Let τ(n) be Ramanujan’s tau function, defined by the discriminant modular form Δ(z) = q∏j=1∞(1 − qj)24 =∑ 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.

中文翻译：

---

算术级数中素数的显式佐藤-泰特猜想

让τ(n)是 Ramanujan 的 tau 函数，由判别模形式定义 Δ(z) = q∏j=1∞(1 - qj)24 =∑ n=1∞τ(n)qn,q = e2π一世z （这是重量为 12 和 1 级的独特的全纯归一化尖瓣新形式）。莱默猜想断言τ(n)≠0对所有人n ≥ 1; 自从τ(n)是乘法的，研究素数就足够了p为此τ(p)可能为零。假设扭曲对称幂的标准猜想大号- 相关的功能τ（包括 GRH），我们证明如果X ≥ 1050， 然后 #{X < p ≤ 2X：τ(p) = 0}≤ 1.22 × 10-5 X3/4 日志 X, 对先前工作中的隐含常数进行了实质性改进。为了实现这一点，在相同的假设下，我们证明了算术级数中素数的佐藤-泰特猜想的显式版本。