Density of the Values Set of the Tau Function
General Mathematics
2014-04-11 v1
Abstract
It is shown that the density of the values set {Tau(n): n <= x} of the nth coefficients Tau(n) of the discriminant function Delta(z), a cusp form of level N = 1 and weight k = 12, has the lower bound #{Tau(n): n <= x} >> x/log x. The currently known density is #{Tau(n) : n <= x} >> x^(1/2+o(1)), and the expected density is #{Tau(n) : n <= x} ~ x. The solutions set of the equation Tau(p) = 0 for all primes p => 2, which arises as a singular case of this analysis, is discussed within.
Cite
@article{arxiv.1404.2789,
title = {Density of the Values Set of the Tau Function},
author = {N. A. Carella},
journal= {arXiv preprint arXiv:1404.2789},
year = {2014}
}
Comments
Eleven Pages. Keywords: Fourier Coefficient, Modular Form, Cusp Form, Tau Function, Lange-Trotter Conjecture