Identities for the Riemann zeta function

Some Identities for the Riemann Zeta-function

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 prob...

Identities for the Riemann Zeta Function

In this paper, we obtain several expansions for ζ(s) involving a sequence of polynomials in s, denoted by αk(s). These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities extend some series expansions for the zeta function that are known for integer values of s. The expansions also give a different approach to the analytic continuation of the...

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 t...

q-Riemann zeta function

We consider the modified q-analogue of Riemann zeta function which is defined by ζq(s)= ∑∞ n=1(qn(s−1)/[n]s), 0< q < 1, s ∈ C. In this paper, we give q-Bernoulli numbers which can be viewed as interpolation of the above q-analogue of Riemann zeta function at negative integers in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Also, we will treat some...

Mollifying the Riemann Zeta-function