Summatory Mobius Function, and Summatory Liouville Function

TL;DR

This paper proves that the summatory Liouville and Mobius functions are both bounded by O(x^0.5) unconditionally and explores their implications for zeta and L-functions.

Contribution

It establishes unconditionally that both summatory functions are of order O(x^0.5) and discusses applications to zeta and L-functions.

Findings

L(x) = O(x^0.5) proven unconditionally

M(x) = O(x^0.5) proven unconditionally

Applications to zeta and L-functions considered

Abstract

The orders of magnitudes of the summatory Liouville function L(x), and the summatory Mobius function M(x), are unconditionally proven to be of the forms L(x) = O(x^.5)), and M(x) = O(x^.5) respectively. Furthermore, applications of these estimates to zeta functions and L-functions are also considered.