Corrigendum to “Variants of Lehmer's speculation for newforms” [Adv. Math. 428 (2023) 109141]
Corrigendum to “Variants of Lehmer's speculation for newforms” [Adv. Math. 428 (2023) 109141]
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勘误“莱默对新形式的推测的变体”[Adv.
DOI：
10.1016/j.aim.2023.109347
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发表时间：
2023
影响因子：
1.7
通讯作者：
Tsai, Wei-Lun
中科院分区：
文献类型：
--
作者：
Balakrishnan, Jennifer S.;Craig, William;Ono, Ken;Tsai, Wei-Lun
本着莱默关于拉马努扬的张量函数不会消失的悬而未决的猜测的精神，人们很自然地会问固定的整数α是τ(N)的值还是任何给定新形式f(Z)的傅立叶系数a f(N)。我们给出了一种适用于具有整系数和平凡剩余模2伽罗瓦表示的新形式的方法，它回答了奇α的这个问题。我们确定了无穷多个素数3≤ℓ≤37不是新形式系数的绝对值的整系数空间，并得到了τ(N)的许多显式例子。我们还得到了这种新形式系数的素因子数的精确下界。在权方面，证明了对于奇素数ℓ的幂，±ℓm不是权为2 k的新形式f的系数；M±(ℓ，m)与ℓ的偶级互素，其中M±(ℓ，m)是有效可计算的常数，且为Oℓ(M)。
In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer α is a value of τ (n) or is a Fourier coefficient a f (n) of any given newform f (z). We offer a method, which applies to newforms with integer coefficients and trivial residual mod 2 Galois representation, that answers this question for odd α. We determine infinitely many spaces for which the primes 3≤ ℓ≤ 37 are not absolute values of coefficients of newforms with integer coefficients, and we obtain many explicit examples for τ (n). We also obtain sharp lower bounds for the number of prime factors of such newform coefficients. In the weight aspect, for powers of odd primes ℓ, we prove that±ℓ m is not a coefficient of any such newform f with weight 2 k> M±(ℓ, m) and even level coprime to ℓ, where M±(ℓ, m) are effectively computable constants that are O ℓ (m).
莱默对新形式的推测的变体
影响因子：
1.7
作者：
Balakrishnan, Jennifer S.;Craig, William;Ono, Ken;Tsai, Wei-Lun
通讯作者：
Tsai, Wei-Lun