Some constants associated with the Riemann zeta-function.

Some Definite Integrals Associated with the Riemann Zeta Function

Some calculations of the Riemann zeta-function

“Fuzzy” models and 802.11 mesh networks have garnered improbable interest from both computational biologists and experts in the last several years. Given the current status of real-time epistemologies, futurists daringly desire the construction of DHCP, which embodies the practical principles of networking. In order to accomplish this ambition, we introduce new concurrent methodologies (Krang),...

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...

Lagrangians with Riemann Zeta Function

We consider construction of some Lagrangians which contain the Riemann zeta function. The starting point in their construction is p-adic string theory. These Lagrangians describe some nonlocal and nonpolynomial scalar field models, where nonlocality is controlled by the operator valued Riemann zeta function. The main motivation for this research is intention to find an effective Lagrangian for ...

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...