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The Stieltjes Constants, Their Relation to the Ηj Coefficients, And

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The Stieltjes Constants, Their Relation to the Ηj Coefficients, And The Stieltjes constants, their relation to the ηj coefficients, and representation of the Hurwitz zeta function Mark W. Coffey Department of Physics Colorado School of Mines Golden, CO 80401 (Received 2009) February 18, 2009 Abstract The Stieltjes constants γk(a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about its only pole at s = 1. We present the relation of γk(1) to the ηj coefficients that appear in the Laurent expansion of the logarithmic derivative of the Riemann zeta function about its pole at s = 1. We obtain novel integral representations of the Stieltjes constants and new decompositions such as S2(n) = Sγ(n)+ SΛ(n) for the crucial oscillatory subsum of the Li criterion for the Riemann hypothesis. The sum Sγ(n) is O(n) and we present various integral representations for it. We present novel series representations of S2(n). We additionally present a rapidly convergent expression for γk = γk(1) and a variety of results pertinent to a parameterized arXiv:0706.0343v2 [math-ph] 25 Feb 2009 representation of the Riemann and Hurwitz zeta functions. Key words and phrases Stieltjes constants, Riemann zeta function, Hurwitz zeta function, Laurent expan- sion, ηj coefficients, von Mangoldt function, Li criterion, Riemann hypothesis 1 Introduction and statement of results In the Laurent expansion of the Riemann zeta function about s = 1, 1 ∞ ( 1)n ζ(s)= + − γ (s 1)n, (1) s 1 n! n − − nX=0 the Stieltjes constants γk [9, 10, 22, 25, 27, 32] can be written in the form N k+1 1 k ln N γk = lim ln m . (2) N→∞ m − k +1 ! mX=1 From the expansion around s = 1 of the logarithmic derivative of the zeta function, ζ′(s) 1 ∞ = η (s 1)p, s 1 < 3, (3) ζ(s) −s 1 − p − | − | − pX=0 we have ∞ η ln ζ(s)= ln(s 1) p−1 (s 1)p. (4) − − − p − pX=1 The constants ηj can be written as k N k+1 ( 1) 1 k ln N ηk = − lim Λ(m) ln m , (5) k! N→∞ m − k +1 ! mX=1 where Λ is the von Mangoldt function [15, 22, 24, 33, 30], defined by Λ(n) = ln p if n = pk for a prime number p and some integer k 1, and Λ(n) = 0 otherwise. The ≥ radius of convergence of the expansion (3) is 3, as the first singularity encountered is the trivial zero of ζ(s) at s = 2. The Dirichlet series corresponding to Eqs. (3) and − (4) valid for Re s> 1 are ζ′(s) ∞ Λ(n) ∞ Λ(n) = and ln ζ(s)= . (6) ζ(s) − ns ns ln n nX=1 nX=2 Indeed, Eqs. (6) hold on σ = Re s = 1 if t = Im s = 0 [33]. 6 2 We note that there are various explicit expressions for the von Mangoldt function. These include LCM(1,...,n) exp[Λ(n)] = , (7) LCM(1,...,n 1) − where LCM is the least common multiple function, and Linnik’s identity [28] (pp. 21-22) Λ(n) ln n/ ln 2 ( 1)k Λ = − τ ′ (n). (8) 1 ≡ ln n − k k kX=1 ′ In regard to Eq. (8), the strict τk and exact τk divisor functions are related by τ ′ (n)= n ,...,n 2; n n = n (9a) k |{ 1 k ≥ 1 ··· k }| k k−ℓ k = ( 1) τℓ(n). (9b) − ℓ! Xℓ=0 In particular, there is a finite sum on the right side of Eq. (8). We recall that the function ln ζ(s) is intimately connected with the prime counting function π(x), the number of primes less than x. We have ∞ π(x) ln ζ(s)= s dx. (10) 2 x(xs 1) Z − Hence the behaviour of the function π(x) is related to the important coefficients ηj. For further background on the classical zeta function we refer to standard texts [15, 22, 24, 33, 30]. ∞ −s The Hurwitz zeta function, defined by ζ(s, a)= n=0(n + a) for Re s> 1 and Re a > 0 extends to a meromorphic function in theP entire complex s-plane. The generalization of the Laurent expansion (1) is 1 ∞ ( 1)n ζ(s, a)= + − γ (a)(s 1)n. (11) s 1 n! n − − nX=0 3 As shown by Eq. (1), by convention one takes γk = γk(1). Having set the notation above we may now state our main results. Proposition 1. For integers k 0 we have ≥ ( 1)k ( 1)k ∞ [1 Λ(m)] − γ η = − − lnk m, (12) k! k − k k! m mX=1 where the sum on the right starts with m = 2 when k > 0. The case of k = 0 yields the identity Corollary 1 1 ∞ [1 Λ(m)] γ = − , (13) 2 m mX=1 where γ is the Euler constant. Put n n S2(n) ηm−1. (14) ≡ − m! mX=1 Let Lα(x) be the Laguerre polynomial of degreee n (e.g., [4, 17]) and P (t)= B (t n 1 1 − [t]) = t [t] 1/2 the first periodized Bernoulli polynomial (e.g., [22, 33]). Then we − − have the series representation given in Proposition 2. For integers n 1 we have (a) ≥ S2(n)= Sγ(n)+ SΛ(n), (15) where n k ∞ ( 1) n 1 1 Sγ (n) − γk−1 = Ln−1(ln t)dP1(t), (16) ≡ (k 1)! k! 1 t kX=1 − Z ∞ [1 Λ(m)] S (n) − L1 (ln m) Λ ≡ m n−1 mX=1 4 ∞ [1 Λ(m)] n + S (n)= n + − L1 (ln m), (17) ≡ 2Λ m n−1 mX=2 (b) Sγ(n)= O(n), (c) the average values M ∞ 1 1 1 2 Sγ(n)= LM−1(ln t)dP1(t), (18a) M M 1 t nX=1 Z 1 M 1 ∞ [1 Λ(m)] S (n)= − L2 (ln m), (18b) M Λ M m M−1 nX=1 mX=1 and (d) for N 1 a fixed integer ≥ N L1 (ln ν) 1 1 S (n)= n−1 L (ln N)+1+ L1 (ln N)+ O L1 (1), γ − ν − n 2N n−1 N 2−ǫ n−1 νX=1 where ǫ> 0 is arbitrary. We have found new integral representations of the Stieltjes constants, given in Proposition 3. Let Re a> 0. Then we have (a) k+1 ∞ k 1 k ln a 2 (y/a i) ln (a iy) γk(a)= ln a + Re − − dy. (19) 2a − k +1 a 0 (1 + y2/a2)(e2πy 1) Z − (b) Let n 1 be an integer. Then we have ≥ n−1 lnk(m + a) lnk(n + a) lnk+1(n + a) γ (a)= + k m + a 2(n + a) − k +1 mX=0 2 ∞ [y/(n + a) i] lnk(n + a iy) + Re − − dy. (20) n + a 0 [1 + y2/(n + a)2](e2πy 1) Z − (c) π 1 ∞ lnm+1(2a 1 it) γm(a)= Re 2 − − dt − 2 m +1 Z0 cosh (πt/2) π m+1 lnj 2 ( 1)j+1 ∞ lnm−j+1(2a 1 it) + m! − Re 2 − − dt. (21) 2 j! (m j + 1)! 0 cosh (πt/2) jX=1 − Z 5 Let ψ(z)=Γ′(z)/Γ(z) be the digamma function, where Γ is the Gamma function [1, 4, 17]. From Proposition 3 follows Corollaries 2 and 3. From the case m = 0 in Proposition 3 we have Corollary 2. (a) 1 2 ∞ (y/a i) γ0(a)= ψ(a)= ln a + Re − dy, (22) − 2a − a 0 (1 + y2/a2)(e2πy 1) Z − (b) for integers n 1, ≥ n−1 1 1 2 ∞ [y/(n + a) i] γ0(a)= ψ(a)= + ln(n+a)+ Re − dy, − m + a 2(n + a)− (n + a) 0 [1 + y2/(n + a)2](e2πy 1) mX=0 Z − (23) and (c) π ∞ ln(2a 1 it) γ0(a)= ψ(a)= Re −2 − dt + ln 2 − − 2 Z0 cosh (πt/2) π ∞ ln[(2a 1)2 + t2] = −2 dt + ln 2. (24) − 4 Z0 cosh (πt/2) Based upon the a = 1 case of Proposition 3 we obtain additional new integral representations of the sum Sγ defined in Eq. (16): Corollary 3. ∞ 1 1 1 Sγ(n)= (y i)Ln−1[ln(1 iy)]+(y + i)Ln−1[ln(1 + iy)] dy, − 0 (1 + y2)(e2πy 1) − − Z − h i (25) and π ∞ dt Sγ(n)= [Ln(2a 1 it)+ Ln(2a 1+ it) 2] 2 4 Z0 − − − − cosh (πt/2) π n lnj 2 n ∞ dt 1F1[j n; j+1; ln(2a 1 it)]+ 1F1[j n; j+1; ln(2a 1+it)] 2 , − 4 j! j ! 0 { − − − − − }cosh (πt/2) jX=1 Z (26) 6 where F is the confluent hypergeometric function, with n F (j n; j + 1; x) = 1 1 j 1 1 − j Ln−j(x). Let d(n) be the number of divisors of n. Then we have Proposition 4. (a) We have for integers m 0 ≥ ∞ lnm n 2 ( 1)m [d(n) ln n 2γ]=( 1)m+1 1+ γ 2γ( 1)mγ − n − − − m +1 m+1 − − m nX=1 m ℓ m + ( 1) γℓγm−ℓ. (27) − ℓ ! Xℓ=0 In particular we have at m =0 Corollary 4. ∞ ∞ 2 1 2 1 [1 Λ(m)] [d(n) ln n 2γ]= 3γ1 γ = 3γ1 − , (28) n − − − − − − 4 m ! nX=1 mX=1 giving Corollary 4’. ∞ d(m) 1 2 2 lim ln n +2γ ln n = γ 2γ1 > 0. m − n→∞ 2 − mX=1 (b) We have for integers m 0 ≥ ∞ lnm n 2( 1)m+1 ( 1)m [d(n) Λ(n) ln n 2γ]=(m +1)η + − γ 2γ( 1)mγ − n − − m+1 m +1 m+1 − − m nX=1 m ℓ m + ( 1) γℓγm−ℓ.
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