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In my thesis, i stumbled across the following problem:

Define $$F(\sigma, T)=\frac{1}{T}\int_{-T}^{T} \frac{1}{|\zeta(\sigma+it)|^2} \mathrm{d}T$$ where $\zeta$ denotes the Riemann zeta function.

Is the statement that $\lim_{T\rightarrow \infty} F(\sigma, T)=O(1)$ equivalent to the statement that $\zeta(s)\neq 0$ for $\Re(s)\geq\sigma$ ?

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  • $\begingroup$ Might it be possible that, for example, there are zeros of $\zeta$ with real part $\frac34$, yet $F(\frac23,T)$ is bounded? $\endgroup$
    – Greg Martin
    Commented Apr 7, 2019 at 17:29
  • $\begingroup$ @GregMartin, i think that's exactly what my question is... $\endgroup$
    – macgucci
    Commented Apr 7, 2019 at 17:30
  • $\begingroup$ Let me rephrase then: what reason do you have to think that $F(\frac23,T)$ should know about zeros with real part $\frac34$? $\endgroup$
    – Greg Martin
    Commented Apr 7, 2019 at 17:32
  • $\begingroup$ @GregMartin, indeed $F(\sigma, T)$ seems to have nothing to do with zeos with real part $>\sigma$. But what of those with real part equal to $\sigma$ ? $\endgroup$
    – macgucci
    Commented Apr 7, 2019 at 17:39

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