I have been reading about Riemann Zeta function $\zeta(s)$ and have been thinking about it for some time. I did some calculations, and reached a conclusion where $\Re(\rho) \le \log_2(3) - 1$ as $\Im(\rho) \to \infty$ where $\rho$'s are the roots of Riemann Zeta function in the critical strip. Anyways, I know its not the place to discuss claimed proofs and similar stuff, but just to give a background of where I am coming from. So straight to the question.

Is there any similar result regarding upper bound ($< 1$) for the real part of the zeros zeta function as their imaginary parts tend to infinity?


  • $\begingroup$ All results on zero-free regions have a strip with width tending to 0 as the imaginary part increases. $\endgroup$ – Charles Matthews Oct 5 '10 at 8:40
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    $\begingroup$ All known results on zero free regions are of the form $\zeta(\sigma+it) \neq 0$ when $\sigma > 1 - f(t)$, for some function $f$ with $\lim_{t \to \infty} f(t) =0$. A bound of the form you describe would be major progress. $\endgroup$ – David E Speyer Oct 5 '10 at 10:49
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    $\begingroup$ Honestly, you almostly certainly have an error. Your past questions have involved a number of elementary errors about analytic number theory and you are working on a problem which is notorious for drawing mathematicians into false proofs. In addition, if you are who I think you are, than 3 of your 6 papers on the arXiv have been withdrawn. (I am glad that you do withdraw errors!) Based on the little I know about you, you really need to find an advisor or mentor who can teach you how to do careful work in analytic number theory. $\endgroup$ – David E Speyer Oct 5 '10 at 11:00
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    $\begingroup$ you may contact Prof.Bhaskar Bagchi of ISI,Bangalore.He was arguably the most interested person in India in RH.or you may contact me at bmat1013@isibang.ac.in. I am from ISI,Bangalore. $\endgroup$ – Koushik Feb 23 '13 at 3:08
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    $\begingroup$ Prof.B.Bagchi spent 20 yrs trying to prove RH. $\endgroup$ – Koushik Feb 23 '13 at 3:08

There is no known non-trivial (less than 1) bound for real parts of Zeta zeros (I guess, it is even called "weak Riemann conjecture" to find such a bound). So, your result is very-very interesting, maybe the most interesting result in mathematics for many years.

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  • $\begingroup$ I was searching for papers having zero free regions... and most of them gave results like $\Re(\rho) > 1 - \beta$. Are you sure there are no such results? (I'll wait before accepting your answer) $\endgroup$ – Roupam Ghosh Oct 5 '10 at 9:36
  • $\begingroup$ EDIT: by "no such results" I mean no results with $\Re(\rho) < c < 1$. Sorry for the typo $\endgroup$ – Roupam Ghosh Oct 5 '10 at 9:39
  • $\begingroup$ I think after Speyer's comments I am ready to accept you answer... Thanks Fedor and Speyer !!! $\endgroup$ – Roupam Ghosh Oct 5 '10 at 11:20
  • $\begingroup$ Here's my paper if anyone is actually interested :DDD docs.google.com/… $\endgroup$ – Roupam Ghosh Nov 11 '10 at 14:54
  • $\begingroup$ If I recall correctly, even proving things like a zero free sliver near Re = 1 would have major ramifications in making effective many of the cryptographic applications of RH (or really GRH). $\endgroup$ – lemiller Jul 27 '16 at 17:30

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