arXiv 2102.001112021-02math.NT

Ramanujan τ函数的偶值

Even values of Ramanujan's tau-function

Jennifer S. Balakrishnan, Ken Ono, Wei-Lun Tsai

备注：: La Matematica (2021)
DOI：: 10.1007/s44007-021-00005-8

摘要：

在莱默的推测精神，拉马努金的tau函数永远不会消失，这是很自然的问是否任何给定的整数$α$是一个值的$τ(n)$。对于奇数$α$，Murty，Murty和Shorey证明了对于足够大的$n$，$τ(n)\neq α$。最近的几篇论文已经确定了奇数$α$的显式例子，它们不是tau值。在这里，我们应用这些结果（最值得注意的是Bennett，Gherga，Patel和Siksek最近的工作）来提供不是tau值的偶数的第一个例子。也就是说，对于素数$\ell$，我们发现，$$ τ(n)\not \in \{ \pm 2\ell \ : \ 3\leq \ell< 100\} \cup \{\pm 2\ell^2 \ : \ 3\leq \ell <100\} \cup \{\pm 2\ell^3 \ : \ 3\leq \ell<100\ {\text {\rm with $\ell\neq 59$}}\}.$$而且，我们得到这样的结果，无限多的权力，每个素数$3\leq \ell<100$。作为一个例子，对于$\ell=97$，我们证明，$$τ(n)\not \in \{ 2\cdot 97^j \ : \ 1\leq j\not \equiv 0\pmod{44}\}\cup \{-2\cdot 97^j \ : \ j\geq 1\}.$$的证明方法适用于作必要的修改，以剩余可约模2伽罗瓦表示的新形式，很容易适应通用的整数系数的新形式。

英文摘要：

In the spirit of Lehmer's speculation that Ramanujan's tau-function never vanishes, it is natural to ask whether any given integer $α$ is a value of $τ(n)$. For odd $α$, Murty, Murty, and Shorey proved that $τ(n)\neq α$ for sufficiently large $n$. Several recent papers have identified explicit examples of odd $α$ which are not tau-values. Here we apply these results (most notably the recent work of Bennett, Gherga, Patel, and Siksek) to offer the first examples of even integers that are not tau-values. Namely, for primes $\ell$ we find that $$ τ(n)\not \in \{ \pm 2\ell \ : \ 3\leq \ell< 100\} \cup \{\pm 2\ell^2 \ : \ 3\leq \ell <100\} \cup \{\pm 2\ell^3 \ : \ 3\leq \ell<100\ {\text {\rm with $\ell\neq 59$}}\}.$$ Moreover, we obtain such results for infinitely many powers of each prime $3\leq \ell<100$. As an example, for $\ell=97$ we prove that $$τ(n)\not \in \{ 2\cdot 97^j \ : \ 1\leq j\not \equiv 0\pmod{44}\}\cup \{-2\cdot 97^j \ : \ j\geq 1\}.$$ The method of proof applies mutatis mutandis to newforms with residually reducible mod 2 Galois representation and is easily adapted to generic newforms with integer coefficients.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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