Analytical continuation of Euler prime product for $\Re(s)>\tfrac{1}{2}$ assuming (RH)
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915900779659264 |
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| author | Kawalec, Artur |
| author_facet | Kawalec, Artur |
| contents | We analytically continue the Euler prime product for $\Re(s)>\tfrac{1}{2}$ (except for its pole $s=1$) assuming (RH) by introducing a new factor to the Euler product. We also discuss how to recover the Mertens's 3rd Theorem at $s=1$ case, and how to apply the same technique to analytically continue other similar Euler products. In the last part, we also construct a simple script in Pari/GP to compute the Euler product and verify the calculations numerically. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28808 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Analytical continuation of Euler prime product for $\Re(s)>\tfrac{1}{2}$ assuming (RH) Kawalec, Artur General Mathematics We analytically continue the Euler prime product for $\Re(s)>\tfrac{1}{2}$ (except for its pole $s=1$) assuming (RH) by introducing a new factor to the Euler product. We also discuss how to recover the Mertens's 3rd Theorem at $s=1$ case, and how to apply the same technique to analytically continue other similar Euler products. In the last part, we also construct a simple script in Pari/GP to compute the Euler product and verify the calculations numerically. |
| title | Analytical continuation of Euler prime product for $\Re(s)>\tfrac{1}{2}$ assuming (RH) |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2603.28808 |