A remark on a theorem of A. E. Ingham.

K G Bhat (1) , K Ramachandra (2)

1 NIAS - National Institute of Advanced Studies
2 TIFR - Tata Institute for Fundamental Research

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 $\zeta(s)$.

Mots clés: Analytic Number Theory; Riemann Zeta-function; Hardy; Hurwitz Zeta-function; Weyl; Littlewood

Domaines: Mathématiques [math]

K G Bhat, K Ramachandra. A remark on a theorem of A. E. Ingham.. Hardy-Ramanujan Journal, 2006, Volume 29 - 2006, pp.37 - 43. ⟨10.46298/hrj.2006.155⟩. ⟨hal-01111487⟩