Riemann zeta function

Riemann zeta function
The Riemann zeta function ζ(z) plotted with domain coloring.
Domain, codomain and image
Domain	$\mathbb {C} \setminus \{1\}$
Codomain	$\mathbb {C}$
Specific values
At zero	$-{\frac {1}{2}}$
Value at +∞	$1$
Value at $2$	${\frac {\pi ^{2}}{6}}$
Value at $-1$	$-{1 \over 12}$
Value at $-2$	$0$

The pole at $z=1$, and two zeros on the critical line.

The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of the Dirichlet series

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

which converges when the real part of s is greater than 1. More general representations of ζ(s) for all s are given below. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.

As a function of a real variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis, which was not available at the time. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers.

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.

Definition

Bernhard Riemann's article On the number of primes below a given magnitude.

The Riemann zeta function ζ(s) is a function of a complex variable s = σ + i t. (The notation s, σ, and t is used traditionally in the study of the zeta function, following Riemann.)

For the special case where $\,{\mathfrak {Re}}(s)\equiv \sigma >1\,$, the zeta function can be expressed by the following integral:

$\zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\,\mathrm {d} \,x\quad {\text{ for }}\quad {\mathfrak {Re}}(s)\equiv \sigma >1$

where

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

is the gamma function.

In the case σ > 1, the integral for ζ(s) always converges, and can be simplified to the following infinite series:

$\zeta (s)=\sum _{n=1}^{\infty }n^{-s}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots \quad {\text{ for }}\quad \sigma \equiv {\mathfrak {Re}}(s)>1\,.$

The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series.

Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to ${\mathfrak {Re}}(s)>1\,$.

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

$\lim _{s\to 1}~(s-1)\,\zeta (s)=1\,.$

Thus the Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.

Specific values

Main article: Particular values of the Riemann zeta function

For any positive even integer 2n:

$\zeta (2n)={\frac {(-1)^{n+1}B_{2n}(2\pi )^{2n}}{2(2n)!}}$

where B_2n is the 2nth 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

$\zeta (-n)=(-1)^{n}{\frac {B_{n+1}}{n+1}}$

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:

• $\zeta (-1)=-{\tfrac {1}{12}}$

This gives a pretext for assigning a finite value to the divergent series 1 + 2 + 3 + 4 + ⋯, which has been used in certain contexts such as string theory.

• $\zeta (0)=-{\tfrac {1}{2}};$

Similarly to the above, this assigns a finite result to the series 1 + 1 + 1 + 1 + ⋯.

• $\zeta {\bigl (}{\tfrac {1}{2}}{\bigr )}\approx -1.46035450880958681289$   (OEIS: A059750)

This is employed in calculating of kinetic boundary layer problems of linear kinetic equations.

• $\zeta (1)=1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\cdots =\infty ;$

If we approach from numbers larger than 1, this is the harmonic series. But its Cauchy principal value

$\lim _{\varepsilon \to 0}{\frac {\zeta (1+\varepsilon )+\zeta (1-\varepsilon )}{2}}$

exists which is the Euler–Mascheroni constant γ = 0.5772….

• $\zeta {\bigl (}{\tfrac {3}{2}}{\bigr )}\approx 2.61237534868548834335;$   (OEIS: A078434)

This is employed in calculating the critical temperature for a Bose–Einstein condensate in a box with periodic boundary conditions, and for spin wave physics in magnetic systems.

• $\zeta (2)=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}\approx 1.64493406684822643647;\!$   (OEIS: A013661)

The demonstration of this equality is known as the Basel problem. The reciprocal of this sum answers the question: What is the probability that two numbers selected at random are relatively prime?

• $\zeta (3)=1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots \approx 1.20205690315959428540;$   (OEIS: A002117)

This number is called Apéry's constant.

• $\zeta (4)=1+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}\approx 1.08232323371113819152;$   (OEIS: A013662)

This appears when integrating Planck's law to derive the Stefan–Boltzmann law in physics.

Taking the limit $n\rightarrow \infty$, one obtains $\zeta (\infty )=1$.

Euler product formula

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

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

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):

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

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 ∏_p ⁠p/p − 1⁠) implies that there are infinitely many primes.

The Euler product formula can be used to calculate the asymptotic probability that s randomly selected integers are set-wise coprime. Intuitively, the probability that any single number is divisible by a prime (or any integer) p is ⁠1/p⁠. Hence the probability that s numbers are all divisible by this prime is ⁠1/p^s⁠, and the probability that at least one of them is not is 1 − ⁠1/p^s⁠. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors n and m if and only if 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,

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

(More work is required to derive this result formally.)

Riemann's functional equation

The zeta function satisfies the functional equation:

$\zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s),$

where Γ(s) is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ(s) has a simple zero at each even negative integer s = −2n, known as the trivial zeros of ζ(s). When s is an even positive integer, the product sin(⁠πs/2⁠)Γ(1 − s) on the right is non-zero because Γ(1 − s) has a simple pole, which cancels the simple zero of the sine factor.

Proof of functional equation

A proof of the functional equation proceeds as follows: We observe that if $\sigma >0$, then $\int _{0}^{\infty }x^{{1 \over 2}{s}-1}e^{-n^{2}\pi x}\,dx={\Gamma \left({s \over 2}\right) \over {n^{s}\pi ^{s \over 2}}}.$ As a result, if $\sigma >1$ then ${\frac {\Gamma \left({\frac {s}{2}}\right)\zeta (s)}{\pi ^{s/2}}}=\sum _{n=1}^{\infty }\int \limits _{0}^{\infty }x^{{s \over 2}-1}e^{-n^{2}\pi x}\,dx=\int _{0}^{\infty }x^{{s \over 2}-1}\sum _{n=1}^{\infty }e^{-n^{2}\pi x}\,dx.$ With the inversion of the limiting processes justified by absolute convergence (hence the stricter requirement on $\sigma$) For convenience, let $\psi (x):=\sum _{n=1}^{\infty }e^{-n^{2}\pi x}$ Then $\zeta (s)={\pi ^{s \over 2} \over \Gamma ({s \over 2})}\int \limits _{0}^{\infty }x^{{1 \over 2}{s}-1}\psi (x)\,dx$ Given that $\sum _{n=-\infty }^{\infty }{e^{-n^{2}\pi x}}={1 \over {\sqrt {x}}}\sum _{n=-\infty }^{\infty }{e^{-n^{2}\pi \over x}}$ Then $2\psi (x)+1={1 \over {\sqrt {x}}}\left\{2\psi \left({1 \over x}\right)+1\right\}$ Hence $\pi ^{-{s \over 2}}\Gamma \left({s \over 2}\right)\zeta (s)=\int _{0}^{1}x^{{s \over 2}-1}\psi (x)\,dx+\int _{1}^{\infty }x^{{s \over 2}-1}\psi (x)\,dx$ This is equivalent to $\int \limits _{0}^{1}x^{{s \over 2}-1}\left\{{1 \over {\sqrt {x}}}\psi \left({1 \over x}\right)+{1 \over 2{\sqrt {x}}}-{1 \over 2}\right\}\,dx+\int \limits _{1}^{\infty }x^{{s \over 2}-1}\psi (x)\,dx$ Or ${\begin{aligned}&{1 \over {s-1}}-{1 \over s}+\int \limits _{0}^{1}x^{{{s} \over 2}-{3 \over 2}}\psi \left({1 \over x}\right)\,dx+\int \limits _{1}^{\infty }x^{{{s} \over 2}-1}\psi (x)\,dx\\[5pt]={}&{1 \over {s({s-1})}}+\int \limits _{1}^{\infty }\left({x^{-{{s} \over 2}-{1 \over 2}}+x^{{{s} \over 2}-1}}\right)\psi (x)\,dx\end{aligned}}$ which is convergent for all s, so holds by analytic continuation. Furthermore, the RHS is unchanged if s is changed to 1 − s. Hence $\pi ^{-{s \over 2}}\Gamma \left({s \over 2}\right)\zeta (s)=\pi ^{-{1 \over 2}+{s \over 2}}\Gamma \left({1 \over 2}-{s \over 2}\right)\zeta (1-s)$ which is the functional equation. E. C. Titchmarsh (1986). The Theory of the Riemann Zeta-function (2nd ed.). Oxford: Oxford Science Publications. pp. 21–22. ISBN 0-19-853369-1. Attributed to Bernhard Riemann.

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. An equivalent relationship had been conjectured by Euler over a hundred years earlier, in 1749, for the Dirichlet eta function (alternating zeta function):

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

Incidentally, this relation gives an equation for calculating ζ(s) in the region 0 < Re(s) < 1, i.e.

$\zeta (s)={\frac {1}{1-{2^{1-s}}}}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{s}}},$

where the η-series is convergent (albeit non-absolutely) in the larger half-plane s > 0 (for a more detailed survey on the history of the functional equation, see e.g. Blagouchine).

Riemann also found a symmetric version of the functional equation applying to the xi-function:

$\xi (s)={\frac {1}{2}}\pi ^{-{\frac {s}{2}}}s(s-1)\Gamma \left({\frac {s}{2}}\right)\zeta (s),\!$

which satisfies:

$\xi (s)=\xi (1-s).\!$

(Riemann's original ξ(t) was slightly different.)

Zeros, the critical line, and the Riemann hypothesis

Main article: Riemann hypothesis

Apart from the trivial zeros, the Riemann zeta function has no zeros to the right of σ = 1 and to the left of σ = 0 (neither can the zeros lie too close to those lines). Furthermore, the non-trivial zeros are symmetric about the real axis and the line σ = ⁠1/2⁠ and, according to the Riemann hypothesis, they all lie on the line σ = ⁠1/2⁠.

This image shows a plot of the Riemann zeta function along the critical line for real values of t running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin.

The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011.

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 impressive results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip {s ∈ ℂ : 0 < Re(s) < 1}, which is called the critical strip. The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has Re(s) = ⁠1/2⁠. In the theory of the Riemann zeta function, the set {s ∈ ℂ : Re(s) = ⁠1/2⁠} is called the critical line. For the Riemann zeta function on the critical line, see Z-function.

The Hardy–Littlewood conjectures

In 1914, Godfrey Harold Hardy proved that ζ (⁠1/2⁠ + it) has infinitely many real zeros.

Hardy and John Edensor 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].

1. For any ε > 0, there exists a T₀(ε) > 0 such that when

   $T\geq T_{0}(\varepsilon )\quad {\text{ and }}\quad H=T^{{\frac {1}{4}}+\varepsilon },$

   the interval (T, T + H] contains a zero of odd order.
2. For any ε > 0, there exists a T₀(ε) > 0 and c_ε > 0 such that the inequality

   $N_{0}(T+H)-N_{0}(T)\geq c_{\varepsilon }H$

   holds when

   $T\geq T_{0}(\varepsilon )\quad {\text{ and }}\quad H=T^{{\frac {1}{2}}+\varepsilon }.$

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

Zero-free region

The location of the Riemann zeta function's zeros is of great importance in the theory of numbers. The prime number theorem is equivalent to the fact that there are no zeros of the zeta function on the Re(s) = 1 line. A better result that follows from an effective form of Vinogradov's mean-value theorem is that ζ (σ + it) ≠ 0 whenever |t| ≥ 3 and

$\sigma \geq 1-{\frac {1}{57.54(\log {|t|})^{\frac {2}{3}}(\log {\log {|t|}})^{\frac {1}{3}}}}.$

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.

Other results

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

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

The critical line theorem 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

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

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.

Various properties

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

Reciprocal

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

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

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.

Universality

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. More recent work has included effective versions of Voronin's theorem and extending it to Dirichlet L-functions.

Estimates of the maximum of the modulus of the zeta function

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

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

Here T is a sufficiently large positive number, 0 < H ≪ ln ln 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 ≫ ln ln T was studied by Kanakanahalli Ramachandra; the case Δ > c, where c is a sufficiently large constant, is trivial.

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

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

hold, where c₁ and c₂ are certain absolute constants.

The argument of the Riemann zeta function

The function

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

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 are the mean value theorems for S(t) and its first integral

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

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

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

contains at least

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

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

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

Representations

Dirichlet series

An extension of the area of convergence can be obtained by rearranging the original series. The series

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

converges for Re(s) > 0, while

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

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

Mellin-type integrals

The Mellin transform of a function f(x) is defined as

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

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

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

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

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

for all s (where H denotes the Hankel contour).

Starting with the integral formula $\zeta (n){\Gamma (n)}=\int _{0}^{\infty }{\frac {x^{n-1}}{e^{x}-1}}\mathrm {d} x,$ one can show by substitution and iterated differentation for natural $k\geq 2$

$\int _{0}^{\infty }{\frac {x^{n}e^{x}}{(e^{x}-1)^{k}}}\mathrm {d} x={\frac {n!}{(k-1)!}}\zeta ^{n}\prod _{j=0}^{k-2}\left(1-{\frac {j}{\zeta }}\right)$

using the notation of umbral calculus where each power $\zeta ^{r}$ is to be replaced by $\zeta (r)$, so e.g. for $k=2$ we have $\int _{0}^{\infty }{\frac {x^{n}e^{x}}{(e^{x}-1)^{2}}}\mathrm {d} x={n!}\zeta (n),$ while for $k=4$ this becomes

$\int _{0}^{\infty }{\frac {x^{n}e^{x}}{(e^{x}-1)^{4}}}\mathrm {d} x={\frac {n!}{6}}{\bigl (}\zeta ^{n-2}-3\zeta ^{n-1}+2\zeta ^{n}{\bigr )}=n!{\frac {\zeta (n-2)-3\zeta (n-1)+2\zeta (n)}{6}}.$

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

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

for values with Re(s) > 1.

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

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

Now we have

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

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.

Theta functions

The Riemann zeta function can be given by a Mellin transform

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

in terms of Jacobi's theta function

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

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:

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

Laurent series

The Riemann zeta function is meromorphic with a single pole of order one at s = 1. It can therefore be expanded as a Laurent series about s = 1; the series development is then

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

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

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

The constant term γ₀ is the Euler–Mascheroni constant.

Integral

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

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

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

Rising factorial

Another series development using the rising factorial valid for the entire complex plane is

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

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 Gauss–Kuzmin–Wirsing operator acting on x^{s − 1}; that context gives rise to a series expansion in terms of the falling factorial.

Hadamard product

On the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansion

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

where the product is over the non-trivial zeros ρ of ζ and the letter γ again denotes the Euler–Mascheroni constant. A simpler infinite product expansion is

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

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 and proven by Helmut Hasse in 1930 (cf. Euler summation):

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

The series only appeared in an appendix to Hasse's paper, and did not become generally known until it was discussed by Jonathan Sondow in 1994.

Hasse also proved the globally converging series

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

in the same publication. Research by Iaroslav Blagouchine has found that a similar, equivalent series was published by Joseph Ser in 1926. Other similar globally convergent series include

${\begin{aligned}\zeta (s)&={\frac {1}{s-1}}\sum _{n=0}^{\infty }H_{n+1}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+2)^{1-s}\\[6pt]\zeta (s)&={\frac {1}{s-1}}\left\{-1+\sum _{n=0}^{\infty }H_{n+2}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+2)^{-s}\right\}\\[6pt]\zeta (s)&={\frac {k!}{(s-k)_{k}}}\sum _{n=0}^{\infty }{\frac {1}{(n+k)!}}\left[{n+k \atop n}\right]\sum _{\ell =0}^{n+k-1}\!(-1)^{\ell }{\binom {n+k-1}{\ell }}(\ell +1)^{k-s},\quad k=1,2,3,\ldots \\[6pt]\zeta (s)&={\frac {1}{s-1}}+\sum _{n=0}^{\infty }|G_{n+1}|\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+1)^{-s}\\[6pt]\zeta (s)&={\frac {1}{s-1}}+1-\sum _{n=0}^{\infty }C_{n+1}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+2)^{-s}\\[6pt]\zeta (s)&={\frac {2(s-2)}{s-1}}\zeta (s-1)+2\sum _{n=0}^{\infty }(-1)^{n}G_{n+2}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+1)^{-s}\\[6pt]\zeta (s)&=-\sum _{l=1}^{k-1}{\frac {(k-l+1)_{l}}{(s-l)_{l}}}\zeta (s-l)+{\frac {k}{s-k}}+k\sum _{n=0}^{\infty }(-1)^{n}G_{n+1}^{(k)}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+1)^{-s}\\[6pt]\zeta (s)&={\frac {(a+1)^{1-s}}{s-1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+1)^{-s},\quad \Re (a)>-1\\[6pt]\zeta (s)&=1+{\frac {(a+2)^{1-s}}{s-1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+2)^{-s},\quad \Re (a)>-1\\[6pt]\zeta (s)&={\frac {1}{a+{\tfrac {1}{2}}}}\left\{-{\frac {\zeta (s-1,1+a)}{s-1}}+\zeta (s-1)+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+2}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+1)^{-s}\right\},\quad \Re (a)>-1\end{aligned}}$

where H_n are the harmonic numbers, $\left[{\cdot \atop \cdot }\right]$ are the Stirling numbers of the first kind, $(s-k)_{k}$ is the Pochhammer symbol, G_n are the Gregory coefficients, G^(k)
_n are the Gregory coefficients of higher order, C_n are the Cauchy numbers of the second kind (C₁ = 1/2, C₂ = 5/12, C₃ = 3/8,...), and ψ_n(a) are the Bernoulli polynomials of the second kind, see Blagouchine's paper.

Peter Borwein has developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series suitable for high precision numerical calculations.

Series representation at positive integers via the primorial

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

Here p_n# is the primorial sequence and J_k is Jordan's totient function.

Series representation by the incomplete poly-Bernoulli numbers

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

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

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.

The Mellin transform of the Engel map

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

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

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

Numerical algorithms

For $v=1,2,3,\dots$ , the Riemann zeta function has for fixed $\sigma _{0}<v$ and for all $\sigma \leq \sigma _{0}$ the following representation in terms of three absolutely and uniformly converging series,$${\begin{aligned}\zeta \left(s\right)&=\sum _{n=1}^{\infty }n^{-s}\sum _{w=0}^{v-1}{\frac {\left({\frac {n}{N}}\right)^{w}}{w!}}e^{-{\frac {n}{N}}}-{\frac {\Gamma \left(1-s+v\right)}{\left(1-s\right)\Gamma \left(v\right)}}N^{1-s}+\sum _{\mu =\pm 1}E_{\mu }\left(s\right)\\E_{\mu }\left(s\right)&=\left(2\pi \right)^{s-1}\Gamma \left(1-s\right)e^{i\mu {\frac {\pi }{2}}\left(1-s\right)}\sum _{m=1}^{\infty }\left[m^{s-1}-\sum _{w=0}^{v-1}{\binom {s-1}{w}}\left(m+{\frac {i\mu }{2\pi N}}\right)^{s-1-w}\left({\frac {-i\mu }{2\pi N}}\right)^{w}\right]\end{aligned}}$$where for positive integer $s=k$ one has to take the limit value $\lim _{s\to k}E_{\mu }\left(s\right)$. The derivatives of $\zeta (s)$ can be calculated by differentiating the above series termwise. From this follows an algorithm which allows to compute, to arbitrary precision, $\zeta (s)$ and its derivatives using at most $C\left(\epsilon \right)\left|\tau \right|^{{\frac {1}{2}}+\epsilon }$ summands for any $\epsilon >0$, with explicit error bounds. For $\zeta (s)$, these are as follows:

For a given argument $s$ with $0\leq \sigma \leq 2$ and $0<t$ one can approximate $\zeta (s)$ to any accuracy $\delta \leq 0.05$ by summing the first series to $n=\left\lceil 3.151\cdot vN\right\rceil$, $E_{1}\left(s\right)$ to $m=\left\lceil N\right\rceil$ and neglecting $E_{-1}\left(s\right)$, if one chooses $v$ as the next higher integer of the unique solution of $x-\max \left({\frac {1-\sigma }{2}},0\right)\ln \left({\frac {1}{2}}+x+\tau \right)=\ln {\frac {8}{\delta }}$ in the unknown $x$, and from this $N=1.11\left(1+{\frac {{\frac {1}{2}}+\tau }{v}}\right)^{\frac {1}{2}}$. For $t=0$ one can neglect $E_{1}\left(s\right)$ altogether. Under the mild condition $\tau >{\frac {5}{3}}\left({\frac {3}{2}}+\ln {\frac {8}{\delta }}\right)$ one needs at most $2+8{\sqrt {1+\ln {\frac {8}{\delta }}+\max \left({\frac {1-\sigma }{2}},0\right)\ln \left(2\tau \right)}}~{\sqrt {\tau }}$ summands. Hence this algorithm is essentially as fast as the Riemann-Siegel formula. Similar algorithms are possible for Dirichlet L-functions.

In February 2020, Sandeep Tyagi showed that a quantum computer can evaluate $\zeta \left(s\right)$ in the critical strip with computational complexity that is polylogarithmic in $t$. Following work by Ghaith Ayesh Hiary, the required exponential sums may be rescaled as $t^{\frac {1}{D}}$, for integer $D\geq 2$.

Applications

The zeta function occurs in applied statistics (see Zipf's law and Zipf–Mandelbrot law).

Zeta function regularization is used as one possible means of regularization of divergent series and divergent integrals in quantum field theory. In one notable example, the Riemann zeta-function shows up explicitly in one method of calculating the Casimir effect. The zeta function is also useful for the analysis of dynamical systems.

Infinite series

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

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

In fact the even and odd terms give the two sums

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

and

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

Parametrized versions of the above sums are given by

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

and

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

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

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

all of which are continuous at $t=1$. Other sums include

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

where Im denotes the imaginary part of a complex number.

There are yet more formulas in the article Harmonic number.

Generalizations

There are a number of related zeta functions that can be considered to be generalizations of the Riemann zeta function. These include the Hurwitz zeta function

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

(the convergent series representation was given by Helmut Hasse in 1930, cf. Hurwitz zeta function), which coincides with the Riemann zeta function when q = 1 (the lower limit of summation in the Hurwitz zeta function is 0, not 1), the Dirichlet L-functions and the Dedekind zeta-function. For other related functions see the articles zeta function and L-function.

The polylogarithm is given by

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

which coincides with the Riemann zeta function when z = 1.

The Lerch transcendent is given by

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

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 Clausen function Cl_s(θ) that can be chosen as the real or imaginary part of Li_s(e^iθ).

The multiple zeta functions are defined by

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

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

See also

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

References

• Apostol, T. M. (2010), "Zeta and Related Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Borwein, Jonathan; Bradley, David M.; Crandall, Richard (2000). "Computational Strategies for the Riemann Zeta Function" (PDF). J. Comp. App. Math. 121 (1–2): 247–296. Bibcode:2000JCoAM.121..247B. doi:10.1016/S0377-0427(00)00336-8.
• Cvijović, Djurdje; Klinowski, Jacek (2002). "Integral Representations of the Riemann Zeta Function for Odd-Integer Arguments". J. Comp. App. Math. 142 (2): 435–439. Bibcode:2002JCoAM.142..435C. doi:10.1016/S0377-0427(02)00358-8. 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.
• Edwards, H. M. (1974). Riemann's Zeta Function. Academic Press. ISBN 0-486-41740-9. Has an English translation of Riemann's paper.
• 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.
• Hardy, G. H. (1949). Divergent Series. Clarendon Press, Oxford.
• Hasse, Helmut (1930). "Ein Summierungsverfahren für die Riemannsche ζ-Reihe". Math. Z. 32: 458–464. doi:10.1007/BF01194645. MR 1545177. (Globally convergent series expression.)
• Ivic, A. (1985). The Riemann Zeta Function. John Wiley & Sons. ISBN 0-471-80634-X.
• 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: W. de Gruyter.
• Mező, István; Dil, Ayhan (2010). "Hyperharmonic series involving Hurwitz zeta function". Journal of Number Theory. 130 (2): 360–369. doi:10.1016/j.jnt.2009.08.005. hdl:2437/90539. MR 2564902.
• Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative number theory. I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge University Press. Ch. 10. ISBN 978-0-521-84903-6.
• Newman, Donald J. (1998). Analytic number theory. Graduate Texts in Mathematics. Vol. 177. Springer-Verlag. Ch. 6. ISBN 0-387-98308-2.
• Raoh, Guo (1996). "The Distribution of the Logarithmic Derivative of the Riemann Zeta Function". Proceedings of the London Mathematical Society. s3–72: 1–27. arXiv:1308.3597. doi:10.1112/plms/s3-72.1.1.
• Riemann, Bernhard (1859). "Über die Anzahl der Primzahlen unter einer gegebenen Grösse". Monatsberichte der Berliner Akademie.. In Gesammelte Werke, Teubner, Leipzig (1892), Reprinted by Dover, New York (1953).
• Sondow, Jonathan (1994). "Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series" (PDF). Proc. Amer. Math. Soc. 120 (2): 421–424. doi:10.1090/S0002-9939-1994-1172954-7.
• 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. Ch. 13.
• Zhao, Jianqiang (1999). "Analytic continuation of multiple zeta functions". Proc. Amer. Math. Soc. 128 (5): 1275–1283. doi:10.1090/S0002-9939-99-05398-8. MR 1670846.

External links

• "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
• 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. 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