Exponential Sums and the Riemann Zeta Function III

Zeros of Partial Sums of the Riemann Zeta-function

We write ρX = βX + iγX for a typical zero of FX(s). The number of these up to height T we denote by NX(T ), and the number of these with βX ≥ σ by NX(σ, T ). We follow the convention that if T is the ordinate of a zero, then NX(T ), say, is defined as limǫ→0+ NX(T + ǫ). There are two natural ways to pose questions about NX(T ), NX(σ, T ), and the distribution of the zeros generally. We can fix ...

On the Riemann Zeta-function and the Divisor Problem Iii

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) and we set T 0 E * (t) dt = 3πT /4 + R(T), then we obtain R(T) = O ε (T 593/912+ε), T 0 R 4 (t) dt ≪ ε T 3+ε , and T 0 R 2 (t) dt = T 2 P 3 (log T) + O ε (T 11/6+ε), whe...

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

Supercomputers and the Riemann Zeta Function

The Riemann Hypothesis, which specifies the location of zeros of the Riemann zeta function, and thus describes the behavior of primes, is one of the most famous unsolved problems in mathematics, and extensive efforts have been made over more than a century to check it numerically for large sets of cases. Recently a new algorithm, invented by the speaker and A. Scho . . nhage, has been implement...