Abstract
Let f(z) =σn=1∞ δ f(n)e 2Φinz Sknew 0(N)) be a newform of even weight k ≥ 2 on τ0(N) without complex multiplication. Let P denote the set of all primes. We prove that the sequence {δf(p)}p does not satisfy Benford's Law in any integer base b ≥ 2. However, given a base b ≥ 2 and a string of digits S in base b, the set Aδf(b,S):= {p prime: the first digits of δf(p) in base b are given by S} has logarithmic density equal to logb(1 + S-1). Thus, {δf(p)}p follows Benford's Law with respect to logarithmic density. Both results rely on the now-proven Sato-Tate Conjecture.
Original language | English (US) |
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Pages (from-to) | 483-494 |
Number of pages | 12 |
Journal | International Journal of Number Theory |
Volume | 12 |
Issue number | 2 |
DOIs | |
State | Published - Mar 1 2016 |
Externally published | Yes |
Keywords
- Benford's Law
- equidistribution mod 1
- modular forms
- Sato-Tate conjecture
ASJC Scopus subject areas
- Algebra and Number Theory