TY - JOUR

T1 - The explicit Sato-Tate conjecture for primes in arithmetic progressions

AU - Hammonds, Trajan

AU - Kothari, Casimir

AU - Luntzlara, Noah

AU - Miller, Steven J.

AU - Thorner, Jesse

AU - Wieman, Hunter

N1 - Publisher Copyright:
© 2021 World Scientific Publishing Company.

PY - 2021/9/1

Y1 - 2021/9/1

N2 - 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.

AB - 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.

KW - modular forms

KW - Sato-Tate conjecture

KW - symmetric power L -functions

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U2 - 10.1142/S179304212150069X

DO - 10.1142/S179304212150069X

M3 - Article

AN - SCOPUS:85104807250

VL - 17

SP - 1905

EP - 1923

JO - International Journal of Number Theory

JF - International Journal of Number Theory

SN - 1793-0421

IS - 8

ER -