Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Between the conjectures of P\'{o}lya and Tur\'{a}n   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 $X\leq 300,001$.

26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con