Academic Journal
Level Curves for Zhang’s Eta Function
| Title: | Level Curves for Zhang’s Eta Function |
|---|---|
| Authors: | Jeffrey Stopple |
| Source: | Experimental Mathematics, vol ahead-of-print, iss ahead-of |
| Publication Status: | Preprint |
| Publisher Information: | Informa UK Limited, 2025. |
| Publication Year: | 2025 |
| Subject Terms: | Numerical and Computational Mathematics, Numerical and computational mathematics, zeros of characteristic polynomials of unitary matrices, Mathematics - Number Theory, Applied Mathematics, General Mathematics, Pure mathematics, Applied mathematics, Pure Mathematics, Mathematical Sciences, zeros of the derivative of the Riemann zeta function, FOS: Mathematics, Number Theory (math.NT), 11M06, 15B52, 30C15, Zeros of the Riemann zeta function |
| Description: | Study of the level curve for the real part of $η(s)=0$ with $η(s)=π^{-s/2}Γ(s/2)ζ^\prime(s)$ gives a new classification of the zeros of $ζ(s)$ and of $ζ^\prime(s)$. We conjecture that for type 2 zeros, $\liminf (β^\prime -1/2)\logγ^\prime = 0$ if and only if $\liminf (γ^+-γ^-)\log γ^\prime=0$, and reduce the conjecture to a lower bound on the curvature of the level curve. We compute and classify $10^6$ zeros of $ζ^\prime(s)$ near $T=10^{10}$. The Riemann Hypothesis is assumed throughout. An appendix develops the analogous classification for characteristic polynomials of unitary matrices. Supersedes arXiv:2009.11886. To appear in Experimental Mathematics |
| Document Type: | Article |
| File Description: | application/pdf |
| Language: | English |
| ISSN: | 1944-950X 1058-6458 |
| DOI: | 10.1080/10586458.2025.2487172 |
| DOI: | 10.48550/arxiv.2503.07696 |
| Access URL: | http://arxiv.org/abs/2503.07696 https://escholarship.org/content/qt8r0775p5/qt8r0775p5.pdf https://escholarship.org/uc/item/8r0775p5 |
| Rights: | CC BY NC ND CC BY |
| Accession Number: | edsair.doi.dedup.....d147ced5d75b5c80dbb054c542f355c8 |
| Database: | OpenAIRE |
| ISSN: | 1944950X 10586458 |
|---|---|
| DOI: | 10.1080/10586458.2025.2487172 |