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The explicit Sato-Tate Conjecture and densities pertaining to Lehmer-type questions


Authors: Jeremy Rouse and Jesse Thorner
Journal: Trans. Amer. Math. Soc. 369 (2017), 3575-3604
MSC (2010): Primary 11F30, 11M41; Secondary 11F33
DOI: https://doi.org/10.1090/tran/6793
Published electronically: December 22, 2016
MathSciNet review: 3605980
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Abstract: Let $ f(z)=\sum _{n=1}^\infty a_f(n)q^n\in S^{\text {new}}_k (\Gamma _0(N))$ be a newform with squarefree level $ N$ that does not have complex multiplication. For a prime $ p$, define $ \theta _p\in [0,\pi ]$ to be the angle for which $ a_f(p)=2p^{( k -1)/2}\cos \theta _p $. Let $ I\subset [0,\pi ]$ be a closed subinterval, and let $ d\mu _{ST}=\frac {2}{\pi }\sin ^2\theta d\theta $ be the Sato-Tate measure of $ I$. Assuming that the symmetric power $ L$-functions of $ f$ satisfy certain analytic properties (all of which follow from Langlands functoriality and the Generalized Riemann Hypothesis), we prove that if $ x$ is sufficiently large, then

$\displaystyle \left \vert\char93 \{p\leq x:\theta _p\in I\} -\mu _{ST}(I)\int _2^x\frac {dt}{\log t}\right \vert\ll \frac {x^{3/4}\log (N k x)}{\log x} $

with an implied constant of $ 3.33$. By letting $ I$ be a short interval centered at $ \frac {\pi }{2}$ and counting the primes using a smooth cutoff, we compute a lower bound for the density of positive integers $ n$ for which $ a_f(n)\neq 0$. In particular, if $ \tau $ is the Ramanujan tau function, then under the aforementioned hypotheses, we prove that

$\displaystyle \lim _{x\to \infty }\frac {\char93 \{n\leq x:\tau (n)\neq 0\}}{x}>1-1.54\times 10^{-13}. $

We also discuss the connection between the density of positive integers $ n$ for which $ a_f(n)\neq 0$ and the number of representations of $ n$ by certain positive-definite, integer-valued quadratic forms.

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Additional Information

Jeremy Rouse
Affiliation: Department of Mathematics, Wake Forest University, PO Box 7388, Winston-Salem, North Carolina 27109

Jesse Thorner
Affiliation: Department of Mathematics and Computer Science, Emory University, 400 Dowman Dr., STE W401, Atlanta, Georgia 30322-2390
Address at time of publication: Department of Mathematics, Stanford University, Stanford, California 94305

DOI: https://doi.org/10.1090/tran/6793
Received by editor(s): May 22, 2013
Received by editor(s) in revised form: June 14, 2015
Published electronically: December 22, 2016
Article copyright: © Copyright 2016 American Mathematical Society