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.

  • $\begingroup$ 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

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


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