New Computations of the Riemann Zeta Function on the Critical Line

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.

Keywords:

• exponential sums
• large values
• the Riemann zeta function
• the van der Corput iteration
• theta algorithm

2000 AMS SUBJECT CLASSIFICATION:

• Primary 11Y35
• Secondary 65Y20

Acknowledgments

The authors are pleased to thank Institute for Advanced Study, Mathematical Sciences Research Institute, and Institute for Computational and Experimental Research in Mathematics, where parts of this work was conducted.

Notes

1 This is the same θ(t) appearing in the definition of Z(t).

2 We assume that Z(t₁) and Z(t₂) are nonzero.

3 The mth gram point g_m is the unique solution the equation for t ⩾ 7. It is called good if ( − 1)^mZ(g_m) > 0. One usually finds a good gram point on testing Z(t) at few consecutive gram points.

Additional information

Funding

GH is partially supported by the National Science Foundation under agreement No. DMS-1406190.