Some Identities for the Riemann Zeta-function Ii

Several identities for the Riemann zeta-function ζ(s) are proved. For example, if φ1(x) := {x} = x− [x], φn(x) := ∫ ∞ 0 {u}φn−1 ( x u ) du u (n ≥ 2), then ζn(s) (−s) = ∫ ∞ 0 φn(x)x −1−s dx (s = σ + it, 0 < σ < 1) and 1 2π ∫ ∞ −∞ |ζ(σ + it)| (σ + t) dt = ∫ ∞ 0 φ n (x)x dx (0 < σ < 1). Let as usual ζ(s) = ∑ ∞ n=1 n −s (Re s > 1) denote the Riemann zeta-function. This note is the continuation of the author’s work [6], where several identities involving ζ(s) were obtained. The basic idea is to use properties of the Mellin transform (f : [0,∞) → R ) (1) F (s) = M[f(x); s] := ∫ ∞ 0 f(x)x dx (s = σ + it, σ > 0), in particular the analogue of the Parseval formula for Mellin transforms, namely (2) 1 2πi ∫ σ+i∞ σ−i∞ |F (s)| ds = ∫ ∞ 0 f(x)x dx. For the conditions under which (2) holds, see e.g., [5] and [10]. If {x} denotes the fractional part of x ({x} = x − [x], where [x] is the greatest integer not exceeding x), we have the classical formula (see e.g., eq. (2.1.5) of E.C. Titchmarsh [11]) (3) ζ(s) s = − ∫ ∞