On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms
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| Format: | Preprint |
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2024
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| _version_ | 1866914677292793856 |
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| author | Gun, Sanoli Naik, Sunil L |
| author_facet | Gun, Sanoli Naik, Sunil L |
| contents | Let $τ$ denote the Ramanujan tau function. One is interested in possible prime values of $τ$ function. Since $τ$ is multiplicative and $τ(n)$ is odd if and only if $n$ is an odd square, we only need to consider $τ(p^{2n})$ for primes $p$ and natural numbers $n \geq 1$. This is a rather delicate question. In this direction, we show that for any $ε> 0$ and integer $n \geq 1$, the largest prime factor of $τ(p^{2n})$, denoted by $P(τ(p^{2n}))$, satisfies $$ P(τ(p^{2n})) ~>~ (\log p)^{1/8}(\log\log p)^{3/8 -ε} $$ for almost all primes $p$. This improves a recent work of Bennett, Gherga, Patel and Siksek. Our results are also valid for any non-CM normalized Hecke eigenforms with integer Fourier coefficients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_07944 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms Gun, Sanoli Naik, Sunil L Number Theory 11F11, 11F30, 11F80, 11N56, 11R45 Let $τ$ denote the Ramanujan tau function. One is interested in possible prime values of $τ$ function. Since $τ$ is multiplicative and $τ(n)$ is odd if and only if $n$ is an odd square, we only need to consider $τ(p^{2n})$ for primes $p$ and natural numbers $n \geq 1$. This is a rather delicate question. In this direction, we show that for any $ε> 0$ and integer $n \geq 1$, the largest prime factor of $τ(p^{2n})$, denoted by $P(τ(p^{2n}))$, satisfies $$ P(τ(p^{2n})) ~>~ (\log p)^{1/8}(\log\log p)^{3/8 -ε} $$ for almost all primes $p$. This improves a recent work of Bennett, Gherga, Patel and Siksek. Our results are also valid for any non-CM normalized Hecke eigenforms with integer Fourier coefficients. |
| title | On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms |
| topic | Number Theory 11F11, 11F30, 11F80, 11N56, 11R45 |
| url | https://arxiv.org/abs/2402.07944 |