Multiplicity Constants for Rough Numbers in Prime Gaps
Abstract
Let pn be the nth prime and gn=pn+1−pn. Following Gafni and Tao, an integer m with pn<m<pn+1 is rough for the gap if its least prime factor is at least gn; let Mn be the number of such integers. Gafni and Tao proved that the number of n with X≤ pn≤ 2X and Mn=0 is O(X/log2 X), and that it is asymptotic to cX/log2 X under a form of the Dickson-Hardy-Littlewood prime tuples conjecture. We study Nr(X), the number of n with X≤ pn≤ 2X and Mn<r. For every r≥ 1 we define a constant cr by an endpoint-conditioned local sieve model, with c1=c, and we prove unconditionally that cr=hr*+Oε(r1/2(log r)1+ε) for r≥ 3, where hr* is the least even h≥ 2 with h∏p<h(1−1/p)≥ r. Consequently cr=Hr(1+O(e−κ√(log r))) for an absolute constant κ>0, where Hr>e solves e−γHr/log Hr=r (γ is Euler's constant), and cr=eγr(log r+loglog r+γ+(loglog r+γ)/log r+O((loglog r)2/(log r)2)). The proof combines the Montgomery-Vaughan moment bounds for reduced residues in short intervals with an exact removal of the endpoint conditioning. For each fixed r, the argument of Gafni and Tao extends to give Nr(X)≪r X/log2 X unconditionally, and Nr(X)∼ crX/log2 X under their conjecture. Exact enumeration of the local model gives c≥ 2.7413.
Related
- Deterministic Determination of Axial Constants, Sectional Regularities and Annihilator Numbers
- Bounds on the Threshold Ramsey Multiplicity of Ramsey Numbers with Many Colors
- On Maximal Prime Gaps
- Prime and Touchard Congruences of Mixed-Type Bell Numbers
- The topological Bárány-Larman conjecture for prime numbers
BibTeX
@misc{ugurdogan2026multiplicity,
title = {{Multiplicity Constants for Rough Numbers in Prime Gaps}},
author = {Bahar Uğurdoğan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10582},
eprint = {2610.10582},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10582},
}
RIS
TY - UNPB
TI - Multiplicity Constants for Rough Numbers in Prime Gaps
AU - Bahar Uğurdoğan
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.10582
UR - https://arxiv.org/abs/2610.10582
AB - Let p_n be the nth prime and g_n=p_n+1−p_n. Following Gafni and Tao, an integer m with p_n<m<p_n+1 is rough for the gap if its least prime factor is at least g_n; let M_n be the number of such integers. Gafni and Tao proved that the number of n with X≤ p_n≤ 2X and M_n=0 is O(X/log^2 X), and that it is asymptotic to cX/log^2 X under a form of the Dickson-Hardy-Littlewood prime tuples conjecture. We study N_r(X), the number of n with X≤ p_n≤ 2X and M_n<r. For every r≥ 1 we define a constant c_r by an endpoint-conditioned local sieve model, with c_1=c, and we prove unconditionally that c_r=h_r^*+O_ε(r^1/2(log r)^1+ε) for r≥ 3, where h_r^* is the least even h≥ 2 with h∏_p<h(1−1/p)≥ r. Consequently c_r=H_r(1+O(e^−κ√(log r))) for an absolute constant κ>0, where H_r>e solves e^−γH_r/log H_r=r (γ is Euler's constant), and c_r=e^γr(log r+loglog r+γ+(loglog r+γ)/log r+O((loglog r)^2/(log r)^2)). The proof combines the Montgomery-Vaughan moment bounds for reduced residues in short intervals with an exact removal of the endpoint conditioning. For each fixed r, the argument of Gafni and Tao extends to give N_r(X)≪_r X/log^2 X unconditionally, and N_r(X)∼ c_rX/log^2 X under their conjecture. Exact enumeration of the local model gives c≥ 2.7413.
ER -