On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917588767866880 |
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| author | Naik, Sunil L |
| author_facet | Naik, Sunil L |
| contents | In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan $Δ$-function, we show that for any $ε> 0$, there exist infinitely many natural numbers $n$ such that $τ(p^n)$ has at least $$ 2^{(1-ε) \frac{\log n}{\log\log n}} $$ distinct prime factors for almost all primes $p$. This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of Modular forms alluded to above. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07942 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms Naik, Sunil L Number Theory 11F11, 11F30, 11N56 In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan $Δ$-function, we show that for any $ε> 0$, there exist infinitely many natural numbers $n$ such that $τ(p^n)$ has at least $$ 2^{(1-ε) \frac{\log n}{\log\log n}} $$ distinct prime factors for almost all primes $p$. This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of Modular forms alluded to above. |
| title | On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms |
| topic | Number Theory 11F11, 11F30, 11N56 |
| url | https://arxiv.org/abs/2402.07942 |