Investigations on the Theory of Riemann Zeta Function II: On the Riemann-Siegel Integral and Hardy’s Z-Function

The Theory of the Riemann Zeta-function

On the Riemann Zeta-function and the Divisor Problem Ii

Let ∆(x) denote the error term in the Dirichlet divisor problem, and E(T ) the error term in the asymptotic formula for the mean square of |ζ( 1 2 + it)|. If E∗(t) = E(t) − 2π∆∗(t/2π) with ∆∗(x) = −∆(x) + 2∆(2x) − 1 2 ∆(4x), then we obtain ∫ T 0 |E(t)| dt ≪ε T 2+ε and ∫ T 0 |E∗(t)| 544 75 dt ≪ε T 601 225 . It is also shown how bounds for moments of |E∗(t)| lead to bounds for moments of |ζ( 1 2 ...

Mollifying 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...

On the Function S ( T ) in the Theory of the Riemann Zeta - Function

The function S(T) is the error term in the formula for the number of zeros of the Riemann zeta-function above the real axis and up to height Tin the complex plane. We assume the Riemann hypothesis, and examine how well S(T) can be approximated by a Dirichlet polynomial in the Lz norm.