On the Order of the Mertens Function

Abstract

We describe a numerical experiment concerning the order of magnitude of q(x) := M (x)/√X, where M(x) is the Mertens function (the summatory function of the Mobius function). It is known that, if the Riemann hypothesis is true and all nontrivial zeros of the Riemann zeta-function are simple, q(x) can be approximated by a series of trigonometric functions of log x. We try to obtain an ω-estimate of the order of q(x) by searching for increasingly large extrema of the sum of the first 10², 10⁴, and 10⁶ terms of this series. Based on the extrema found in the range 10⁴ ≤ x ≤ 10¹⁰¹⁰ we conjecture that q(x) = ω±,().

Keywords:

• Mertens function
• Mobius function
• Mertens hypothesis