On the series expansion of the secondary zeta function
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929304852496384 |
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| author | Kawalec, Artur |
| author_facet | Kawalec, Artur |
| contents | In article, we explore the secondary zeta function $Z(s)$, which is defined as a generalized zeta type of series over imaginary parts of non-trivial zeros of the Riemann zeta function $ζ(s)$. This function has been analytically continued as a meromorphic function in $\mathbb{C}$ with one double pole and an infinity of simple poles. The secondary zeta function is of interest because it can naturally represent an analytical formula for non-trivial zeros of the Riemann zeta function that we will explore, and we show that the non-trivial zeros can be generated directly from primes by introducing a new form of an explicit formula written in terms of the prime zeta function. Additionally, we will also give several new series expansions for $Z(s)$ and numerically compute these coefficients to high precision, and also develop several new methods to analytically extend $Z(s)$ to larger domains and develop algorithms to compute them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15741 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the series expansion of the secondary zeta function Kawalec, Artur Number Theory In article, we explore the secondary zeta function $Z(s)$, which is defined as a generalized zeta type of series over imaginary parts of non-trivial zeros of the Riemann zeta function $ζ(s)$. This function has been analytically continued as a meromorphic function in $\mathbb{C}$ with one double pole and an infinity of simple poles. The secondary zeta function is of interest because it can naturally represent an analytical formula for non-trivial zeros of the Riemann zeta function that we will explore, and we show that the non-trivial zeros can be generated directly from primes by introducing a new form of an explicit formula written in terms of the prime zeta function. Additionally, we will also give several new series expansions for $Z(s)$ and numerically compute these coefficients to high precision, and also develop several new methods to analytically extend $Z(s)$ to larger domains and develop algorithms to compute them. |
| title | On the series expansion of the secondary zeta function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2403.15741 |