The explicit Sato–Tate conjecture for primes in arithmetic progressions
نویسندگان
چکیده
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 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 multiplicative, it suffices to study primes $p$ which $\tau(p)$ might possibly zero. Assuming standard conjectures twisted symmetric power $L$-functions associated $\tau$ (including GRH), we prove 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 implied constant in previous work. To achieve this, under same hypotheses, an explicit version Sato-Tate arithmetic progressions.
منابع مشابه
The abc conjecture and non-Wieferich primes in arithmetic progressions
Article history: Received 18 June 2012 Revised 12 September 2012 Accepted 6 October 2012 Available online xxxx Communicated by Greg Martin MSC: 11A41 11B25
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ژورنال
عنوان ژورنال: International Journal of Number Theory
سال: 2021
ISSN: ['1793-7310', '1793-0421']
DOI: https://doi.org/10.1142/s179304212150069x