The Riemann zeta function 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 ζ ( s ) = ∑ n = 1 ∞ 1 n s displaystyle zeta (s)=sum _ n=1 ^ infty frac 1 n^ s when the real part of s is not 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 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 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 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.Contents1 Definition 2 Specific values 3 Euler product formula 4 Riemann's functional equation 5 Zeros, the critical line, and the Riemann hypothesis5.1 The Hardy–Littlewood conjectures 5.2 Zero-free region 5.3 Other results6 Various properties6.1 Reciprocal 6.2 Universality 6.3 Estimates of the maximum of the modulus of the zeta function 6.4 The argument of the Riemann zeta function7 Representations7.1 Dirichlet series 7.2 Mellin-type integrals 7.3 Theta functions 7.4 Laurent series 7.5 Integral 7.6 Rising factorial 7.7 Hadamard Hadamard product 7.8 Globally convergent series 7.9 Series representation at positive integers via the primorial 7.10 Series representation by the incomplete poly-Bernoulli numbers8 Numerical algorithms 9 Applications9.1 Infinite series10 Generalizations 11 Fractional derivative 12 See also 13 Notes 14 References 15 External linksDefinitionBernhard Riemann's article On the number of primes below a given magnitude.The Riemann zeta function Riemann zeta function ζ(s) is a function of a complex variable s = σ + it. (The notation s, σ, and t is used traditionally in the study of the zeta function, following Riemann.) The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case: ζ ( s ) = ∑ n = 1 ∞ n − s = 1 1 s + 1 2 s + 1 3 s + ⋯ σ = Re ⁡ ( s ) > 1. displaystyle zeta (s)=sum _ n=1 ^ infty n^ -s = frac 1 1^ s + frac 1 2^ s + frac 1 3^ s +cdots qquad sigma =operatorname Re (s)>1. It can also be defined by the integral ζ ( s ) = 1 Γ ( s ) ∫ 0 ∞ 1 e x − 1 x s d x x displaystyle zeta (s)= frac 1 Gamma (s) int _ 0 ^ infty frac 1 e^ x -1 ,x^ s frac mathrm d x x where Γ ( s ) = ∫ 0 ∞ e − x x s d x x displaystyle Gamma (s)=int _ 0 ^ infty e^ -x ,x^ s frac mathrm d x x is the gamma function. The Riemann zeta function 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 Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev Chebyshev extended the definition to 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 → 1 ( s − 1 ) ζ ( s ) = 1. displaystyle lim _ sto 1 (s-1)zeta (s)=1. Thus the Riemann zeta function 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 n displaystyle mathrm n .Main article: Particular values of the Riemann zeta function For any positive even integer 2n: ζ ( 2 n ) = ( − 1 ) n + 1 B 2 n ( 2 π ) 2 n 2 ( 2 n ) ! displaystyle zeta (2n)= frac (-1)^ n+1 B_ 2n (2pi )^ 2n 2(2n)! where B2n 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 Special values of L-functions. For nonpositive integers, one has ζ ( − n ) = ( − 1 ) n B n + 1 n + 1 displaystyle zeta (-n)=(-1)^ n frac B_ n+1 n+1 for n ≥ 0 (using the NIST convention that B1 = −1/2) In particular, ζ vanishes at the negative even integers because Bm = 0 for all odd m other than 1. Via analytic continuation, one can show that: ζ ( − 1 ) = − 1 12 displaystyle zeta (-1)=- tfrac 1 12 This gives a way to assign a finite result to the divergent series 1 + 2 + 3 + 4 + ⋯, which has been used in certain contexts such as string theory. ζ ( 0 ) = − 1 2 ; displaystyle zeta (0)=- tfrac 1 2 ; Similarly to the above, this assigns a finite result to the series 1 + 1 + 1 + 1 + ⋯. ζ ( 1 2 ) ≈ − 1.46035450880958681289 displaystyle zeta bigl ( tfrac 1 2 bigr ) approx -1.46035450880958681289   ( A059750)This is employed in calculating of kinetic boundary layer problems of linear kinetic equations. ζ ( 1 ) = 1 + 1 2 + 1 3 + ⋯ = ∞ ; displaystyle zeta (1)=1+ tfrac 1 2 + tfrac 1 3 +cdots =infty ; if we approach from numbers larger than 1. Then this is the harmonic series. But its Cauchy principal value lim ε → 0 ζ ( 1 + ε ) + ζ ( 1 − ε ) 2 displaystyle lim _ varepsilon to 0 frac zeta (1+varepsilon )+zeta (1-varepsilon ) 2 ζ ( 3 2 ) ≈ 2.61237534868548834335 ; displaystyle zeta bigl ( tfrac 3 2 bigr ) approx 2.61237534868548834335;   ( A078434)This is employed in calculating the critical temperature for a Bose–Einstein condensate Bose–Einstein condensate in a box with periodic boundary conditions, and for spin wave physics in magnetic systems. ζ ( 2 ) = 1 + 1 2 2 + 1 3 2 + ⋯ = π 2 6 ≈ 1.64493406684822643647 ; displaystyle zeta (2)=1+ frac 1 2^ 2 + frac 1 3^ 2 +cdots = frac pi ^ 2 6 approx 1.64493406684822643647;!   ( 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? ζ ( 3 ) = 1 + 1 2 3 + 1 3 3 + ⋯ ≈ 1.20205690315959428540 ; displaystyle zeta (3)=1+ frac 1 2^ 3 + frac 1 3^ 3 +cdots approx 1.20205690315959428540;   ( A002117)This number is called Apéry's constant. ζ ( 4 ) = 1 + 1 2 4 + 1 3 4 + ⋯ = π 4 90 ≈ 1.08232323371113819152 ; displaystyle zeta (4)=1+ frac 1 2^ 4 + frac 1 3^ 4 +cdots = frac pi ^ 4 90 approx 1.08232323371113819152;   ( A0013662) Euler product formula The connection between the zeta function and prime numbers was discovered by Euler, who proved the identity ∑ n = 1 ∞ 1 n s = ∏ p  prime 1 1 − p − s , 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): ∏ p  prime 1 1 − p − s = 1 1 − 2 − s ⋅ 1 1 − 3 − s ⋅ 1 1 − 5 − s ⋅ 1 1 − 7 − s ⋅ 1 1 − 11 − s ⋯ 1 1 − p − s ⋯ 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 ∏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/ps, and the probability that at least one of them is not is 1 − 1/ps. 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, ∏ p  prime ( 1 − 1 p s ) = ( ∏ p  prime 1 1 − p − s ) − 1 = 1 ζ ( s ) . 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) . (More work is required to derive this result formally.) Riemann's functional equation The zeta function satisfies the functional equation: ζ ( s ) = 2 s π s − 1   sin ⁡ ( π s 2 )   Γ ( 1 − s )   ζ ( 1 − s ) , 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 gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function 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 equationA proof of the functional equation proceeds as follows: We observe that, if σ > 0 displaystyle sigma >0 ∫ 0 ∞ x 1 2 s − 1 e − n 2 π x d x = Γ ( s 2 ) n s π s 2 displaystyle int limits _ 0 ^ infty x^ 1 over 2 s -1 e^ - n ^ 2 pi x ,dx= Gamma ( s over 2 ) over n^ s pi ^ s over 2 As a result, if σ > 1 displaystyle sigma >1 Γ ( s 2 ) ζ ( s ) π s 2 = ∑ n = 1 ∞ ∫ 0 ∞ x s 2 − 1 e − n 2 π x d x = ∫ 0 ∞ x s 2 − 1 ∑ n = 1 ∞ e − n 2 π x d x displaystyle Gamma ( s over 2 )zeta (s) over pi ^ s over 2 = sum _ n=1 ^ infty int limits _ 0 ^ infty x^ s over 2 -1 e^ -n^ 2 pi x ,dx = int limits _ 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 σ displaystyle sigma ) For convenience, let ψ ( x ) := ∑ n = 1 ∞ e − n 2 π x displaystyle psi (x):=sum _ n=1 ^ infty e^ -n^ 2 pi x Then ζ ( s ) = π s 2 Γ ( s 2 ) ∫ 0 ∞ x 1 2 s − 1 ψ ( x ) d x displaystyle 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 ∑ n = − ∞ ∞ e − n 2 π x = 1 x ∑ n = − ∞ ∞ e − n 2 π x displaystyle 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 ψ ( x ) + 1 = 1 x 2 ψ ( 1 x ) + 1 displaystyle 2psi (x)+1= 1 over sqrt x 2psi left( 1 over x right)+1 Hence π − s 2 Γ ( s 2 ) ζ ( s ) = ∫ 0 1 x s 2 − 1 ψ ( x ) d x + ∫ 1 ∞ x s 2 − 1 ψ ( x ) d x displaystyle pi ^ - s over 2 Gamma left( s over 2 right)zeta (s)=int limits _ 0 ^ 1 x^ s over 2 -1 psi (x),dx+int limits _ 1 ^ infty x^ s over 2 -1 psi (x),dx This is equivalent to ∫ 0 1 x s 2 − 1 1 x ψ ( 1 x ) + 1 2 x − 1 2 d x + ∫ 1 ∞ x s 2 − 1 ψ ( x ) d x displaystyle int limits _ 0 ^ 1 x^ s over 2 -1 1 over sqrt x psi left( 1 over x right)+ 1 over 2 sqrt x - 1 over 2 ,dx+int limits _ 1 ^ infty x^ s over 2 -1 psi (x),dx Or 1 s − 1 − 1 s + ∫ 0 1 x s 2 − 3 2 ψ ( 1 x ) d x + ∫ 1 ∞ x s 2 − 1 ψ ( x ) d x displaystyle 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 = 1 s ( s − 1 ) + ∫ 1 ∞ ( x − s 2 − 1 2 + x s 2 − 1 ) ψ ( x ) d x displaystyle = 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 Which is convergent for all s, so holds by analytic continuation. Furthermore, the RHS is unchanged if s is changed to 1-s. Hence π − s 2 Γ ( s 2 ) ζ ( s ) = π − 1 2 + s 2 Γ ( 1 2 − s 2 ) ζ ( 1 − s ) displaystyle 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 Oxford Science Publications. pp. 21–22. ISBN 0-19-853369-1.  access-date= requires url= (help) 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 Dirichlet eta function (alternating zeta function): η ( s ) = ∑ n = 1 ∞ ( − 1 ) n + 1 n s = ( 1 − 2 1 − s ) ζ ( s ) . displaystyle 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. ζ ( s ) = 1 1 − 2 1 − s ∑ n = 1 ∞ ( − 1 ) n + 1 n s , displaystyle 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. Riemann also found a symmetric version of the functional equation applying to the xi-function: ξ ( s ) = 1 2 π − s 2 s ( s − 1 ) Γ ( s 2 ) ζ ( s ) , displaystyle xi (s)= frac 1 2 pi ^ - frac s 2 s(s-1)Gamma left( frac s 2 right)zeta (s),! which satisfies: ξ ( s ) = ξ ( 1 − s ) . displaystyle xi (s)=xi (1-s).! (Riemann's original ξ(t) was slightly different.) Zeros, the critical line, and the Riemann hypothesis Main article: Riemann hypothesisApart from the trivial zeros, the Riemann zeta function 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 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 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 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].For any ε > 0, there exists a T0(ε) > 0 such that when T ≥ T 0 ( ε )  and  H = T 1 4 + ε , displaystyle Tgeq T_ 0 (varepsilon )quad text and quad H=T^ frac 1 4 +varepsilon , the interval (T, T + H] contains a zero of odd order. For any ε > 0, there exists a T0(ε) > 0 and cε > 0 such that the inequality N 0 ( T + H ) − N 0 ( T ) ≥ c ε H displaystyle N_ 0 (T+H)-N_ 0 (T)geq c_ varepsilon H holds when T ≥ T 0 ( ε )  and  H = T 1 2 + ε displaystyle Tgeq 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 σ ≥ 1 − 1 57.54 ( log ⁡ t ) 2 3 ( log ⁡ log ⁡ t ) 1 3 . displaystyle 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 (γn) contains the imaginary parts of all zeros in the upper half-plane in ascending order, then lim n → ∞ ( γ n + 1 − γ n ) = 0. displaystyle lim _ nrightarrow 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 ( A058303). The fact that ζ ( s ) = ζ ( s ¯ ) ¯ 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. 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 Möbius function μ(n): 1 ζ ( s ) = ∑ n = 1 ∞ μ ( n ) n s 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 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 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 ≤ H ζ ( 1 2 + i t ) , G ( s 0 ; Δ ) = max s − s 0 ≤ Δ ζ ( s ) . displaystyle F(T;H)=max _ t-Tleq H leftzeta left( tfrac 1 2 +itright)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 ) ≥ T − c 1 , G ( s 0 ; Δ ) ≥ T − c 2 , 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 S ( t ) = 1 π arg ⁡ ζ ( 1 2 + i t ) displaystyle S(t)= frac 1 pi arg zeta left( tfrac 1 2 +itright) 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 ) = ∫ 0 t S ( u ) d u 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 H ≥ T 27 82 + ε displaystyle Hgeq T^ frac 27 82 +varepsilon contains at least H ln ⁡ T 3 e − c ln ⁡ ln ⁡ T 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 Atle Selberg for the case H ≥ T 1 2 + ε . displaystyle Hgeq 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 ζ ( s ) = 1 s − 1 ∑ n = 1 ∞ ( n ( n + 1 ) s − n − s n s ) 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 ζ ( s ) = 1 s − 1 ∑ n = 1 ∞ n ( n + 1 ) 2 ( 2 n + 3 + s ( n + 1 ) s + 2 − 2 n − 1 − s n s + 2 ) 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) 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 integralsThis section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2014) (Learn how and when to remove this template message)The Mellin transform of a function f(x) is defined as ∫ 0 ∞ f ( x ) x s d x x 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 Γ ( s ) ζ ( s ) = ∫ 0 ∞ x s − 1 e x − 1 d x , displaystyle 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 ⁡ ( π s ) Γ ( s ) ζ ( s ) = i ∮ H ⁡ ( − x ) s − 1 e x − 1 d x displaystyle 2sin(pi s)Gamma (s)zeta (s)=ioint _ 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 ζ ( n ) Γ ( n ) = ∫ 0 ∞ x n − 1 e x − 1 d x , displaystyle 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 ≥ 2 displaystyle kgeq 2 ∫ 0 ∞ x n e x ( e x − 1 ) k d x = n ! ( k − 1 ) ! ζ n ∏ j = 0 k − 2 ( 1 − j ζ ) displaystyle 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 ζ r displaystyle zeta ^ r is to be replaced by ζ ( r ) displaystyle zeta (r) , so e.g. for k = 2 displaystyle k=2 we have ∫ 0 ∞ x n e x ( e x − 1 ) 2 d x = n ! ζ ( n ) , displaystyle int _ 0 ^ infty frac x^ n e^ x (e^ x -1)^ 2 mathrm d x= n! zeta (n), while for k = 4 displaystyle k=4 this becomes ∫ 0 ∞ x n e x ( e x − 1 ) 4 d x = n ! 6 ( ζ n − 2 − 3 ζ n − 1 + 2 ζ n ) = n ! ζ ( n − 2 ) − 3 ζ ( n − 1 ) + 2 ζ ( n ) 6 . displaystyle int _ 0 ^ infty frac x^ n e^ x (e^ x -1)^ 4 mathrm d x= frac n! 6 bigl ( zeta ^ n-2 -3zeta ^ n-1 +2zeta ^ n bigr ) =n! frac zeta (n-2)-3zeta (n-1)+2zeta (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 ⁡ ζ ( s ) = s ∫ 0 ∞ π ( x ) x ( x s − 1 ) d x , displaystyle ln zeta (s)=sint _ 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 ) = ∑ π ( x 1 n ) n . displaystyle J(x)=sum frac pi left(x^ frac 1 n right) n . Now we have ln ⁡ ζ ( s ) = s ∫ 0 ∞ J ( x ) x − s − 1 d x . displaystyle ln zeta (s)=sint _ 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 Riemann zeta function can be given formally by a divergent Mellin transform 2 π − s 2 Γ ( s 2 ) ζ ( s ) = ∫ 0 ∞ θ ( i t ) t s 2 − 1 d t , displaystyle 2pi ^ - frac s 2 Gamma left( frac s 2 right)zeta (s)=int _ 0 ^ infty theta (it)t^ frac s 2 -1 ,mathrm d t, in terms of Jacobi's theta function θ ( τ ) = ∑ n = − ∞ ∞ e π i n 2 τ . displaystyle theta (tau )=sum _ n=-infty ^ infty e^ pi in^ 2 tau . However this integral does not converge for any value of s and so needs to be regularized: this gives the following expression for the zeta function: π − s 2 Γ ( s 2 ) ζ ( s ) = 1 s − 1 − 1 s + 1 2 ∫ 0 1 ( θ ( i t ) − t − 1 2 ) t s 2 − 1 d t + 1 2 ∫ 1 ∞ ( θ ( i t ) − 1 ) t s 2 − 1 d t . 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. Laurent seriesThis section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2014) (Learn how and when to remove this template message)The Riemann zeta function Riemann zeta function is meromorphic with a single pole of order one at s = 1. It can therefore be expanded as a Laurent series Laurent series about s = 1; the series development is then ζ ( s ) = 1 s − 1 + ∑ n = 0 ∞ ( − 1 ) n γ n n ! ( s − 1 ) n . displaystyle zeta (s)= frac 1 s-1 +sum _ n=0 ^ infty frac (-1)^ n gamma _ n n! (s-1)^ n . The constants γn here are called the Stieltjes constants Stieltjes constants and can be defined by the limit γ n = lim m → ∞ ( ( ∑ k = 1 m ( ln ⁡ k ) n k ) − ( ln ⁡ m ) n + 1 n + 1 ) . displaystyle gamma _ n =lim _ mrightarrow 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) ζ ( s ) = 1 2 + 1 s − 1 − 2 s ∫ 0 ∞ sin ⁡ ( s arctan ⁡ t ) ( 1 + t 2 ) s 2 ( e π t + 1 ) d t , displaystyle zeta (s)= frac 1 2 + frac 1 s-1 -2^ s !int _ 0 ^ infty !!! frac sin(sarctan t) left(1+t^ 2 right)^ frac s 2 left(e^ pi t +1right) ,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[citation needed] ζ ( s ) = s s − 1 − ∑ n = 1 ∞ ( ζ ( s + n ) − 1 ) s ( s + 1 ) ⋯ ( s + n − 1 ) ( n + 1 ) ! . 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 Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the Gauss–Kuzmin–Wirsing operator acting on xs − 1; that context gives rise to a series expansion in terms of the falling factorial. Hadamard Hadamard product On the basis of Weierstrass's factorization theorem, Hadamard Hadamard gave the infinite product expansion ζ ( s ) = e ( log ⁡ ( 2 π ) − 1 − γ 2 ) s 2 ( s − 1 ) Γ ( 1 + s 2 ) ∏ ρ ( 1 − s ρ ) e s ρ , displaystyle zeta (s)= frac e^ left(log(2pi )-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 ζ ( s ) = π s 2 ∏ ρ ( 1 − s ρ ) 2 ( s − 1 ) Γ ( 1 + s 2 ) . 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 and proven by Helmut Hasse Helmut Hasse in 1930 (cf. Euler summation): ζ ( s ) = 1 1 − 2 1 − s ∑ n = 0 ∞ 1 2 n + 1 ∑ k = 0 n ( n k ) ( − 1 ) k ( k + 1 ) s . 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 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 ζ ( s ) = 1 s − 1 ∑ n = 0 ∞ 1 n + 1 ∑ k = 0 n ( n k ) ( − 1 ) k ( k + 1 ) s − 1 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, but research by Iaroslav Blagouchine has found that this latter series was actually first published by Joseph Ser in 1926. New proofs for both of these results were offered by Demetrios Kanoussis in 2017. Peter Borwein has developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function Dirichlet eta function to produce a very rapidly convergent series suitable for high precision numerical calculations. Series representation at positive integers via the primorial ζ ( k ) = 2 k 2 k − 1 + ∑ r = 2 ∞ ( p r − 1 # ) k J k ( p r # ) k = 2 , 3 , … . 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 pn# is the primorial sequence and Jk 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 ζ ( s ) = ∑ n = 0 ∞ B n , ≥ 2 ( s ) ( W k ( − 1 ) ) n n ! , displaystyle zeta (s)=sum _ n=0 ^ infty B_ n,geq 2 ^ (s) frac (W_ k (-1))^ n n! , where k ∈ −1, 0 , Wk is the kth branch of the Lambert W-function, and B(μ) n, ≥2 is an incomplete poly-Bernoulli number. Numerical algorithms For v = 1 , 2 , 3 , … displaystyle v=1,2,3,dots σ 0 < v displaystyle sigma _ 0 0 displaystyle epsilon >0 , with explicit error bounds. For ζ ( s ) displaystyle zeta (s) , these are as follows: For a given argument s displaystyle s with 0 ≤ σ ≤ 2 displaystyle 0leq sigma leq 2 and 0 < t displaystyle 0 5 3 ( 3 2 + ln ⁡ 8 δ ) displaystyle tau > frac 5 3 left( frac 3 2 +ln frac 8 delta right) one needs at most 2 + 8 1 + ln ⁡ 8 δ + max ( 1 − σ 2 , 0 ) ln ⁡ ( 2 τ )   τ displaystyle 2+8 sqrt 1+ln frac 8 delta +max left( frac 1-sigma 2 ,0right)ln left(2tau right) ~ sqrt tau summands. Hence this algorithm is essentially as fast as the Riemann-Siegel formula. Similar algorithms are possible for Dirichlet L-functions. Applications The zeta function occurs in applied statistics (see Zipf's law 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. ∑ n = 2 ∞ ( ζ ( n ) − 1 ) = 1 displaystyle sum _ n=2 ^ infty bigl ( zeta (n)-1 bigr ) =1 In fact the even and odd terms give the two sums ∑ n = 1 ∞ ( ζ ( 2 n ) − 1 ) = 3 4 displaystyle sum _ n=1 ^ infty bigl ( zeta (2n)-1 bigr ) = frac 3 4 and ∑ n = 1 ∞ ( ζ ( 2 n + 1 ) − 1 ) = 1 4 displaystyle sum _ n=1 ^ infty bigl ( zeta (2n+1)-1 bigr ) = frac 1 4 Parametrized versions of the above sums are given by ∑ n = 1 ∞ ( ζ ( 2 n ) − 1 ) t 2 n = t 2 t 2 − 1 + 1 2 ( 1 − π t ⋅ cot ⁡ ( t π ) ) displaystyle sum _ n=1 ^ infty (zeta (2n)-1),t^ 2n = frac t^ 2 t^ 2 -1 + frac 1 2 left(1-pi ,tcdot cot left(t,pi right)right) and ∑ n = 1 ∞ ( ζ ( 2 n + 1 ) − 1 ) t 2 n = t 2 t 2 − 1 + 1 2 ( ψ 0 ( t ) + ψ 0 ( − t ) ) − γ 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 t < 2 displaystyle t<2 and where ψ displaystyle psi and γ displaystyle gamma ∑ n = 1 ∞ ζ ( 2 n ) − 1 n t 2 n = log ⁡ ( 1 − t 2 sinc ( π t ) ) displaystyle sum _ n=1 ^ infty frac zeta (2n)-1 n ,t^ 2n =log left( dfrac 1-t^ 2 text sinc (pi ,t) right) all of which are continuous at t = 1 displaystyle t=1 . Other sums include ∑ n = 2 ∞ ζ ( n ) − 1 n = 1 − γ displaystyle sum _ n=2 ^ infty frac zeta (n)-1 n =1-gamma ∑ n = 2 ∞ ζ ( n ) − 1 n ( ( 3 2 ) n − 1 − 1 ) = 1 3 ln ⁡ π displaystyle sum _ n=2 ^ infty frac zeta (n)-1 n left(left( tfrac 3 2 right)^ n-1 -1right)= frac 1 3 ln pi ∑ n = 1 ∞ ( ζ ( 4 n ) − 1 ) = 7 8 − π 4 ( e 2 π + 1 e 2 π − 1 ) . displaystyle sum _ n=1 ^ infty bigl ( zeta (4n)-1 bigr ) = frac 7 8 - frac pi 4 left( frac e^ 2pi +1 e^ 2pi -1 right). ∑ n = 2 ∞ ζ ( n ) − 1 n Im ⁡ ( ( 1 + i ) n − ( 1 + i n ) ) = π 4 displaystyle 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 ζ ( s , q ) = ∑ k = 0 ∞ 1 ( k + q ) s displaystyle zeta (s,q)=sum _ k=0 ^ infty frac 1 (k+q)^ s (the convergent series representation was given by Helmut Hasse Helmut Hasse in 1930, cf. Hurwitz zeta function), which coincides with the Riemann zeta function when q = 1 (note that the lower limit of summation in the Hurwitz zeta function 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 L i s ( z ) = ∑ k = 1 ∞ z k k s displaystyle mathrm Li _ s (z)=sum _ k=1 ^ infty frac z^ k k^ s Φ ( z , s , q ) = ∑ k = 0 ∞ z k ( k + q ) s displaystyle Phi (z,s,q)=sum _ k=0 ^ infty frac z^ k (k+q)^ s which coincides with the Riemann zeta function Riemann zeta function when z = 1 and q = 1 (note that the lower limit of summation in the Lerch transcendent is 0, not 1). The Clausen function Cls(θ) that can be chosen as the real or imaginary part of Lis(eiθ). The multiple zeta functions are defined by ζ ( s 1 , s 2 , … , s n ) = ∑ k 1 > k 2 > ⋯ > k n > 0 k 1 − s 1 k 2 − s 2 ⋯ k n − s n . 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 of these functions are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics. Fractional derivative In the case of the Riemann zeta function, a difficulty is represented by the fractional differentiation in the complex plane. The Ortigueira generalization of the classical Caputo fractional derivative solves this problem. The α-order fractional derivative of the Riemann zeta function is given by  ζ ( α ) ( s ) = e i π α ∑ n = 2 ∞ log α ⁡ n n s   . displaystyle zeta ^ (alpha ) (s)=e^ ipi alpha sum _ n=2 ^ infty frac log ^ alpha n n^ s . Given that α is a fractional number such that ⌊ α ⌋ > 0 displaystyle lfloor alpha rfloor >0 , the half-plane of convergence is Re s > 1+α. See also1 + 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 functionNotes^ "Jupyter Notebook Viewer". Nbviewer.ipython.org. Retrieved 2017-01-04.  ^ This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function Riemann zeta function that is considered by many mathematicians to be the most important unsolved problem in pure mathematics.Bombieri, Enrico. 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"On the probability that k positive integers are relatively prime". Journal of Number Theory. 4 (5): 469–473. Bibcode:1972JNT.....4..469N. doi:10.1016/0022-314X(72)90038-8.  ^ Diamond, Harold G. (1982). "Elementary methods in the study of the distribution of prime numbers". Bulletin of the American Mathematical Society. 7 (3): 553–89. doi:10.1090/S0273-0979-1982-15057-1. MR 0670132.  ^ Ford, K. (2002). "Vinogradov's integral and bounds for the Riemann zeta function". Proc. London Math. Soc. 85 (3): 565–633. doi:10.1112/S0024611502013655.  ^ Voronin, S. M. (1975). "Theorem on the Universality of the Riemann Zeta Function". Izv. Akad. Nauk SSSR, Ser. Matem. 39: 475–486.  Reprinted in Math. USSR Izv. (1975) 9: 443–445. ^ Ramūnas Garunkštis; Antanas Laurinčikas; Kohji Matsumoto; Jörn Steuding; Rasa Steuding (2010). "Effective uniform approximation by the Riemann zeta-function". Publicacions Matemàtiques. 54: 209–219. doi:10.1090/S0025-5718-1975-0384673-1. 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The Theory of the Riemann Zeta Function, (2nd rev. ed.). Oxford 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 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 linksHazewinkel, Michiel, ed. (2001) , "Zeta-function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4  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 Functionv t eL-functions in number theoryAnalytic examplesRiemann zeta function Dirichlet L-functions L-functions of Hecke characters Automorphic L-functions Selberg classAlgebraic examplesDedekind zeta functions Artin L-functions Hasse–Weil L-functions Motivic L-functionsTheoremsAnalytic class number formula Riemann–von Mangoldt formula Weil conjecturesAnalytic conjecturesRiemann hypothesis Generalized Riemann hypothesis Lindelöf hypothesis Ramanujan–Petersson conjecture Artin conjectureAlgebraic conjecturesBirch and Swinnerton-Dyer conjecture Deligne's conjecture Beilinson conjectures Bloch–Kato conjecture Langlands conjecturep-adic L-functionsMain conjecture of Iwasawa theory Selmer group Euler systemAuthority controlLCCN: sh85052354 GND: 4308419-9 SUDOC: 031709117 BNF: cb12287377j (data) NDL: 0057