Between the 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 Pólya conjecture, and the non-negativity of $L(X,1)$ is the Turán 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 $L(X,\frac{1}{2})\le 0$ for all $X\ge 17$: this has been verified for $X\le 300,001$.

5 pages