Maximal order for divisor functions and zeros of the Riemann zeta-function
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910719818072064 |
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| author | Akatsuka, Hirotaka |
| author_facet | Akatsuka, Hirotaka |
| contents | Ramanujan investigated maximal order for the number of divisors function by introducing some notion such as (superior) highly composite numbers. He also studied maximal order for other arithmetic functions including the sum of powers of divisors function. In this paper we relate zero-free regions for the Riemann zeta-function to maximal order for the sum of powers of divisors function. In particular, we give equivalent conditions for the Riemann hypothesis in terms of the sum of the $1/2$-th powers of divisors function. As a by-product, we also give equivalent conditions for the Riemann hypothesis in terms of the partial Euler product for the Riemann zeta-function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19259 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximal order for divisor functions and zeros of the Riemann zeta-function Akatsuka, Hirotaka Number Theory 11N56, 11M26 Ramanujan investigated maximal order for the number of divisors function by introducing some notion such as (superior) highly composite numbers. He also studied maximal order for other arithmetic functions including the sum of powers of divisors function. In this paper we relate zero-free regions for the Riemann zeta-function to maximal order for the sum of powers of divisors function. In particular, we give equivalent conditions for the Riemann hypothesis in terms of the sum of the $1/2$-th powers of divisors function. As a by-product, we also give equivalent conditions for the Riemann hypothesis in terms of the partial Euler product for the Riemann zeta-function. |
| title | Maximal order for divisor functions and zeros of the Riemann zeta-function |
| topic | Number Theory 11N56, 11M26 |
| url | https://arxiv.org/abs/2411.19259 |