ACTA ARITHMETICA LXXXII.2 (1997) Hypergeometric series and the Riemann zeta function by Wenchang Chu (Roma) For infinite series related to the Riemann zeta function, De Doelder [4] established numerous interesting identities through evaluating improper in- tegrals. Some of them are rederived in [2] by means of the Parseval identity for Fourier series. A couple of the most elegant examples may be reproduced as ∞ 2 X 1 + 1 + ... + 1 17π4 17 (0.1a) 2 n = = ζ(4), n 360 4 n=1 ∞ X 1 + 1 + ... + 1 π4 5 (0.1b) 2 n = = ζ(4), n3 72 4 n=1 where the latter is due to Euler (1775). Other formulas and the related references may be found in Berndt [1, Ch. 9]. Since the first glance at the paper of Borweins [2], the author has been stimulated, by the beauty of these formulas, to search for a different way of dealing with such kind of infinite series. The present paper will show that the hypergeometric method can fulfill that purpose, which may be illustrated by the following simple example. For complex parameters {x, c}, consider the binomial expansion ∞ X c (1 + x)c = xn. n n=0 The coefficient of cm+1 gives a general infinite series identity m+1 ∞ ln (1 + x) X em(n) = (−1)m+n xn+1, (m + 1)! n + 1 n=0 1991 Mathematics Subject Classification: Primary 11M06; Secondary 33C20. Key words and phrases: the Riemann zeta function, the harmonic numbers, hyperge- ometric series, the gamma function, symmetric functions. [103] 104 W. C. Chu where em(n) is the mth elementary symmetric function in {1, 1/2,..., 1/n}. Its specification, for x = m = 1, leads us to a strange summation formula ∞ 1 1 ln2 2 X 1 + + ... + = (−1)n 2 n+1 2 n + 2 n=0 due to De Doelder [4, Eq. 25]. This method works for most of the hypergeometric summation theorems. With the notation from Slater [7], the generalized hypergeometric series (unilateral and bilateral) read as ∞ u0, u1, . , up X (u0)n (u1)n ... (up)n (0.2a) F ; z = zn, 1+p q v , . , v n!(v ) ... (v ) 1 q n=0 1 n q n ∞ u1, u2, . , up X (u1)n (u2)n ... (up)n (0.2b) H ; z = zn, p q v , v , . , v (v ) (v ) ... (v ) 1 2 q n=−∞ 1 n 2 n q n where the shifted factorial is defined, for a complex number w, by w(w + 1) ... (w + n − 1), n = 1, 2,..., (w)n = 1, n = 0, (w − 1)(w − 2) ... (w + n), n = −1, −2,... On the one hand, the hypergeometric series may be expressed in terms of the partial sums of the Riemann zeta function through the symmetric functions generated by finite products (cf. [6]) n Y x2 (0.3a) (1 + x/k) = 1 + xH + (H2 − H00) + ..., n 2 n n k=1 n Y x2 (0.3b) (1 − x/k)−1 = 1 + xH + (H2 + H00) + ..., n 2 n n k=1 n Y y y2 (0.3c) 1 + = 1 + yO + (O2 − O00) + ..., 2k − 1 n 2 n n k=1 n −1 Y y y2 (0.3d) 1 − = 1 + yO + (O2 + O00) + ..., 2k − 1 n 2 n n k=1 whose first two terms n n X 1 X 1 (0.4a) H = ,O = , n k n 2k − 1 k=1 k=1 n n X 1 X 1 (0.4b) H00 = ,O00 = n k2 n (2k − 1)2 k=1 k=1 are closely related to the harmonic numbers. Riemann zeta function 105 On the other hand, most of the hypergeometric summation theorems (cf. [3]) evaluate infinite series as a fraction of Γ -functions Γ (a) Γ (b) ...Γ (c) Γ a, b, . , c = , A, B, . , C Γ (A) Γ (B) ...Γ (C) which may be expanded as multivariate formal power series. In fact, recall- ing, for the Γ -function (cf. [5]), the Weierstrass product ∞ Y Γ (z) = z−1 {(1 + 1/n)z/(1 + z/n)} n=1 and the logarithmic derivative ∞ Γ 0(z) X z − 1 (0.5) = −γ + Γ (z) (n + 1) (n + z) n=0 with the Euler constant n X 1 γ = lim − ln n n→∞ k k=1 we can derive the following expansions: ∞ X σk (0.6a) Γ (1 − z) = exp zk , k k=1 ∞ √ X τk (0.6b) Γ 1 − z = π exp zk , 2 k k=1 where the Riemann zeta sequences {σk, τk} are defined by (0.7a) σ1 = γ, σm = ζ(m), m = 2, 3,... m (0.7b) τ1 = γ + 2 ln 2, τm = (2 − 1)ζ(m), m = 2, 3,... For a specific hypergeometric summation theorem, both power series expan- sions mentioned above should coincide. The coefficients of a single monomial xiyjzk ... in both power series would result in a summation formula (trivial or non-trivial) similar to (0.1). By means of hypergeometric theorems, this approach will be extensively explored to establish several further infinite summation formulas. Some examples are ∞ X H00 7π4 (0.8a) n+1 = , (n + 1)2 360 n=0 ∞ 4 X H2nH2n+1 π (0.8b) = , (2n + 1)2 64 n=0 106 W. C. Chu ∞ 2 X OnOn+1 π (0.8c) = ln2 2. (2n + 1)2 8 n=0 Because the hypergeometric method is quite mechanical and systematic, the tedious demonstration for summation formulas will not be presented in detail except for necessity. Instead, we will use a self-explanatory nota- tion [xiyjzk] to indicate extracting the coefficients of the monomial xiyjzk from power series expansions. Throughout the paper, the Euler summation formulas ∞ ∞ X 1 π2 X (−1)n π (0.9a) = , = , n2 6 2n + 1 4 n=1 n=0 ∞ ∞ X 1 π4 X (−1)n π3 (0.9b) = , = , n4 90 (2n + 1)3 32 n=1 n=0 will frequently be applied without indication. 1. The Gauss summation theorem. Recall the Gauss theorem [7, p. 28] 1 − z, 1 − x − y − z F x, y ; 1 = Γ . 2 1 1 − z 1 − x − z, 1 − y − z We may reformulate it, by means of (0.6a), as a functional equation between two multivariate infinite series ∞ n 1 − z, 1 − x − y − z X Q (1 + x/i)(1 + y/i) Γ = 1 + xy i=1 1 − x − z, 1 − y − z 2 Qn+1 n=0 (n + 1) j=1 (1 − z/j) ∞ X σk = exp [zk + (x + y + z)k − (x + z)k − (y + z)k] k k=1 = exp σ2xy + σ3xy(x + y + 2z) 2 2 2 3 + σ4xy x + y + 3xz + 3yz + 3z + 2 xy + ... , where the right hand side of the first equality may be expanded, via (0.3), as a power series. Then term-by-term comparison of the coefficients of the two power series results in an infinite number of summation formulas. Example 1.1. Summation formulas [4, Eq. 2] related to ζ(3): ∞ X Hn (1.1a) [x2y] = σ = ζ(3), (n + 1)2 3 n=0 ∞ X Hn+1 (1.1b) [xyz] = 2σ = 2ζ(3). (n + 1)2 3 n=0 Riemann zeta function 107 Example 1.2. Summation formulas [2, Eqs. 2–3] related to ζ(4): ∞ 4 X HnHn+1 π (1.2a) [x2yz] = 3σ = , (n + 1)2 4 30 n=0 ∞ X H2 1 11π4 (1.2b) [x2y2] n = (σ2 + 3σ ) = , (n + 1)2 2 2 4 360 n=1 ∞ 2 4 X H 17σ4 17π (1.2c) n+1 = = . (n + 1)2 4 360 n=0 Example 1.3. Summation formulas [2, Eq. 4] and [4, Eq. 7] related to ζ(4): ∞ 4 X Hn σ4 π (1.3a) = = , (n + 1)3 4 360 n=0 ∞ 4 X Hn+1 5σ4 π (1.3b) = = . (n + 1)3 4 72 n=0 Example 1.4. More summation formulas related to ζ(4): ∞ X H00 3 π4 (1.4a) n = σ = , (n + 1)2 4 4 120 n=0 ∞ X H00 7 7π4 (1.4b) n+1 = σ = . (n + 1)2 4 4 360 n=0 S k e t c h o f p r o o f. The coefficients of xy in both formal power series expansions lead us to Euler’s formula ζ(2) = π2/6. In Example 1.2, the third formula follows from the previous two identities since 1 2 2H 1 H2 = H + = H2 + n + n+1 n n + 1 n n + 1 (n + 1)2 1 = H2 + 2H (H − H ) + n n n+1 n (n + 1)2 1 = 2H H − H2 + . n n+1 n (n + 1)2 Linear combinations of the identities of Example 1.2 yield the formulas in Example 1.3. Other two terms ∞ X H2 + H00 π4 (1.5a) [xyz2] n n = 6σ = , n2 4 15 n=1 ∞ X H2 − H00 π4 (1.5b) [x3y] n n = 2σ = (n + 1)2 4 45 n=0 can be used to derive, without difficulty, the formulas in the last example. 108 W. C. Chu 2. The Kummer summation theorem. The Kummer theorem [7, p. 51] F x, y ; −1 = Γ 1 + x/2, 1 + x − y 2 1 1 + x − y 1 + x, 1 + x/2 − y may be restated, through (0.6a), as 1 + x/2, 1 + x − y Γ 1 + x, 1 + x/2 − y ∞ n X Q (1 + x/i)(1 + y/i) = 1 − xy (−1)n i=1 2 Qn+1 n=0 (n + 1) j=1 [1 − (y − x)/j] ∞ X σk = exp (−1)k [xk + (2x − 2y)k − (2x)k − (x − 2y)k] k · 2k k=1 σ σ σ = exp − 2 xy + 3 xy(3x − 2y) + 4 xy(9xy − 7x2 − 4y2) + ..