I have been reading about Riemann Zeta function and have been thinking about it for some time.

Has anything been published regarding upper bound for the real part of zeta function zeros as the imaginary part of the zeros tend to infinity?


  • $\begingroup$ Well, the conjecture is that all the real parts of critical strip zeroes are equal to 1/2. anyway, have a look at (sadly now nearing its end) ZetaGrid. $\endgroup$ Oct 5 '10 at 8:35
  • $\begingroup$ I feel bad about ZetaGrid too... :( $\endgroup$ Oct 5 '10 at 8:52

The term in analytic number theory is "zero-free regions". Any proof of the prime number theorem will produce such a region, and the region is equivalent to the error term in bounds for $\pi(x)$ and to the lower bounds in nonvanishing theorems $|\zeta (1+it)|> 0$. At present, the known zero-free regions are asymptotic to the line $Re(s)=1$: at height $h$ all zeros are at least at distance $d(h)$ from the line with $\lim_{|h| \to \infty} d(h) = 0$.

(Added: on the subject of later improvements, if any region not asymptotic to $Re(s)=1$ were demonstrated it would be a giant advance in number theory, comparable to Wiles' proof of the modularity conjecture and Fermat's Last Theorem. In the analogous function field case there is an algebraic technique for boosting Beta < 1 to Beta < (1/2)+epsilon, and the latter is the Riemann hypothesis. You can be sure that if a zero free region were proven that had finite distance from the boundary of the critical strip, we would all have heard about it. Some known bounds are listed at http://www.math.uiuc.edu/~ford/wwwpapers/zeros.pdf )

  • 1
    $\begingroup$ Great answer... I wanted to know exactly this... "At present, the known zero-free regions are asymptotic to the line Re(s)=1" $\endgroup$ Oct 6 '10 at 4:44
  • $\begingroup$ Im sorry but there are better bounds known then asymptotic to Re(s) = 1 ? See the wiki pages on zeta , RH and PNT. $\endgroup$
    – mick
    Dec 10 '13 at 12:17

I would suggest you read the book "Riemann's Zeta Function" by H.M.Edwards.

As to your question, this is one theorem (from the above book)

De La Vallee Poussin's Theorem (1899) : There exist constants $c > 0, K > 1$ such that

$$ \beta < 1 - \frac{c}{\log \gamma}$$

for all roots $\rho = \beta + i \gamma$, in the range $\gamma > K$.

I am pretty sure there will be more such theorems (the book itself mentions that the above theorem has been improved upon). The book also mentions that (as of 1974) that the bound $\beta < 1$ (real part of the root) has not been improved!

Hope that helps.

  • 2
    $\begingroup$ I second the recommendation of Edwards's book. $\endgroup$
    – Matt E
    Oct 6 '10 at 4:19
  • $\begingroup$ Thanks for the recommendation!!! I have the Edwards book. I'll look into it. $\endgroup$ Oct 6 '10 at 4:42

De La Vallee-Pousin's theorem has been improved by Korobov and Vinogradov in the 50's and I believe their result is the strongest known asymptotic zero free region, cf. The Riemann zeta-function: Theory and applications by Alexandar Ivic. One can find more recent papers but my impression is that they don't touch the main exponents and so relatively tractable problems seem to be improving the constants, or giving explicit bounds on the constants etc.

  • 1
    $\begingroup$ What is the title of Ivic's book, if you don't mind? $\endgroup$ Dec 23 '10 at 7:47
  • 1
    $\begingroup$ J.M., the book title is The Riemann zeta-function: Theory and applications by Alexandar Ivic. (Actually the c in Ivic has an accent over it). $\endgroup$
    – timur
    Dec 24 '10 at 3:10

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