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Proportions of the non-trivial zeros of the Riemann zeta function
Abstract
Let C0=3/2−1/√2cot(1/√2) =0.67250… and C1=(C0+1)/2= 0.83625…. Recently, it is obatained by Alpöge and Furman that more than 67.25\% of the non-trivial zeros of the Riemann zeta function are simple and on the critical line, and more than 83.62\% are distinct. Later, Lamzouri gave a different and more direct proof. In this article, by refining the method of Lamzouri, we slightly improve the bounds C0 and C1 in these two results to C0+δ0 and C1+δ0/2 with δ0=6.66624…×10−8, respectively.
Related
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- More than 8/9 zeros of the Riemann zeta function are simple or on the critical line
- One-level density of zeros of the Riemann zeta function weighted by its special values
- Half-sized gaps between zeros of the Riemann zeta function
- Simple critical zeros and distinct zeros of the Riemann zeta function in short intervals
BibTeX
@misc{wang2026proportions,
title = {{Proportions of the non-trivial zeros of the Riemann zeta function}},
author = {Biao Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.24167},
eprint = {2609.24167},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24167},
}
RIS
TY - UNPB
TI - Proportions of the non-trivial zeros of the Riemann zeta function
AU - Biao Wang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2609.24167
UR - https://arxiv.org/abs/2609.24167
AB - Let C_0=3/2−1/√2cot(1/√2) =0.67250… and C_1=(C_0+1)/2= 0.83625…. Recently, it is obatained by Alpöge and Furman that more than 67.25\% of the non-trivial zeros of the Riemann zeta function are simple and on the critical line, and more than 83.62\% are distinct. Later, Lamzouri gave a different and more direct proof. In this article, by refining the method of Lamzouri, we slightly improve the bounds C_0 and C_1 in these two results to C_0+δ_0 and C_1+δ_0/2 with δ_0=6.66624…×10^−8, respectively.
ER -