Experiments with zeta zeros and Perron's formula

TL;DR

This paper explores how sums over the zeros of the Riemann zeta function, derived via Perron's formula, can be used to count primes, squarefree integers, and evaluate arithmetic functions through experimental methods.

Contribution

It demonstrates the practical use of zeta zeros in computational number theory for counting and evaluating arithmetical functions, with a focus on experimental results.

Findings

Sums over zeta zeros can effectively count primes and squarefree integers.

Perron's formula provides a framework for linking zeta zeros to arithmetic functions.

Experimental results support the utility of zeta zeros in computational number theory.

Abstract

Of what use are the zeros of the Riemann zeta function? We can use sums involving zeta zeros to count the primes up to $x$. Perron's formula leads to sums over zeta zeros that can count the squarefree integers up to $x$, or tally Euler's $\phi$ function and other arithmetical functions. This is largely a presentation of experimental results.