Modular forms and an explicit Chebotarev variant of the Brun–Titchmarsh theorem

Abstract

We prove an explicit Chebotarev variant of the Brun–Titchmarsh theorem. This leads to explicit versions of the best known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that limx→∞#{1≤n≤x∣τ(n)≠0}x>1-1.15×10-12,lim⁡x→∞#{1≤n≤x∣τ(n)≠0}x>1−1.15×10−12,\begin{aligned} \lim _{x \rightarrow \infty } \frac{\#\{1 \le n \le x \mid \tau (n) \ne 0\}}{x} > 1-1.15 \times 10^{-12}, \end{aligned}where τ(n)τ(n)\tau (n) is Ramanujan’s tau-function. This is the first known positive unconditional lower bound for the proportion of positive integers n such that τ(n)≠0τ(n)≠0\tau (n) \ne 0.

Full text

D. Hu et al. Res. Number Theory (2023) 9:46
https://doi.org/10.1007/s40993-023-00451-z
RESEARCH
Modular forms and an explicit Chebotarev
variant of the Brun–Titchmarsh theorem
Daniel Hu1,HariR.Iyer
2* and Alexander Shashkov3
*Correspondence:
hiyer@college.harvard.edu
1Department of Mathematics,
Princeton University, Princeton,
NJ, USA 2Department of
Mathematics, Harvard University,
Cambridge, MA, USA
Full list of author information is
available at the end of the article
Abstract
We prove an explicit Chebotarev variant of the Brun–Titchmarsh theorem. This leads to
explicit versions of the best known unconditional upper bounds toward conjectures of
Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular,
we prove that
lim
x→∞
#{1≤n≤x|τ(n)= 0}
x>1−1.15 ×10−12,
where τ(n) is Ramanujan’s tau-function. This is the first known positive unconditional
lower bound for the proportion of positive integers nsuch that τ(n)= 0.
Mathematics Subject Classification: Primary: 11R44, Secondary: 11N36, 11F30
Contents
1 Introduction ......................................... 2
Outline of the paper ..................................... 7
2 Preliminaries ......................................... 7
2.1 Notation ........................................ 7
2.2 Preliminary estimates ................................. 9
2.3 Auxiliary results .................................... 11
3 Reduction to the abelian case ................................ 13
4 Sums over integral ideals .................................. 14
4.1 Bounding A(x;a, n)................................... 14
4.2 Bounding V(z)..................................... 16
5 Implementing the Selberg sieve ............................... 18
6 Brun–Titchmarsh for abelian extensions ......................... 21
7 Upper bounds for the Lang–Trotter conjecture ...................... 22
7.1 Sieving πf(x, a)byprimes ............................... 22
7.2 Reduction to a Chebotarev problem ......................... 24
7.3 Bounding M(L/K)................................... 27
7.4 Lang–Trotter type bounds .............................. 28
8 Proof of Theorems 1.2 and 1.4 ............................... 31
8.1 Auxiliary results on bounds for prime-counting functions ............. 31
8.2 Proof of Theorem 1.2 ................................. 32
8.3 Proof of Theorem 1.4 ................................. 33
9 Proof of Theorems 1.3 and 1.5 ............................... 34
References ............................................ 36
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