Even Values of Ramanujan’s Tau-Function
Even Values of Ramanujan’s Tau-Function
复制标题
拉马努金 Tau 函数的偶数值
DOI：
10.1007/s44007-021-00005-8
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Tsai, Wei-Lun
中科院分区：
文献类型：
--
作者：
Balakrishnan, Jennifer S.;Ono, Ken;Tsai, Wei-Lun
In the spirit of Lehmer’s speculation that Ramanujan’s tau-function never vanishes, it is natural to ask whether any given integeris a value of. For odd, Murty, Murty, and Shorey proved thatfor sufficiently largen. Several recent papers have identified explicit examples of oddwhich 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 primeswe find that $$\begin{aligned} \tau (n)\not \in \{ \pm 2\ell \ : \ 3\le \ell< 100\}&\cup&\{\pm 2\ell ^2 \ : \ 3\le \ell<100\}\\&\cup&\{\pm 2\ell ^3 \ : \ 3\le \ell <100\ {\text {with }\ell \ne 59}\}. \end{aligned}$$Moreover, we obtain such results for infinitely many powers of each prime. As an example, forwe prove that $$\begin{aligned} \tau (n)\not \in \{ 2\cdot 97^j \ : \ 1\le j\not \equiv 0\pmod {44}\}\cup \{-2\cdot 97^j \ : \ j\ge 1\}. \end{aligned}$$The method of proof appliesmutatis mutandisto newforms with residually reducible mod 2 Galois representation and is easily adapted to generic newforms with integer coefficients.
In the spirit of Lehmer’s speculation that Ramanujan’s tau-function never vanishes, it is natural to ask whether any given integeris a value of. For odd, Murty, Murty, and Shorey proved thatfor sufficiently largen. Several recent papers have identified explicit examples of oddwhich 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 primeswe find that $$\begin{aligned} \tau (n)\not \in \{ \pm 2\ell \ : \ 3\le \ell< 100\}&\cup&\{\pm 2\ell ^2 \ : \ 3\le \ell<100\}\\&\cup&\{\pm 2\ell ^3 \ : \ 3\le \ell <100\ {\text {with }\ell \ne 59}\}. \end{aligned}$$Moreover, we obtain such results for infinitely many powers of each prime. As an example, forwe prove that $$\begin{aligned} \tau (n)\not \in \{ 2\cdot 97^j \ : \ 1\le j\not \equiv 0\pmod {44}\}\cup \{-2\cdot 97^j \ : \ j\ge 1\}. \end{aligned}$$The method of proof appliesmutatis mutandisto newforms with residually reducible mod 2 Galois representation and is easily adapted to generic newforms with integer coefficients.
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DOI：
10.1007/s11139-021-00388-w
发表时间：
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期刊：
The Ramanujan Journal
影响因子：
--
作者：
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通讯作者：
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