New computations of the Riemann zeta function on the critical line

TL;DR

This paper introduces advanced computational techniques for evaluating the Riemann zeta function near large zeros, utilizing a fast quadratic exponential sum algorithm and a multi-evaluation method for efficiency.

Contribution

It presents novel algorithms for high-precision computation of the zeta function, improving efficiency and accuracy in the critical line region.

Findings

Successful computation of zeta function values near large zeros

Implementation of a fast quadratic exponential sum algorithm

Efficient multi-evaluation method for small ranges

Abstract

We present highlights of computations of the Riemann zeta function around large values and high zeros. The main new ingredient in these computations is an implementation of the second author's fast algorithm for numerically evaluating quadratic exponential sums. In addition, we use a new simple multi-evaluation method to compute the zeta function in a very small range at little more than the cost of evaluation at a single point.