The explicit Sato–Tate conjecture for primes in arithmetic progressions
| Autor: | Hunter Wieman, Casimir Kothari, Jesse Thorner, Noah Luntzlara, Trajan Hammonds, Steven J. Miller |
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| Rok vydání: | 2021 |
| Předmět: |
Algebra and Number Theory
Mathematics - Number Theory 11F30 11M41 11N13 Computer Science::Information Retrieval Mathematics::Number Theory Sato–Tate conjecture Modular form Astrophysics::Instrumentation and Methods for Astrophysics Holomorphic function Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) Ramanujan's sum Combinatorics symbols.namesake Discriminant FOS: Mathematics symbols Computer Science::General Literature Number Theory (math.NT) Ramanujan tau function Mathematics |
| Zdroj: | International Journal of Number Theory. 17:1905-1923 |
| ISSN: | 1793-7310 1793-0421 |
| DOI: | 10.1142/s179304212150069x |
| Popis: | Let $\tau(n)$ be Ramanujan's tau function, defined by the discriminant modular form \[ \Delta(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\ \sum_{n=1}^{\infty}\tau(n) q^n \,,q=e^{2\pi i z} \] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer's conjecture asserts that $\tau(n)\neq 0$ for all $n\geq 1$; since $\tau(n)$ is multiplicative, it suffices to study primes $p$ for which $\tau(p)$ might possibly be zero. Assuming standard conjectures for the twisted symmetric power $L$-functions associated to $\tau$ (including GRH), we prove that if $x\geq 10^{50}$, then \[ \#\{x < p\leq 2x: \tau(p) = 0\} \leq 1.22 \times 10^{-5} \frac{x^{3/4}}{\sqrt{\log x}},\] a substantial improvement on the implied constant in previous work. To achieve this, under the same hypotheses, we prove an explicit version of the Sato-Tate conjecture for primes in arithmetic progressions. Comment: 16 pages, fixed typographical errors and minor computational details. To be published in the International Journal of Number Theory |
| Databáze: | OpenAIRE |
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