# On Robin's inequality and the zeros of the Riemann zeta function [on hold]

Let $$\zeta$$ denote the Riemann zeta function. By an argument of Robin http://zakuski.utsa.edu/~jagy/Robin_1984.pdf, we know that $$\zeta(\rho)=0$$ for some $$\rho$$ with $$\Re(\rho) \in (1/2, 1/2 + \beta]$$, where $$0<\beta\leq 1/2$$, if and only if there exists some positive constants $$\beta'$$ and $$c$$ such that $$$$\sum_{d|N} d \geq e^{\gamma}N\log \log N + \frac{cN\log \log N}{(\log N)^{\beta'}}$$$$ for infinitely many positive integers $$N$$, where $$\gamma$$ denotes the Euler-Mascheroni constant and $$\beta'$$ can be taken to have any value satisfying $$1/2 -\beta <\beta'<1/2$$.

Notice that if it could be shown that the above inequality is false for some $$\beta'$$, then it must also be false for any $$\beta''<\beta'$$, implying that if $$\zeta(\rho)\neq 0$$ for $$\Re(\rho)=1/2 + \beta$$, then $$\zeta(\rho)\neq 0$$ for $$1/2<\Re(\rho)\leq 1/2 + b$$, where $$0.

Is this a known result ?

## put on hold as off-topic by Yemon Choi, Jan-Christoph Schlage-Puchta, Dima Pasechnik, András Bátkai, Alexey Ustinov27 mins ago

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Jan-Christoph Schlage-Puchta, Dima Pasechnik, András Bátkai, Alexey Ustinov
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• I don't see how this follows from Robin's result - it concerns the existence of zeros with real part in the strip $(1/2,1/2+\beta]$, so I'm not sure how you're using the assumption that $\zeta$ has no zeros on the line $\Re(\rho)=1/2+\beta$? – Thomas Bloom Mar 10 at 9:34
• I'm voting to close this question because of the sock-puppetry (visible to users with 10k+ rep) – Yemon Choi yesterday