Some Identities for the Riemann Zeta-function
نویسندگان
چکیده
Several identities for the Riemann zeta-function ζ(s) are proved. For example, if s = σ + it and σ > 0, then ∞ −∞ (1 − 2 1−s)ζ(s) s 2 dt = π σ (1 − 2 1−2σ)ζ(2σ). Let as usual ζ(s) = ∞ n=1 n −s (ℜe s > 1) denote the Riemann zeta-function. The motivation for this note is the quest to evaluate explicitly integrals of |ζ(1 2 + it)| 2k , k ∈ N, weighted by suitable functions. In particular, the problem is to evaluate in closed form ∞ 0 (3 − √ 8 cos(t log 2)) k |ζ(1 2 + it)| 2k dt (1 4 + t 2) k (k ∈ N). When k = 1, 2 this may be done, thanks to the identities which will be established below. The first identity in question is given by THEOREM 1. Let s = σ + it. Then for σ > 0 we have (1) ∞ −∞ (1 − 2 1−s)ζ(s) s 2 dt = π σ (1 − 2 1−2σ)ζ(2σ). Since lim s→1 (s − 1)ζ(s) = 1, then setting in (1) σ = 1 2 we obtain the following Corollary 1. (2) ∞ 0 (3 − √ 8 cos(t log 2))|ζ(1 2 + it)| 2 dt 1 4 + t 2 = π log 2. Another identity, which relates directly the square of ζ(s) to a Mellin-type integral , is contained in
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