The explicit Sato-Tate conjecture for primes in arithmetic progressions

Trajan Hammonds, Casimir Kothari, Noah Luntzlara, Steven J. Miller, Jesse Thorner, Hunter Wieman

Research output: Contribution to journalArticlepeer-review


Let τ(n) be Ramanujan's tau function, defined by the discriminant modular form Δ(z) = q-j=1∞(1 - qj)24 =ζ n=1∞τ(n)qn,q = e2πiz (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer's conjecture asserts that τ(n)=0 for all n ≥ 1; since τ(n) is multiplicative, it suffices to study primes p for which τ(p) might possibly be zero. Assuming standard conjectures for the twisted symmetric power L-functions associated to τ (including GRH), we prove that if x ≥ 1050, then #{x < p ≤ 2x:τ(p) = 0}≤ 1.22 × 10-5 x3/4 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.

Original languageEnglish (US)
Pages (from-to)1905-1923
Number of pages19
JournalInternational Journal of Number Theory
Issue number8
StatePublished - Sep 1 2021


  • modular forms
  • Sato-Tate conjecture
  • symmetric power L -functions

ASJC Scopus subject areas

  • Algebra and Number Theory


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