5
$\begingroup$

Sometime ago, I read somewhere (should be in Titschmarsh) that, if $N(\sigma, T)$ denotes the number of zeros of the Riemann zeta function $\zeta(s)$ with $\Re(s)\geq \sigma>1/2$ up to height $T>0$, then some result of Ingham says $$N(\sigma, T)\ll T^{\frac{3(1-\sigma)}{2-\sigma}}\log^{5}T.$$

Where can I find a proof of this result? A Google search didn't yield much.

$\endgroup$
  • $\begingroup$ Results of this type are called zero-density theorems, if you want to look for further developments. $\endgroup$ – Greg Martin Feb 9 at 0:48
3
$\begingroup$

According to Titchmarsh p.236, the result appears in

Ingham, A. E. On the estimation of N(σ,T). Quart. J. Math., Oxford Ser. 11, (1940). 291–292.

Link: https://doi.org/10.1093/qmath/os-11.1.201

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.