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

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 $X\leq 300,001$.