The Farey Sequence and the Mertens Function

TL;DR

This paper explores the deep connections between Farey sequences, the Mertens function, and the Riemann hypothesis, proposing new functions and conjectures that relate these fundamental number theory concepts.

Contribution

It introduces new functions related to Farey sequences that mirror the Mertens function and proposes a novel conjecture linking Farey sequences to the Riemann hypothesis beyond existing theorems.

Findings

Mikolas theorem on Mertens function sums

Properties of new Farey sequence functions analogous to Mertens

A conjecture connecting Farey sequences, Mertens function, and the Riemann hypothesis

Abstract

Franel and Landau derived an arithmetic statement involving the Farey sequence that is equivalent to the Riemann hypothesis. Since there is a relationship between the Mertens function and the Riemann hypothesis, there should be a relationship between the Mertens function and the Farey sequence. Functions of subsets of the fractions in Farey sequences that are analogous to the Mertens function are introduced. Mikolas proved that the sum of certain Mertens function values is 1. Results analogous to Mikolas theorem are the defining property of these functions. A relationship between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem is postulated. This conjecture involves a theorem of Mertens and the second Chebyshev function.