A Direct Proof of the Simplicity of Non-Trivial Zeros of the Riemann Zeta Function
Abstract
The Riemann zeta function ζ(s) is central to analytic number theory and the study of prime distribution. Its non-trivial zeros, located in the critical strip 0 < ℜ(s) < 1, are of fundamental importance. While the Riemann Hypothesis concerns their location on the critical line ℜ(s) = 1 2 , another deep question concerns their simplicity: whether each zero is of multiplicity one. Recent works (Alpöge-Furman 2026, Lamzouri 2026, Goldston et al. 2025, Chavez 2024) have established that a large proportion of zeros are simple, or provided conditional results under conjectures. In this paper, has been presented a new unconditional proof that every non-trivial zero of ζ(s) is simple. The presented approach uses integral representations I(s) and I ′ (s), a reductio ad absurdum argument showing they cannot coincide in the critical strip, and de l'Hôpital's principle at a zero. This analytic framework leads to a contradiction if a multiple zero exists, thereby establishing the simplicity of all non-trivial zeros.
Related
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BibTeX
@misc{mansoursalem2026direct,
title = {{A Direct Proof of the Simplicity of Non-Trivial Zeros of the Riemann Zeta Function}},
author = {Mansour Salem, S Hashad},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05751250v1},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - A Direct Proof of the Simplicity of Non-Trivial Zeros of the Riemann Zeta Function
AU - Mansour Salem, S Hashad
PY - 2026
DA - 2026/09/16
PB - HAL
UR - https://hal.science/hal-05751250v1
AB - The Riemann zeta function ζ(s) is central to analytic number theory and the study of prime distribution. Its non-trivial zeros, located in the critical strip 0 < ℜ(s) < 1, are of fundamental importance. While the Riemann Hypothesis concerns their location on the critical line ℜ(s) = 1 2 , another deep question concerns their simplicity: whether each zero is of multiplicity one. Recent works (Alpöge-Furman 2026, Lamzouri 2026, Goldston et al. 2025, Chavez 2024) have established that a large proportion of zeros are simple, or provided conditional results under conjectures. In this paper, has been presented a new unconditional proof that every non-trivial zero of ζ(s) is simple. The presented approach uses integral representations I(s) and I ′ (s), a reductio ad absurdum argument showing they cannot coincide in the critical strip, and de l'Hôpital's principle at a zero. This analytic framework leads to a contradiction if a multiple zero exists, thereby establishing the simplicity of all non-trivial zeros.
ER -