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Primes in short intervals assuming the Riemann hypothesis
Abstract
In 2001 Baker, Harman, and Pintz proved, for x sufficiently large, the existence of at least one prime in the sub-linear interval [x,x+x1/2 +1/40]. But just how large is sufficiently large? In the current note I shall make the Baker--Harman--Pintz theorem effective, but at an extremely high price. Contingent upon assuming validity of the Riemann hypothesis (RH), I shall show that the Baker--Harman--Pintz theorem holds for x ≥ 3.5× 1090. Furthermore, again assuming RH, I shall also show that for any 0< m≤15.831 and x ≥ 1 there is at least one prime in the interval [x,x+x1/2 +1/m], and that for M≥ 2.29 and y ≥ 1 there is at least one prime in the interval [yM, (y+1)M].
Related
- The Waring-Goldbach problem in short intervals under the Generalized Riemann Hypothesis
- Biases in the distribution of primes in short intervals
- The Poisson Tail Conjecture for primes in short intervals
- Simple critical zeros and distinct zeros of the Riemann zeta function in short intervals
- Almost primes in short intervals and averages of long Dirichlet polynomial approximations to (ζ'/ζ)k
BibTeX
@misc{visser2026primes,
title = {{Primes in short intervals assuming the Riemann hypothesis}},
author = {Matt Visser},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27830},
eprint = {2610.27830},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27830},
}
RIS
TY - UNPB
TI - Primes in short intervals assuming the Riemann hypothesis
AU - Matt Visser
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27830
UR - https://arxiv.org/abs/2610.27830
AB - In 2001 Baker, Harman, and Pintz proved, for x sufficiently large, the existence of at least one prime in the sub-linear interval [x,x+x^1/2 +1/40]. But just how large is sufficiently large? In the current note I shall make the Baker--Harman--Pintz theorem effective, but at an extremely high price. Contingent upon assuming validity of the Riemann hypothesis (RH), I shall show that the Baker--Harman--Pintz theorem holds for x ≥ 3.5× 10^90. Furthermore, again assuming RH, I shall also show that for any 0< m≤15.831 and x ≥ 1 there is at least one prime in the interval [x,x+x^1/2 +1/m], and that for M≥ 2.29 and y ≥ 1 there is at least one prime in the interval [y^M, (y+1)^M].
ER -