Effective exponential bounds on the prime gaps

TL;DR

This work develops explicit, effective exponential bounds on the prime gaps by translating fully effective exponential bounds for the first Chebyshev function $\vartheta(x)$ into bounds for the relative gap $g_n/p_n$. It introduces a monotone decaying function $f(x)=a(\ln x)^b\exp(-c\sqrt{\ln x})$ and, via $x_* = \max\{x_0, \exp([2b/c]^2)\}$, derives two concrete bound forms for $g_n/p_n$—a ratio bound ${g_n}/{p_n} < {2a (\ln p_n)^b \exp(-c\sqrt{\ln p_n})}/{1- a (\ln p_n)^b \exp(-c\sqrt{\ln p_n})}$ (when $f_{peak}\le 1$) and a simpler bound ${g_n}/{p_n} < 3a (\ln p_n)^b \exp(-c\sqrt{\ln p_n})$ (in general). The paper organizes results into widely applicable, intermediate, and asymptotically stringent bounds, providing explicit computation rules for $x_*$ (and sometimes $x_{**}$) and showcasing concrete numerical forms such as $\exp(-\sqrt{\ln p_n}/3)$ and $\exp(-\sqrt{\ln p_n}/2)$. This yields practical, verifiable exponential bounds on prime gaps with known coefficients and validity ranges, contributing a valuable toolkit for both analytic and computational number theory. The approach underscores the central role of the exponential factor in achieving stringent, explicit bounds on $g_n$ from bounds on $\vartheta(x)$.

Abstract

Over the last 50 years a large number of effective exponential bounds on the first Chebyshev function $\vartheta(x)$ have been obtained. Specifically we shall be interested in effective exponential bounds of the form \[ |\vartheta(x)-x| < a \;x \;(\ln x)^b \; \exp\left(-c\; \sqrt{\ln x}\right); \qquad (x \geq x_0). \] Herein we shall convert these effective bounds on $\vartheta(x)$ into effective exponential bounds on the prime gaps $g_n = p_{n+1}-p_n$. Specifically we shall establish a number of effective exponential bounds of the form \[ {g_n\over p_n} < { 2a \;(\ln p_n)^b \; \exp\left(-c\; \sqrt{\ln p_n}\right) \over 1- a \;(\ln p_n)^b \; \exp\left(-c\; \sqrt{\ln p_n}\right)}; \qquad (x \geq x_*); \] and \[ {g_n\over p_n} < 3a \;(\ln p_n)^b \; \exp\left(-c\; \sqrt{\ln p_n}\right); \qquad (x \geq x_*); \] for some effective computable $x_*$. It is the explicit presence of the exponential factor, with known coefficients and known range of validity for the bound, that makes these bounds particularly interesting.