PreprintCC BY 4.0arXiv
Joint universality of the Riemann zeta-function with arbitrary real shift parameters
Abstract
We prove that the shifts ζ(s+idjτ), 1≤ j≤ n, are jointly universal for any nonzero real parameters dj with pairwise distinct absolute values. These restrictions are best possible. As an immediate consequence, we obtain generalized strong recurrence for arbitrary finite families of nonzero real parameters. Our proof uses Emsalem's generalization of the six exponentials theorem and a new variant of Pecherskii's rearrangement theorem.
Related
- Currently, there are no reasons to doubt the Riemann Hypothesis: The zeta function beyond the realm of computation
- More than 8/9 zeros of the Riemann zeta function are simple or on the critical line
- Ramanujan Conjecture and Riemann Hypothesis for the unitary groups over function fields
- On the p-variation of Riemann's "nondifferentiable'' function
- An Omega result for partial sums of the Riemann zeta function
BibTeX
@misc{pankowski2026joint,
title = {{Joint universality of the Riemann zeta-function with arbitrary real shift parameters}},
author = {Łukasz Pańkowski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41600},
eprint = {2610.41600},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41600},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Joint universality of the Riemann zeta-function with arbitrary real shift parameters
AU - Łukasz Pańkowski
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41600
UR - https://arxiv.org/abs/2610.41600
AB - We prove that the shifts ζ(s+id_jτ), 1≤ j≤ n, are jointly universal for any nonzero real parameters d_j with pairwise distinct absolute values. These restrictions are best possible. As an immediate consequence, we obtain generalized strong recurrence for arbitrary finite families of nonzero real parameters. Our proof uses Emsalem's generalization of the six exponentials theorem and a new variant of Pecherskii's rearrangement theorem.
ER -