Between the conjectures of P\'{o}lya and Tur\'{a}n

TL;DR

This paper investigates the sign behavior of a sum involving the Liouville function, proposing that at the critical point 1/2, the sum remains non-positive for all sufficiently large X, supported by computational evidence.

Contribution

It introduces a new conjecture that the sum involving the Liouville function at 1/2 is always non-positive beyond a certain point, extending the understanding of sign constancy in number theory.

Findings

Evidence supports the conjecture for X ≤ 300,001.

The sum L(X, 1/2) is the most promising candidate for constant sign behavior.

The paper links the conjecture to classical conjectures of Pólya and Turán.

Abstract

This paper is concerned with the constancy in the sign of $L(X, \alpha) = \sum_{1}^{X} \frac{\lambda(n)}{n^{\alpha}}$, where $\lambda(n)$ the Liouville function. The non-positivity of $L(X, 0)$ is the P\'{o}lya conjecture, and the non-negativity of $L(X, 1)$ is the Tur\'{a}n conjecture --- both of which are false. By constructing an auxiliary function, evidence is provided that $L(X, \frac{1}{2})$ is the best contender for constancy in sign. The core of this paper is the conjecture that $L(X, \frac{1}{2}) \leq 0$ for all $X\geq 17$: this has been verified for $X\leq 300,001$.