LOG-TANGENT INTEGRALS AND THE RIEMANN ZETA FUNCTION

VALUES OF THE RIEMANN ZETA FUNCTION AND INTEGRALS INVOLVING log ( 2 sinhf ) AND

Integrals involving the functions log (2 sinh(0/2)) and log(2sin(0/2)) are studied, particularly their relationship to the values of the Riemann zeta function at integral arguments. For example general formulae are proved which contain the known results Γ log 2 (2sin(0/2))d0 = 7π 3 /108, Jo Γ 01og 2 (2sin(0/2))d0 = 17τr 4 /6480, Jo / (log 4 (2sin(0/2))-^0 2 log 2 (2sin(0/2)))d0 = 253π 5 /3240, ...

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

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