 2543–2550.
  doi:10.1090/S0002-9939-97-04102-6
  .
• ISBN 0-486-41740-9
  – via archive.org. Has an English translation of Riemann's paper.
• doi:10.24033/bsmf.545
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• Hardy, G.H. (1949). Divergent Series. Oxford, UK: Clarendon Press.
• S2CID 120392534
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• ISBN 0-471-80634-X
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• Motohashi, Y. (1997). Spectral Theory of the Riemann Zeta-Function. Cambridge University Press.
  ISBN 0521445205
  .
• Karatsuba, A.A.
  ; Voronin, S.M. (1992). The Riemann Zeta-Function. Berlin, DE: W. de Gruyter.
• ISBN 978-0-521-84903-6
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• ISBN 0-387-98308-2
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• Raoh, Guo (1996). "The distribution of the logarithmic derivative of the Riemann zeta function". Proceedings of the London Mathematical Society. S3–72: 1–27.
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• Trinity College, Dublin (maths.tcd.ie). Also available in Riemann, Bernhard
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• Sondow, Jonathan (1994). "Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series" (PDF).
  doi:10.1090/S0002-9939-1994-1172954-7 – via American Mathematical Society
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• Titchmarsh, E.C. (1986). Heath-Brown (ed.). The Theory of the Riemann Zeta Function (2nd rev. ed.). Oxford University Press.
• Whittaker, E.T.; Watson, G.N. (1927). A Course in Modern Analysis (4th ed.). Cambridge University Press. Chapter 13.
• Zhao, Jianqiang (1999). "Analytic continuation of multiple zeta functions".
  MR 1670846
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External links

• Media related to Riemann zeta function at Wikimedia Commons
• "Zeta-function". Encyclopedia of Mathematics. EMS Press. 2001 [1994].
• Riemann Zeta Function, in Wolfram Mathworld — an explanation with a more mathematical approach
• Tables of selected zeros Archived 17 May 2009 at the Wayback Machine
• Prime Numbers Get Hitched A general, non-technical description of the significance of the zeta function in relation to prime numbers.
• X-Ray of the Zeta Function Visually oriented investigation of where zeta is real or purely imaginary.
• Formulas and identities for the Riemann Zeta function functions.wolfram.com
• Riemann Zeta Function and Other Sums of Reciprocal Powers, section 23.2 of Abramowitz and Stegun
• Frenkel, Edward. "Million Dollar Math Problem" (video). Brady Haran. Archived from the original on 11 December 2021. Retrieved 11 March 2014.
• Mellin transform and the functional equation of the Riemann Zeta function—Computational examples of Mellin transform methods involving the Riemann Zeta Function
• Visualizing the Riemann zeta function and analytic continuation a video from 3Blue1Brown