# Can exceptionally large primes be used to get information on the roots of $\zeta$?

Take a list $L$ of roots of the $\zeta$ function, like the ones provided by Andrew Odlyzko and plug it into the Prime Counting Function $\pi(x)$ given by $$\pi(x) \approx \operatorname{R}(x^1) - \sum_{\rho\in L}\operatorname{R}(x^{\rho}) \tag{1}$$ with $\operatorname{R}(z) = \sum_{n=1}^{\infty} \frac{ \mu (n)}{n} \operatorname{li}(z^{1/n})$ . By that we can somehow determine primes up to certain extend and accuracy.

Now, projects like GIMPS and others provide us with a bunch of exceptionally large prime numbers, see the largest here.

$\hskip.5in$ Can these large prime numbers be used to get information on the roots of $\zeta$?

And I can imagine that information might come in many ways. For example:

• accuracy of known roots
• bounds on the imaginary part of yet unknown $\rho$'s
• bounds on the real part
• ...
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I don't believe this is possible at all. The existence of large primes doesn't tell us anything we didn't know before, as we expect about 1 out of every $\log x$ integers to be prime anyway. It is conceivable that if we knew $\pi(x)$ in a certain range, then we might be able to fish out bounds on the zeros up to a certain height - but knowing the existence of a single prime doesn't help. –  Eric Naslund Jun 29 '13 at 16:35

hmm, what about the Divergence of the Derivative of the Prime Counting Function. Knowing a large prime $p$ should lead to a divergent expression for $\pi'(p)$ and this depends on the roots $\rho$. So they, the roots, have to behave in such a way the $\pi'(p)$ diverges. –  draks ... Jun 6 '13 at 20:53