Effective exponential bounds on the prime gaps

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.