New computations of the Riemann zeta function on the critical line
This work provides faster numerical verification of the Riemann hypothesis for high zeros, benefiting number theorists.
The authors performed computations of the Riemann zeta function at large heights and high zeros using a fast algorithm for quadratic exponential sums and a multi-evaluation method, achieving efficient evaluation over small ranges at nearly single-point cost.
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.