On the Zeros of the Riemann Zeta Function with Two Ordinate Shifts
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914272349519872 |
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| author | Ebadi, Ali |
| author_facet | Ebadi, Ali |
| contents | We prove that for any fixed real numbers y_1, y_2 not equal to 0, and constant C > 0, there exists a threshold T_* = T_*(y_1, y_2, C) > 0 such that for all T >= T_*, the interval [T, T(1 + epsilon)], with epsilon = exp(-C sqrt(log T)), contains at least one gamma satisfying zeta(1/2 + i gamma) = 0, zeta(1/2 + i (gamma + y_1)) != 0, and zeta(1/2 + i (gamma + y_2)) != 0.
This extends earlier work by Banks (for a single shift y) to two distinct shifts y_1, y_2. Our argument is based on the behavior of zeta and L functions in zero-free regions via Perron's formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15610 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Zeros of the Riemann Zeta Function with Two Ordinate Shifts Ebadi, Ali Number Theory 11M26 We prove that for any fixed real numbers y_1, y_2 not equal to 0, and constant C > 0, there exists a threshold T_* = T_*(y_1, y_2, C) > 0 such that for all T >= T_*, the interval [T, T(1 + epsilon)], with epsilon = exp(-C sqrt(log T)), contains at least one gamma satisfying zeta(1/2 + i gamma) = 0, zeta(1/2 + i (gamma + y_1)) != 0, and zeta(1/2 + i (gamma + y_2)) != 0. This extends earlier work by Banks (for a single shift y) to two distinct shifts y_1, y_2. Our argument is based on the behavior of zeta and L functions in zero-free regions via Perron's formula. |
| title | On the Zeros of the Riemann Zeta Function with Two Ordinate Shifts |
| topic | Number Theory 11M26 |
| url | https://arxiv.org/abs/2601.15610 |