Benford's Law for coefficients of newforms

Marie Jameson, Jesse Thorner, Lynnelle Ye

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)483-494
Number of pages12
JournalInternational Journal of Number Theory
Issue number2
StatePublished - Mar 1 2016
Externally publishedYes


  • Benford's Law
  • equidistribution mod 1
  • modular forms
  • Sato-Tate conjecture

ASJC Scopus subject areas

  • Algebra and Number Theory


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