Abstract. In this paper, we show that Riemann hypothesis (concerning zeros of the zeta function in the critical strip) is equivalent to the analytic continuation of Euler products obtained by restricting the Euler zeta product to suitable subsets Mk, k ≥ 1 of the set of prime numbers. Each of these Euler product defines so a partial zeta function ζk(s) equal to a Dirichlet series of the form ∑ ...

Recently, λ-Bernoulli and λ-Euler numbers are studied in [5, 10]. The purpose of this paper is to present a systematic study of some families of the q-extensions of the λBernoulli and the λ-Euler numbers by using the bosonic p-adic q-integral and the fermionic p-adic q-integral. The investigation of these λ-q-Bernoulli and λ-q-Euler numbers leads to interesting identities related to these objec...

The study on degenerate versions of some special numbers and polynomials, which began with Carlitz's pioneering work, has regained recent interests mathematicians. Motivated by this, we introduce Hermite polynomials as a version the ordinary polynomials. Recently, introduced was ?-umbral calculus where usual exponential function appearing in generating Sheffer sequence is replaced function. The...

In this paper, a discontinuity in beams whose intensity is adjusted by the spring stiffness factor is modeled using a torsional spring. Adapting two analyses in strong and weak forms for discontinuous beams, the improved governing differential equations and the modified stiffness matrix are derived respectively. In the strong form, two different solution methods have been presented to make an a...

In this paper, a discontinuity in beams whose intensity is adjusted by the spring stiffness factor is modeled using a torsional spring. Adapting two analyses in strong and weak forms for discontinuous beams, the improved governing differential equations and the modified stiffness matrix are derived respectively. In the strong form, two different solution methods have been presented to make an a...

We investigate Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials using the Lipschitz summation formula and obtain their integral representations. We give some explicit formulas at rational arguments for these polynomials in terms of the Hurwitz zeta function. We also derive the integral representations for the classical Bernoulli and Euler polynomials and related known ...