An amortized-complexity method to compute the Riemann zeta function

An amortized-complexity method to compute the Riemann zeta function

A practical method to compute the Riemann zeta function is presented. The method can compute ζ(1/2 + it) at any T 1/4 points in [T, T + T 1/4] using an average time of T 1/4+o(1) per point. This is the same complexity as the Odlyzko-Schönhage algorithm over that interval. Although the method far from competes with the Odlyzko-Schönhage algorithm over intervals much longer than T 1/4, it still h...

Fast methods to compute the Riemann zeta function

The Riemann zeta function on the critical line can be computed using a straightforward application of the Riemann-Siegel formula, Schönhage’s method, or Heath-Brown’s method. The complexities of these methods have exponents 1/2, 3/8 (=0.375), and 1/3 respectively. In this paper, three new fast and potentially practical methods to compute zeta are presented. One method is very simple. Its comple...

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

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 ...