Fourier coefficients associated with the Riemann zeta-function

Logarithmic Fourier integrals for the Riemann Zeta Function

We use symmetric Poisson-Schwarz formulas for analytic functions f in the half-plane Re(s) > 1 2 with f(s) = f(s) in order to derive factorisation theorems for the Riemann zeta function. We prove a variant of the Balazard-Saias-Yor theorem and obtain explicit formulas for functions which are important for the distribution of prime numbers. In contrast to Riemann’s classical explicit formula, th...

Some Definite Integrals Associated with the Riemann Zeta Function

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 ...

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