On some connections between zeta-zeros and square-free integers
The main purpose of the paper is to present a complex explicit formula for square-free integers and an application to the proof of the classical explicit formula for the number of square-free integers not exceeding \(x\). More precisely, it is constructed and studied in detail a meromorphic function defined in terms of a certain sum involving all non- real zeros of the Riemann zeta-function. This function has simple poles with residues \(|\mu(n)|\) (\(\mu\) is the Möbius function) at all numbers \(\log n\), where \(n\) is square-free or \(n\) is equal to 1, and is regular everywhere else.
explicit formula for square-free integers
scientific article; zbMATH DE number 164923