An alternative to Riemann-Siegel type formulas

TL;DR

This paper introduces elementary, unsmoothed formulas for computing the Riemann zeta and Dirichlet L-functions, achieving near square-root complexity with explicit error control, offering a practical alternative to traditional methods.

Contribution

It presents new elementary formulas for zeta and L-functions that match the efficiency of Riemann-Siegel formulas without smoothing, with explicit error bounds and practical parameter guidance.

Findings

Formulas achieve near square-root complexity with logarithmic loss.

The approach yields a convexity bound comparable to Riemann-Siegel.

Explicit remainder terms are easy to estimate and control.

Abstract

Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet $L$-functions to a power-full modulus, are derived by elementary means (Taylor expansions and the geometric series). The formulas enable square-root of the analytic conductor complexity, up to logarithmic loss, and have an explicit remainder term that is easy to control. The formula for zeta yields a convexity bound of the same strength as that from the Riemann-Siegel formula, up to a constant factor. Practical parameter choices are discussed.