Hardy Ramanujan Journal |

Referring to a theorem of A. E. Ingham, that for all $N\geq N_0$ (an absolute constant), the inequality $N^3\leq p\leq(N+1)^3$ is solvable in a prime $p$, we point out in this paper that it is implicit that he has actually proved that $\pi(x+h)-\pi(x) \sim h(\log x)^{-1}$ where $h=x^c$ and $c (>\frac{5}{8})$ is any constant. Further, we point out that even this stronger form can be proved without using the functional equation of $\zeta(s)$.

Source : oai:HAL:hal-01111487v1

Volume: Volume 29

Published on: January 1, 2006

Submitted on: March 3, 2015

Keywords: Littlewood, Weyl, Hurwitz Zeta-function, Hardy, Riemann Zeta-function,Analytic Number Theory,[MATH] Mathematics [math]

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