Chapter 25 Zeta and Related Functions T. M. Apostol1
Notation 602 25.12 Polylogarithms ...... 610 25.1 Special Notation ...... 602 25.13 Periodic Zeta Function ...... 612 25.14 Lerch’s Transcendent ...... 612 Riemann Zeta Function 602 25.15 Dirichlet L-functions ...... 612 25.2 Definition and Expansions ...... 602 25.3 Graphics ...... 603 Applications 613 25.4 Reflection Formulas ...... 603 25.16 Mathematical Applications ...... 613 25.5 Integral Representations ...... 604 25.17 Physical Applications ...... 614 25.6 Integer Arguments ...... 605 25.7 Integrals ...... 606 Computation 614 25.8 Sums ...... 606 25.18 Methods of Computation ...... 614 25.9 Asymptotic Approximations ...... 606 25.19 Tables ...... 614 25.10 Zeros ...... 606 25.20 Approximations ...... 615 25.21 Software ...... 615 Related Functions 607 25.11 Hurwitz Zeta Function ...... 607 References 615
1California Institute of Technology, Pasadena, California. Copyright c 2009 National Institute of Standards and Technology. All rights reserved.
601 602 Zeta and Related Functions
∞ Notation 1 X (−1)n 25.2.4 ζ(s) = + γ (s − 1)n, 0, s − 1 n! n n=0 25.1 Special Notation where
m ! (For other notation see pp. xiv and 873.) X (ln k)n (ln m)n+1 25.2.5 γn = lim − . m→∞ k n + 1 k, m, n nonnegative integers. k=1 p prime number. ∞ X x real variable. 25.2.6 ζ0(s) = − (ln n)n−s, 1. a real or complex parameter. n=2 s = σ + it complex variable. 25.2.7 ∞ z = x + iy complex variable. (k) k X k −s γ Euler’s constant (§5.2(ii)). ζ (s) = (−1) (ln n) n , 1, k = 1, 2, 3,... . ψ(x) digamma function Γ0(x)/ Γ(x) except in n=2 §25.16. See §5.2(i). For further expansions of functions similar to (25.2.1) (Dirichlet series) see §27.4. This includes, for Bn,Bn(x) Bernoulli number and polynomial (§24.2(i)). example, 1/ ζ(s). Ben(x) periodic Bernoulli function Bn(x − bxc). m | n m divides n. 25.2(iii) Representations by the primes on function symbols: derivatives with Euler–Maclaurin Formula respect to argument. N X 1 N 1−s Z ∞ x − bxc The main function treated in this chapter is the Rie- ζ(s) = + − s dx, 25.2.8 ks s − 1 xs+1 mann zeta function ζ(s). This notation was introduced k=1 N in Riemann(1859). 0, N = 1, 2, 3,... .
The main related functions are the Hurwitz zeta N X 1 N 1−s 1 function ζ(s, a), the dilogarithm Li2(z), the polylog- ζ(s) = + − N −s ks s − 1 2 arithm Lis(z) (also known as Jonqui`ere’s function k=1 φ(z, s)), Lerch’s transcendent Φ(z, s, a), and the Dirich- n X s + 2k − 2 B2k let L-functions L(s, χ). + N 1−s−2k 25.2.9 2k − 1 2k k=1 Z ∞ s + 2n Be2n+1(x) − s+2n+1 dx, 2n + 1 N x Riemann Zeta Function −2n; n, N = 1, 2, 3,... . n 1 1 X s + 2k − 2 B2k 25.2 Definition and Expansions ζ(s) = + + s − 1 2 2k − 1 2k k=1 25.2(i) Definition 25.2.10 Z ∞ s + 2n Be2n+1(x) − s+2n+1 dx, When 1, 2n + 1 1 x ∞ X 1 −2n, n = 1, 2, 3,... . 25.2.1 ζ(s) = . ns n=1 For B2k see §24.2(i), and for Ben(x) see §24.2(iii). Elsewhere ζ(s) is defined by analytic continuation. It is a meromorphic function whose only singularity in C is 25.2(iv) Infinite Products a simple pole at s = 1, with residue 1. Y 25.2.11 ζ(s) = (1 − p−s)−1, 1, 25.2(ii) Other Infinite Series p product over all primes p. ∞ 1 X 1 s −s−(γs/2) 25.2.2 (2π) e Y s s/ρ ζ(s) = −s s , 1. 25.2.12 1 − 2 (2n + 1) ζ(s) = 1 1 − e , n=0 2(s − 1) Γ s + 1 ρ 2 ρ ∞ n−1 1 X (−1) product over zeros ρ of ζ with <ρ > 0 (see §25.10(i)); γ 25.2.3 ζ(s) = , 0. 1 − 21−s ns n=1 is Euler’s constant (§5.2(ii)). 25.3 Graphics 603
25.3 Graphics
1 Figure 25.3.4: Z(t), 0 ≤ t ≤ 50. Z(t) and ζ 2 + it have the same zeros. See §25.10(i).
Figure 25.3.1: Riemann zeta function ζ(x) and its derivative ζ0(x), −20 ≤ x ≤ 10.
Figure 25.3.5: Z(t), 1000 ≤ t ≤ 1050.
Figure 25.3.2: Riemann zeta function ζ(x) and its derivative ζ0(x), −12 ≤ x ≤ −2.
Figure 25.3.6: Z(t), 10000 ≤ t ≤ 10050.
25.4 Reflection Formulas
For s 6= 0, 1, −s 1 25.4.1 ζ(1 − s) = 2(2π) cos 2 πs Γ(s) ζ(s), s−1 1 25.4.2 ζ(s) = 2(2π) sin 2 πs Γ(1 − s) ζ(1 − s). Figure 25.3.3: Modulus of the Riemann zeta function Equivalently, | ζ(x + iy)|, −4 ≤ x ≤ 4, −10 ≤ y ≤ 40. 25.4.3 ξ(s) = ξ(1 − s), 604 Zeta and Related Functions where ξ(s) is Riemann’s ξ-function, defined by: 1 Z ∞ xs−1 1 1 −s/2 25.5.1 25.4.4 ξ(s) = s(s − 1) Γ s π ζ(s). ζ(s) = x dx, 1. 2 2 Γ(s) 0 e − 1 For s 6= 0, 1 and k = 1, 2, 3,... , 1 Z ∞ exxs 25.5.2 ζ(s) = dx, 1. 25.4.5 x 2 Γ(s + 1) 0 (e − 1) (−1)k ζ(k)(1 − s) Z ∞ s−1 k m 1 x 2 X X k m k−m 1 25.5.3 ζ(s) = dx, 0. = <(c ) cos πs (1 − 21−s) Γ(s) ex + 1 (2π)s m r 2 0 m=0 r=0 1 Z ∞ exxs k−m 1 (r) (m−r) ζ(s) = dx, + =(c ) sin πs Γ (s) ζ (s), 25.5.4 1−s x 2 2 (1 − 2 ) Γ(s + 1) 0 (e + 1) where 0. 1 ∞ 1 25.4.6 c = − ln(2π) − 2 πi. Z x − bxc − 25.5.5 2 ζ(s) = −s s+1 dx, −1 < −1.
n Z ∞ n ! s−1 1 1 X B2m Γ(s + 2m − 1) 1 1 1 1 X B2m x ζ(s) = + + + − + − x2m−1 dx, 25.5.7 2 s − 1 (2m)! Γ(s) Γ(s) ex − 1 x 2 (2m)! ex m=1 0 m=1 −(2n + 1), n = 1, 2, 3,... .
For θ3 see §20.2(i). For similar representations involving 1 Z ∞ xs−1 25.5.8 other theta functions see Erd´elyi et al. (1954a, p. 339). ζ(s) = −s dx, 1. 2(1 − 2 ) Γ(s) 0 sinh x In (25.5.15)–(25.5.19), 0 < 1. Γ(s + 1) 0 (sinh x) (25.5.16) is also valid for 0 <
25.5(iii) Contour Integrals 25.6.5 ∞ ∞ 1 X X 1 25.5.20 ζ(k + 1) = ... , (0+) s−1 k! n1 ··· nk(n1 + ··· + nk) Γ(1 − s) Z z n1=1 nk=1 ζ(s) = dz, s 6= 1, 2,... , −z k = 1, 2, 3,... . 2πi −∞ e − 1 where the integration contour is a loop around the neg- 25.6.6 ative real axis; it starts at −∞, encircles the origin once (−1)k+1(2π)2k+1 Z 1 in the positive direction without enclosing any of the ζ(2k + 1) = B (t) cot(πt) dt, 2(2k + 1)! 2k+1 points z = ±2πi, ±4πi, . . . , and returns to −∞. Equiv- 0 alently, k = 1, 2, 3,... . 25.5.21 Z 1 Z 1 1 Γ(1 − s) Z (0+) zs−1 25.6.7 ζ(2) = dx dy. 1 − xy ζ(s) = 1−s −z dz, s 6= 1, 2,... . 0 0 2πi(1 − 2 ) −∞ e + 1 ∞ X 1 The contour here is any loop that encircles the origin 25.6.8 ζ(2) = 3 . 2 2k in the positive direction not enclosing any of the points k=1 k k ±πi, ±3πi,.... ∞ 5 X (−1)k−1 25.6.9 ζ(3) = . 25.6 Integer Arguments 2 3 2k k=1 k k ∞ 25.6(i) Function Values 36 X 1 25.6.10 ζ(4) = . 17 4 2k 25.6.1 k=1 k k 1 π2 π4 π6 ζ(0) = − , ζ(2) = , ζ(4) = , ζ(6) = . 2 6 90 945 25.6(ii) Derivative Values (2π)2n 25.6.2 ζ(2n) = |B2n| , n = 1, 2, 3,... . 2(2n)! 25.6.11 0 1 ζ (0) = − 2 ln(2π). Bn+1 25.6.3 ζ(−n) = − , n = 1, 2, 3,... . 25.6.12 00 1 2 1 2 1 2 n + 1 ζ (0) = − 2 (ln(2π)) + 2 γ − 24 π + γ1, 25.6.4 ζ(−2n) = 0, n = 1, 2, 3,... . where γ1 is given by (25.2.5).
With c defined by (25.4.6) and n = 1, 2, 3,..., k m 2(−1)n X X k m 25.6.13 (−1)k ζ(k)(−2n) = =(ck−m)Γ(r)(2n + 1) ζ(m−r)(2n + 1), (2π)2n+1 m r m=0 r=0 k m 2(−1)n X X k m 25.6.14 (−1)k ζ(k)(1 − 2n) = <(ck−m)Γ(r)(2n) ζ(m−r)(2n), (2π)2n m r m=0 r=0 (−1)n+1(2π)2n 25.6.15 ζ0(2n) = (2n ζ0(1 − 2n) − (ψ(2n) − ln(2π)) B ) . 2(2n)! 2n
25.6(iii) Recursion Formulas 25.6.18 n 1 1 2 X n + 4 ζ(4n) + 2 (ζ(2n)) = ζ(2k) ζ(4n − 2k), k=1 n−1 25.6.16 1 X n ≥ 1. n + 2 ζ(2n) = ζ(2k) ζ(2n − 2k), n ≥ 2. k=1 3 m + n + 2 ζ(2m + 2n + 2) m n ! 25.6.17 25.6.19 X X n = + ζ(2k) ζ(2m + 2n + 2 − 2k), 3 X k=1 k=1 n + 4 ζ(4n + 2) = ζ(2k) ζ(4n + 2 − 2k), n ≥ 1. k=1 m ≥ 0, n ≥ 0, m + n ≥ 1. 606 Zeta and Related Functions
25.6.20 25.9 Asymptotic Approximations n−1 1 2n X 2n−2k 2 (2 − 1) ζ(2n) = (2 − 1) ζ(2n − 2k) ζ(2k), If x ≥ 1, y ≥ 1, 2πxy = t, and 0 ≤ σ ≤ 1, then as k=1 t → ∞ with σ fixed, n ≥ 2. X 1 X 1 For related results see Basu and Apostol(2000). ζ(σ + it) = + χ(s) ns n1−s 25.9.1 1≤n≤x 1≤n≤y 1 −σ σ−1 2 −σ 25.7 Integrals + O x + O y t , where s = σ + it and For definite integrals of the Riemann zeta function see s− 1 1 1 1 25.9.2 2 Prudnikov et al. (1986b, §2.4), Prudnikov et al. (1992a, χ(s) = π Γ 2 − 2 s / Γ 2 s . §3.2), and Prudnikov et al. (1992b, §3.2). 1 p If σ = 2 , x = y = t/(2π), and m = bxc, then (25.9.1) becomes
m 25.8 Sums X 1 ζ 1 + it = 2 1 +it n 2 ∞ 25.9.3 n=1 X m 25.8.1 (ζ(k) − 1) = 1. 1 X 1 −1/4 + χ 2 + it 1 + O t . k=2 2 −it n=1 n ∞ X Γ(s + k) (ζ(s + k) − 1) For other asymptotic approximations see Berry and 25.8.2 (k + 1)! k=0 Keating(1992), Paris and Cang(1997); see also Paris = Γ(s − 1), s 6= 1, 0, −1, −2,... . and Kaminski(2001, pp. 380–389).
∞ X Γ(s + k) ζ(s + k) 25.8.3 = (1 − 2−s) ζ(s), s 6= 1. k! Γ(s)2s+k 25.10 Zeros k=0 25.8.4 25.10(i) Distribution ∞ n−1 X (−1)k Y (ζ(nk) − 1) = ln Γ 2 − e(2j+1)πi/n , The product representation (25.2.11) implies ζ(s) 6= 0 k k=1 j=0 for 1. Also, ζ(s) 6= 0 for functional equation (25.4.1) X 25.8.5 ζ(k)zk = −γz − z ψ(1 − z), |z| < 1. implies ζ(−2n) = 0 for n = 1, 2, 3,... . These are called k=2 the trivial zeros. Except for the trivial zeros, ζ(s) 6= 0 ∞ for
25.10(ii) Riemann–Siegel Formula Related Functions Riemann developed a method for counting the total number N(T ) of zeros of ζ(s) in that portion of the critical strip with 0 < t < T . By comparing N(T ) 25.11 Hurwitz Zeta Function with the number of sign changes of Z(t) we can de- cide whether ζ(s) has any zeros off the line in this re- 25.11(i) Definition gion. Sign changes of Z(t) are determined by multiply- The function ζ(s, a) was introduced in Hurwitz(1882) ing (25.9.3) by exp(iϑ(t)) to obtain the Riemann–Siegel and defined by the series expansion formula: 25.11.1 m X cos(ϑ(t) − t ln n) ∞ 25.10.3 Z(t) = 2 + R(t), X 1 1/2 ζ(s, a) = , 25.11(ii) Graphics Figure 25.11.1: Hurwitz zeta function ζ(x, a), a = 0.3, Figure 25.11.2: Hurwitz zeta function ζ(x, a), −19.5 ≤ 0.5, 0.8, 1, −20 ≤ x ≤ 10. The curves are almost indis- x ≤ 10, 0.02 ≤ a ≤ 1. tinguishable for −14 < x < −1, approximately. 608 Zeta and Related Functions 25.11(iii) Representations by the Euler–Maclaurin Formula N X 1 (N + a)1−s Z ∞ x − bxc 25.11.5 ζ(s, a) = + − s dx, s 6= 1, 25.11.7 n Z ∞ 1 1 1 1 + a X s + 2k − 2 B2k 1 s + 2n Be2n+1(x) ζ(s, a) = + + + − dx, as (1 + a)s 2 s − 1 2k − 1 2k (1 + a)s+2k−1 2n + 1 (x + a)s+2n+1 k=1 1 s 6= 1, a > 0, n = 1, 2, 3,... , For Ben(x) see §24.2(iii). 25.11(iv) Series Representations 1 25.11.13 ζ(0, a) = 2 − a. ∞ X (−1)n ζ s, 1 a = ζ s, 1 a + 1 + 2s , Bn+1(a) 25.11.8 2 2 2 (n + a)s 25.11.14 ζ(−n, a) = − , n = 0, 1, 2,... . n=0 n + 1 1−s Z ∞ Z ∞ ln a 1 a a Be2(x) ln(x + a) Be2(x) ζ0(s, a) = − + − + s(s + 1) dx − (2s + 1) dx, 25.11.19 s 2 s+2 s+2 a 2 s − 1 (s − 1) 0 (x + a) 0 (x + a) k k−1 r Z ∞ k (ln a) 1 a X (ln a) Be2(x)(ln(x + a)) (−1)k ζ(k)(s, a) = + + k!a1−s − s(s + 1) dx as 2 s − 1 r!(s − 1)k−r+1 (x + a)s+2 r=0 0 25.11.20 Z ∞ k−1 Z ∞ k−2 Be2(x)(ln(x + a)) Be2(x)(ln(x + a)) + k(2s + 1) s+2 dx − k(k − 1) s+2 dx, 0 (x + a) 0 (x + a) 25.11.21 n+1 k−1 h (ψ(2n) − ln(2πk)) B2n(h/k) (ψ(2n) − ln(2π)) B2n (−1) π X 2πrh r ζ0 1 − 2n, = − + sin ψ(2n−1) k 2n 2nk2n (2πk)2n k k r=1 k−1 (−1)n+12 · (2n − 1)! X 2πrh r ζ0(1 − 2n) + cos ζ0 2n, + , (2πk)2n k k k2n r=1 where h, k are integers with 1 ≤ h ≤ k and n = 1, 2, 3,... . 2n−1 0 B2n ln 2 (2 − 1) ζ (1 − 2n) 25.11.22 ζ0 1 − 2n, 1 = − − , n = 1, 2, 3,... . 2 n · 4n 22n−1 25.11.23 π(9n − 1) B B ln 3 (−1)n ψ(2n−1) 1 32n−1 − 1 ζ0(1 − 2n) ζ0 1 − 2n, 1 = − √ 2n − 2n − √ 3 − , n = 1, 2, 3,... . 3 8n 3(32n−1 − 1) 4n · 32n−1 2 3(6π)2n−1 2 · 32n−1 k−1 X r 25.11.24 ζ0 s, = (ks − 1) ζ0(s) + ks ζ(s) ln k, s 6= 1, k = 1, 2, 3,... . k r=1 25.11(vii) Integral Representations 1 Z ∞ xs−1e−ax 25.11.25 ζ(s, a) = −x dx, 25.11(viii) Further Integral Representations Z ∞ s−1 −ax 1 x e −s 1 1 3 1 25.11.31 dx = 4 ζ s, 4 + 4 a − ζ s, 4 + 4 a , 25.11(ix) Integrals uniformly with respect to bounded nonnegative values of α. See Prudnikov et al. (1990, §2.3), Prudnikov et al. As a → ∞ in the sector | ph a| ≤ π − δ(< π), with (1992a, §3.2), and Prudnikov et al. (1992b, §3.2). s(6= 1) and δ fixed, we have the asymptotic expansion 25.11.43 25.11(x) Further Series Representations 1−s ∞ a 1 X B2k Γ(s + 2k − 1) ζ(s, a)− − a−s ∼ a1−s−2k. 25.11.35 s − 1 2 (2k)! Γ(s) k=1 ∞ n X (−1) 1 Similarly, as a → ∞ in the sector | ph a| ≤ 2 π − δ(< (n + a)s 1 n=0 2 π), ∞ 1 Z xs−1e−ax 0 1 1 2 1 1 1 2 = dx ζ (−1, a) − + a − − a + a ln a −x 12 4 12 2 2 Γ(s) 0 1 + e −s 1 1 25.11.44 ∞ = 2 ζ s, a − ζ s, (1 + a) , X B2k+2 2 2 ∼ − a−2k, (2k + 2)(2k + 1)2k 0, 25.12.6 By (25.12.2) 1 2 Li2(x) + Li2(1 − x) = π − (ln x) ln(1 − x), 0 < x < 1. 6 ∞ Z θ When z = eiθ, 0 ≤ θ ≤ 2π,(25.12.1) becomes 25.12.9 X sin(nθ) 1 2 = − ln 2 sin 2 x dx. ∞ ∞ n 0 X cos(nθ) X sin(nθ) n=1 25.12.7 Li eiθ = + i . 2 n2 n2 n=1 n=1 The right-hand side is called Clausen’s integral. The cosine series in (25.12.7) has the elementary sum For graphics see Figures 25.12.1 and 25.12.2, and for ∞ 2 2 X cos(nθ) π πθ θ further properties see Maximon(2003), Kirillov(1995), 25.12.8 = − + . n2 6 2 4 n=1 Lewin(1981), Nielsen(1909), and Zagier(1989). Figure 25.12.1: Dilogarithm function Li2(x), −20 ≤ x < Figure 25.12.2: Absolute value of the dilogarithm func- 1. tion |Li2(x + iy)|, −20 ≤ x ≤ 20, −20 ≤ y ≤ 20. Prin- cipal value. There is a cut along the real axis from 1 to ∞. 25.12(ii) Polylogarithms Further properties include 25.12.12 For real or complex s and z the polylogarithm Lis(z) is s−1 ∞ 1 X (ln z)n defined by Lis(z) = Γ(1 − s) ln + ζ(s − n) , ∞ z n! X zn n=0 25.12.10 Li (z) = . s ns s 6= 1, 2, 3,... , | ln z| < 2π, n=1 and For each fixed complex s the series defines an ana- 25.12.13 lytic function of z for |z| < 1. The series also converges (2π)seπis/2 when |z| = 1, provided that z Z ∞ xs−1 25.12(iii) Fermi–Dirac and Bose–Einstein 25.12.11 Lis(z) = x dx, Integrals Γ(s) 0 e − z valid when 25.14(ii) Properties 1 Z ∞ ts 25.12.14 F (x) = dt, s > −1, s t−x With the conditions of (25.14.1) and m = 1, 2, 3,... , Γ(s + 1) 0 e + 1 Z ∞ s m−1 n 1 t m X z Gs(x) = dt, 25.14.4 t−x Φ(z, s, a) = z Φ(z, s, a + m) + s . 25.12.15 Γ(s + 1) 0 e − 1 (a + n) n=0 s > −1, x < 0; or s > 0, x ≤ 0, 1 Z ∞ xs−1e−ax respectively. Sometimes the factor 1/ Γ(s + 1) is omit- Φ(z, s, a) = −x dx, 25.14.5 Γ(s) 0 1 − ze ted. See Cloutman(1989) and Gautschi(1993). 25.15.9 L(1, χ) 6= 0 if χ 6= χ1, The Riemann hypothesis is equivalent to the state- where χ1 is the principal character (mod k). This re- ment sult plays an important role in the proof of Dirichlet’s 1 + 25.16.4 ψ(x) = x + O x 2 , x → ∞, theorem on primes in arithmetic progressions (§27.11). Related results are: for every > 0. k 1 X − rχ(r), χ 6= χ1, 25.15.10 L(0, χ) = k r=1 0, χ = χ1. 25.16(ii) Euler Sums Euler sums have the form Applications ∞ X h(n) 25.16.5 H(s) = , ns 25.16 Mathematical Applications n=1 where h(n) is given by (25.11.33). 25.16(i) Distribution of Primes H(s) is analytic for k ∞ Z ∞ 1 X X 1 Be2k+1(x) 25.16.6 H(s) = − ζ0(s) + γ ζ(s) + ζ(s + 1) + ζ(1 − 2r) ζ(s + 2r) + dx, 2 ns x2k+2 r=1 n=1 n k ∞ Z ∞ 1 ζ(s) X s + 2r − 2 s + 2k X 1 Be2k+1(x) 25.16.7 H(s) = ζ(s + 1) + − ζ(1 − 2r) ζ(s + 2r) − dx. 2 s − 1 2r − 1 2k + 1 n xs+2k+1 r=1 n=1 n For integer s (≥ 2), H(s) can be evaluated in terms Also, of the zeta function: 25.16.10 25.16.8 H(2) = 2 ζ(3),H(3) = 5 ζ(4), 1 B 4 H(−2a) = ζ(1 − 2a) = − 2a , a = 1, 2, 3,... . 2 4a a−2 a + 2 1 X H(a) = ζ(a + 1)− ζ(r + 1) ζ(a − r), H(s) has a simple pole with residue ζ(1 − 2r) (= 25.16.9 2 2 r=1 − B2r /(2r)) at each odd negative integer s = 1 − 2r, a = 2, 3, 4,... . r = 1, 2, 3,... . 614 Zeta and Related Functions 1 H(s) is the special case H(s, 1) of the function ζ 2 + it . Details are provided in Haselgrove and Miller ∞ n (1960). See also Allasia and Besenghi(1989), Butzer X 1 X 1 25.16.11 H(s, z) = , <(s + z) > 1, and Hauss(1992), Kerimov(1980), and Yeremin et al. ns mz n=1 m=1 (1985). Calculations relating to derivatives of ζ(s) which satisfies the reciprocity law and/or ζ(s, a) can be found in Apostol(1985a), Choud- 25.16.12 H(s, z) + H(z, s) = ζ(s) ζ(z) + ζ(s + z), hury(1995), Miller and Adamchik(1998), and Yeremin et al. (1988). when both H(s, z) and H(z, s) are finite. For the Hurwitz zeta function ζ(s, a) see Spanier and For further properties of H(s, z) see Apostol and Vu Oldham(1987, p. 653). (1984). Related results are: For dilogarithms and polylogarithms see Jacobs and ∞ 2 X h(n) 17 Lambert(1972), Os´acar et al. (1995), and Spanier and 25.16.13 = ζ(4), n 4 Oldham(1987, pp. 231–232). n=1 ∞ r For Fermi–Dirac and Bose–Einstein integrals see X X 1 5 25.16.14 = ζ(3), Cloutman(1989), Gautschi(1993), Mohankumar and rk(r + k) 4 r=1 k=1 Natarajan(1997), Natarajan and Mohankumar(1993), ∞ r Paszkowski(1988, 1991), Pichon(1989), and Sagar X X 1 3 25.16.15 = ζ(3). (1991a,b). r2(r + k) 4 r=1 k=1 For further generalizations, see Flajolet and Salvy 25.18(ii) Zeros (1998). Most numerical calculations of the Riemann zeta func- tion are concerned with locating zeros of ζ 1 + it in 25.17 Physical Applications 2 an effort to prove or disprove the Riemann hypothesis, Analogies exist between the distribution of the zeros of which states that all nontrivial zeros of ζ(s) lie on the 1 ζ(s) on the critical line and of semiclassical quantum critical line 25.20 Approximations §25.2 Apostol(1976, Chapter 12). For (25.2.2)– (25.2.7) see also Hardy(1912). For (25.2.8)– • Cody et al. (1971) gives rational approximations (25.2.10) see also Knopp(1948, p. 533). (25.2.9) for ζ(s) in the form of quotients of polynomials follows from (25.2.8) by repeated integration by or quotients of Chebyshev series. The ranges cov- parts. For (25.2.11), (25.2.12) see also Titchmarsh ered are 0.5 ≤ s ≤ 5, 5 ≤ s ≤ 11, 11 ≤ s ≤ 25, (1986b, p. 30). 25 ≤ s ≤ 55. Precision is varied, with a maximum of 20S. §25.3 These graphics were constructed at NIST. • Piessens and Branders(1972) gives the coefficients §25.4 Apostol(1976, Chapter 12). of the Chebyshev-series expansions of s ζ(s + 1) §25.5 Apostol(1976, Chapter 12), Erd´elyi et al. and ζ(s + k), k = 2, 3, 4, 5, 8, for 0 ≤ s ≤ 1 (23D). (1953a, Chapter I). For (25.5.2) and (25.5.4) integrate (25.5.1) and (25.5.3) by parts. For (25.5.5) see Titchmarsh(1986b, p. 15). (25.5.6) • Luke(1969b, p. 306) gives coefficients in comes from (25.5.1) by using the identity e−x = Chebyshev-series expansions that cover ζ(s) for (1 − e−x)/(ex − 1) in the integral Γ(s) = 0 ≤ s ≤ 1 (15D), ζ(s + 1) for 0 ≤ s ≤ 1 (20D), ∞ R e−xxs−1 dx together with (5.5.1). (25.5.7) and ln ξ 1 + ix (§25.4) for −1 ≤ x ≤ 1 (20D). 0 2 follows from (25.5.6) because Γ(s + 2m − 1) = For errata see Piessens and Branders(1972). R ∞ −x s+2m−2 0 e x dx. For (25.5.10) and (25.5.11) • Morris(1979) gives rational approximations for see Lindel¨of(1905, p. 103). For (25.5.12) see Sri- vastava and Choi(2001, p. 12). For (25.5.13) Li2(x)(§25.12(i)) for 0.5 ≤ x ≤ 1. Precision is varied with a maximum of 24S. see Titchmarsh(1986b, p. 22). For (25.5.14)– (25.5.19) see de Bruijn(1937). For (25.5.21) see • Antia(1993) gives minimax rational approxima- Erd´elyi et al. (1953a, p. 32). tions for Γ(s + 1)F (x), where F (x) is the Fermi– s s §25.6 For (25.6.1)–(25.6.4) see Apostol(1976, pp. 266– Dirac integral (25.12.14), for the intervals −∞ < 268). For (25.6.5) see Mordell(1958). For (25.6.6) x ≤ 2 and 2 ≤ x < ∞, with s = − 1 , 1 , 3 , 5 . For 2 2 2 2 see N¨orlund(1924, p. 66). For (25.6.7) see Apostol each s there are three sets of approximations, with (1983). For (25.6.8)–(25.6.10) see van der Poorten relative maximum errors 10−4, 10−8, 10−12. (1980, pp. 271, 274). For (25.6.11)–(25.6.14) see Apostol(1985a). For (25.6.15) see Miller and 25.21 Software Adamchik(1998). For (25.6.16)–(25.6.20) see Basu and Apostol(2000). See http://dlmf.nist.gov/25.21. §25.8 Titchmarsh(1986b, Chapter IV), Adamchik and Srivastava(1998), Erd´elyi et al. (1953a, pp. 45 and 51). For (25.8.2) see Landau(1953, p. 274). For (25.8.3) see Srivastava(1988). For (25.8.7), References (25.8.8) divide by x in (25.8.5), (25.8.6) and inte- grate. For (25.8.9) see Srivastava and Choi(2001, General References p. 212). For (25.8.10) see Ewell(1990). §25.9 Titchmarsh(1986b, Chapter XV), Berry(1995). The main references used in writing this chapter are Apostol(1976), Erd´elyi et al. (1953a), and Titch- §25.10 Apostol(1976, Chapter 12), Titchmarsh marsh(1986b). For additional bibliographic reading see (1986b, pp. 89 and 263). Edwards(1974), Ivi´c(1985), Karatsuba and Voronin (1992). §25.11 Apostol(1976, Chapter 12). Analytic prop- erties of ζ(s, a) with respect to a follow from (25.11.30). For (25.11.5)–(25.11.6) see Apostol Sources (1985a). For (25.11.7) take N = 1 in (25.11.5) and integrate by parts. For (25.11.8)–(25.11.9) The following list gives the references or other indica- see Srivastava and Choi(2001, p. 89). For tions of proofs that were used in constructing the various (25.11.10) use Taylor’s theorem (§§1.4(vi), 1.10(i)) sections of this chapter. These sources supplement the and (25.11.17). For (25.11.11) apply (25.2.2) and references that are quoted in the text. (25.11.1). For (25.11.12) see Erd´elyi et al. (1953a, 616 Zeta and Related Functions p. 45). For (25.11.13) and (25.11.14) see Apostol (25.11.36) see Apostol(1976). For (25.11.37)– (1976, pp. 268, 264). For (25.11.15) use (25.11.1) (25.11.40) see Adamchik and Srivastava(1998). and analytic continuation. For (25.11.16) see For (25.11.41) and (25.11.42) see Apostol(1952). Apostol(1976, p. 263). For (25.11.17) differen- For (25.11.43) see Paris(2005b). For (25.11.44) tiate (25.11.1). For (25.11.18) see Erd´elyi et al. and (25.11.45) see Elizalde(1986). The graphics (1953a, p. 26). For (25.11.19)–(25.11.23) see were constructed at NIST. Apostol(1985a, p. 231) and Miller and Adam- chik(1998). For (25.11.24) use (25.11.15) with §25.12 Erd´elyi et al. (1953a, pp. 27, 29), Maximon a = 1/k, multiply by ks and differentiate. (2003). For (25.12.13) see Erd´elyi et al. (1953a, For (25.11.25) see Srivastava and Choi(2001, p. 31) with change of notation. The graphics were p. 89) For (25.11.26) see Berndt(1972). For constructed at NIST. (25.11.27) and (25.11.28) argue as indicated above §25.13 Apostol(1976, Chapter 13). for (25.5.6) and (25.5.7). For (25.11.29) see Lin- del¨of(1905, p. 106). For (25.11.30) assume 1, a 6= 0, −1, −2,... . n (n + a)s n=1 n=0 −1/4 where R(t) = O t as t → ∞. ζ(s, a) has a meromorphic continuation in the s- The error term R(t) can be expressed as an asymp- plane, its only singularity in C being a simple pole at totic series that begins s = 1 with residue 1. As a function of a, with s (6= 1) 25.10.4 √ fixed, ζ(s, a) is analytic in the half-plane 0. The 1/4 1 2π cos t − (2m + 1) 2πt − π Riemann zeta function is a special case: R(t) = (−1)m−1 √ 8 t cos 2πt 25.11.2 ζ(s, 1) = ζ(s). −3/4 + O t . For most purposes it suffices to restrict 0 < 0, a > 0, N = 0, 1, 2, 3,... . (n + a)s s − 1 (x + a)s+1 n=0 N 1 1 a Z ∞ B (x) 25.11.6 e2 ζ(s, a) = s + − s(s + 1) s+2 dx, s 6= 1, −1, a > 0. a 2 s − 1 0 (x + a) −2n. 0, s 6= 1, 0 < a ≤ 1. 25.11.15 k−1 ∞ −s X n 2 Γ(s) X 1 1 ζ(s, ka) = k ζ s, a + , s 6= 1, k = 1, 2, 3,... . ζ(1 − s, a) = cos πs − 2nπa , k 25.11.9 (2π)s ns 2 n=0 n=1 1, 0 < a ≤ 1. 25.11.16 k ∞ h 2 Γ(s) X πs 2πrh r X Γ(n + s) ζ 1 − s, = cos − ζ s, , ζ(s, a) = ζ(n + s)(1 − a)n, k (2πk)s 2 k k 25.11.10 n! Γ(s) r=1 n=0 s 6= 0, 1; h, k integers, 1 ≤ h ≤ k. s 6= 1, |a − 1| < 1. 1 When a = 2 ,(25.11.10) reduces to (25.8.3); compare 25.11(vi) Derivatives (25.11.11). a-Derivative ∂ 25.11(v) Special Values 25.11.17 ζ(s, a) = −s ζ(s + 1, a), s 6= 0, 1; 0. ∂a Throughout this subsection 0. s-Derivatives 1 s In (25.11.18)–(25.11.24) primes on ζ denote deriva- 25.11.11 ζ s, 2 = (2 − 1) ζ(s), s 6= 1. tives with respect to s. Similarly in §§25.11(viii) and 25.11.12 25.11(xii). (−1)n+1 ψ(n)(a) ζ(n + 1, a) = , n = 1, 2, 3,... . 25.11.18 ζ0(0, a) = ln Γ(a) − 1 ln(2π), a > 0. n! 2 −1, s 6= 1, a > 0. −1, s 6= 1, a > 0. 25.11 Hurwitz Zeta Function 609 1, 0. Γ(s) 0 1 − e Z ∞ x − bxc − 1 25.11.26 2 ζ(s, a) = −s s+1 dx, −1 < −1, s 6= 1, 0. 2 s − 1 Γ(s) 0 e − 1 x 2 e 1−s n 1 a X Γ(s + 2k − 1) B2k ζ(s, a) = a−s + + a−2k−s+1 2 s − 1 Γ(s) (2k)! k=1 25.11.28 Z ∞ n ! 1 1 1 1 X B2k + − + − x2k−1 xs−1e−ax dx, −(2n + 1), s 6= 1, 0. Γ(s) ex − 1 x 2 (2k)! 0 k=1 1 a1−s Z ∞ sin(s arctan(x/a)) 25.11.29 ζ(s, a) = a−s + + 2 dx, s 6= 1, 0. 2 2 s/2 2πx 2 s − 1 0 (a + x ) (e − 1) Γ(1 − s) Z (0+) eazzs−1 25.11.30 ζ(s, a) = z dz, s 6= 1, 0, 2πi −∞ 1 − e where the integration contour is a loop around the negative real axis as described for (25.5.20). 0, −1. Γ(s) 0 2 cosh x Z a n Bn+1 X n Bk+1(a) xn ψ(x) dx = (−1)n−1 ζ0(−n) + (−1)nh(n) − (−1)k h(k) an−k n + 1 k k + 1 0 k=0 25.11.32 n X n + (−1)k ζ0(−k, a)an−k, n = 1, 2,... , 0, k k=0 where n X 25.11.33 h(n) = k−1. k=1 Z a Bn+1 − Bn+1(a) 25.11.34 n ζ0(1 − n, x) dx = ζ0(−n, a) − ζ0(−n) + , n = 1, 2,... , 0. 0 n(n + 1) 610 Zeta and Related Functions 0; or 1, 25.11.45 ∞ n k X 2B2k+2 n=1 r=1 ∼ a−(2k−1). (2k + 2)(2k + 1)2k(2k − 1) where χ(n) is a Dirichlet character (mod k)(§27.8). k=1 See also Srivastava and Choi(2001). For the more general case ζ0(−m, a), m = 1, 2,... , see Elizalde(1986). 25.11(xi) Sums For an exponentially-improved form of (25.11.43) see Paris(2005b). 25.11.37 ∞ X (−1)k ζ(nk, a) = −n ln Γ(a) 25.12 Polylogarithms k k=1 25.12(i) Dilogarithms n−1 Y (2j+1)πi/n + ln Γ a − e , The notation Li2(z) was introduced in Lewin(1981) for j=0 a function discussed in Euler(1768) and called the dilog- n = 2, 3, 4,... , 0, |z| < |a|. Other notations and names for Li2(z) include S2(z) (K¨olbig et al. (1970)), Spence function Sp(z)(’t Hooft ∞ X k and Veltman(1979)), and L (z)(Maximon(2003)). 25.11.39 ζ k + 1, 3 = 8G, 2 2k 4 In the complex plane Li (z) has a branch point at k=2 2 where G is Catalan’s constant: z = 1. The principal branch has a cut along the in- terval [1, ∞) and agrees with (25.12.1) when |z| ≤ 1; ∞ X (−1)n see also §4.2(i). The remainder of the equations in this 25.11.40 G = = 0.91596 55941 772 .... (2n + 1)2 subsection apply to principal branches. n=0 For further sums see Prudnikov et al. (1990, pp. 396– 25.12.3 z 1 397) and Hansen(1975, pp. 358–360). Li (z) + Li = − (ln(1 − z))2, z ∈ \[1, ∞). 2 2 z − 1 2 C 25.11(xii) a-Asymptotic Behavior 25.12.4 1 1 1 Li (z) + Li = − π2 − (ln(−z))2, z ∈ \[0, ∞). As a → 0 with s (6= 1) fixed, 2 2 z 6 2 C 2 m−1 25.11.41 ζ(s, a + 1) = ζ(s) − s ζ(s + 1)a + O a . m X 2πik/m Li2(z ) = m Li2 ze , As β → ±∞ with s fixed, 1, 25.12.5 k=0 25.11.42 ζ(s, α + iβ) → 0, m = 1, 2, 3,... , |z| < 1. 25.12 Polylogarithms 611 1. For other values of Li e2πia + eπis Li e−2πia = ζ(1 − s, a), s s Γ(s) z, Li (z) is defined by analytic continuation. s valid when 0, =a > 0 or 1, =a = 0. When The notation φ(z, s) was used for Li (z) in Truesdell s s = 2 and e2πia = z,(25.12.13) becomes (25.12.4). (1945) for a series treated in Jonqui`ere(1889), hence See also Lewin(1981), K¨olbig(1986), Maximon the alternative name Jonqui`ere’s function. The special (2003), Prudnikov et al. (1990, §§1.2 and 2.5), Prud- case z = 1 is the Riemann zeta function: ζ(s) = Li (1). s nikov et al. (1992a, §3.3), and Prudnikov et al. (1992b, Integral Representation §3.3). 0 and |ph(1 − z)| < π, or 1 and The Fermi–Dirac and Bose–Einstein integrals are de- z = 1. (In the latter case (25.12.11) becomes (25.5.1)). fined by 612 Zeta and Related Functions 0, 0, z ∈ C\[1, ∞). In terms of polylogarithms 25.14.6 x x Z ∞ x 25.12.16 Fs(x) = − Lis+1(−e ),Gs(x) = Lis+1(e ). 1 −s z Φ(z, s, a) = a + s dx For a uniform asymptotic approximation for Fs(x) 2 0 (a + x) see Temme and Olde Daalhuis(1990). Z ∞ sin(x ln z − s arctan(x/a)) − 2 dx, 2 2 s/2 2πx 0 (a + x ) (e − 1) 0 if |z| < 1; 1 if |z| = 1, 0. 25.13 Periodic Zeta Function For these and further properties see Erd´elyi et al. The notation F (x, s) is used for the polylogarithm (1953a, pp. 27–31). 2πix Lis e with x real: 25.15 Dirichlet L-functions ∞ X e2πinx 25.13.1 F (x, s) = , 25.15(i) Definitions and Basic Properties ns n=1 The notation L(s, χ) was introduced by Dirichlet(1837) where 1 if x is an integer, 0 otherwise. for the meromorphic continuation of the function de- F (x, s) is periodic in x with period 1, and equals fined by the series ζ(s) when x is an integer. Also, ∞ X χ(n) Γ(1 − s) 25.15.1 L(s, χ) = , 1, F (x, s) = eπi(1−s)/2 ζ(1 − s, x) ns (2π)1−s n=1 25.13.2 where χ(n) is a Dirichlet character (mod k)(§27.8). + eπi(s−1)/2 ζ(1 − s, 1 − x) , For the principal character χ1 (mod k), L(s, χ1) is an- 0 < x < 1, 1, alytic everywhere except for a simple pole at s = 1 with residue φ(k)/k, where φ(k) is Euler’s totient function 25.13.3 (§27.2). If χ 6= χ , then L(s, χ) is an entire function of Γ(s) 1 ζ(1 − s, x) = e−πis/2 F (x, s) + eπis/2 F (−x, s) , s. s −1 (2π) Y χ(p) 25.15.2 L(s, χ) = 1 − , 1, 0 < x < 1, 0. ps p with the product taken over all primes p, beginning with 25.14 Lerch’s Transcendent p = 2. This implies that L(s, χ) 6= 0 if 1. Equations (25.15.3) and (25.15.4) hold for all s if 25.14(i) Definition χ 6= χ1, and for all s (6= 1) if χ = χ1: k−1 ∞ n X z −s X r Φ(z, s, a) = , 25.15.3 L(s, χ) = k χ(r) ζ s, , 25.14.1 (a + n)s k n=0 r=1 Y χ0(p) a 6= 0, −1, −2,..., |z| < 1; 1, |z| = 1. 25.15.4 L(s, χ) = L(s, χ ) 1 − , 0 ps For other values of z, Φ(z, s, a) is defined by analytic p|k continuation. This is the notation used in Erd´elyi where χ0 is a primitive character (mod d) for some pos- et al. (1953a, p. 27). Lerch(1887) used K(a, x, s) = itive divisor d of k (§27.8). 2πix Φ e , s, a . When χ is a primitive character (mod k) the L- The Hurwitz zeta function ζ(s, a)(§25.11) and the functions satisfy the functional equation: polylogarithm Lis(z)(§25.12(ii)) are special cases: 25.15.5 ks−1 Γ(s) 25.14.2 ζ(s, a) = Φ(1, s, a), 1, a 6= 0, −1, −2,... , L(1 − s, χ) = e−πis/2 + χ(−1)eπis/2 (2π)s 25.14.3 Lis(z) = z Φ(z, s, 1), 1, |z| ≤ 1. × G(χ) L(s, χ), Applications 613 where χ is the complex conjugate of χ, and with the digamma function used elsewhere in this chap- k ter), given by X 2πir/k 25.15.6 G(χ) = χ(r)e . ∞ X X r=1 25.16.1 ψ(x) = ln p, 25.15(ii) Zeros m=1 pm≤x Since L(s, χ) 6= 0 if 1, (25.15.5) shows that for a which is related to the Riemann zeta function by primitive character χ the only zeros of L(s, χ) for 1, and can be extended In studying the distribution of primes p ≤ x, Chebyshev meromorphically into the half-plane −2k for ev- (1851) introduced a function ψ(x) (not to be confused ery positive integer k by use of the relationsAbramowitz and Stegun §25.15 Apostol(1976, Chapter 12), Apostol(1985b). 1, collapse the integration path onto the real For (25.15.9) see Apostol(1976, pp. 142, 149). axis, apply (25.11.25) and (5.5.3) followed by an- alytic continuation. For (25.11.31) use (25.11.25). §25.16 Apostol(1976, Chapter 13). For (25.16.2) see For (25.11.32)–(25.11.34) see Adamchik(1998). Apostol(2000). For (25.16.4) see Ingham(1932, For (25.11.35) use (25.11.25) and (25.11.8). For p. 84). For (25.16.5)–(25.16.15) see Apostol and Vu(1984) and Basu and Apostol(2000).