PreprintCC BY 4.0arXiv
Optimising Selberg's method for critical zeros
Abstract
We revisit Zhuravlev's 1974 quantitative form of Selberg's sign-change method for the zeros of the Riemann zeta function on the critical line. Zhuravlev's work, originally in Russian and little known in the West, established an explicit positive proportion of such zeros. Using modern techniques we optimise the arithmetic mean value at the heart of the method and obtain the strongest result yet known from Selberg's method, proving that more than 6.78% of the non-trivial zeros of the Riemann zeta function lie on the critical line.
Related
- Counting multigeodesics and the Selberg-Delange method
- Zeta zeros on the critical line
- More than 8/9 zeros of the Riemann zeta function are simple or on the critical line
- Simple critical zeros and distinct zeros of the Riemann zeta function in short intervals
- More than two thirds of the zeta zeros are simple and on the critical line
BibTeX
@misc{pearcecrump2026optimising,
title = {{Optimising Selberg's method for critical zeros}},
author = {Andrew Pearce-Crump},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.15329},
eprint = {2609.15329},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15329},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimising Selberg's method for critical zeros
AU - Andrew Pearce-Crump
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2609.15329
UR - https://arxiv.org/abs/2609.15329
AB - We revisit Zhuravlev's 1974 quantitative form of Selberg's sign-change method for the zeros of the Riemann zeta function on the critical line. Zhuravlev's work, originally in Russian and little known in the West, established an explicit positive proportion of such zeros. Using modern techniques we optimise the arithmetic mean value at the heart of the method and obtain the strongest result yet known from Selberg's method, proving that more than 6.78% of the non-trivial zeros of the Riemann zeta function lie on the critical line.
ER -