I would like to obtain large numbers for a particular Euler product given by:
$\zeta(s) = {\displaystyle \prod_{p} \left( 1-p^{-s} \right)^{-1} }$
for $s = 1 + ia$
$i$ being the imaginary unit.
In particular, I am interested in finding a large value of $\zeta(s)$ for large values - $a$ ~ $o(10^{33})$.
My current method is to loop through different values of $a$ in a brute force way. I hoped there might be some useful mathematical approach to make this search more efficient.
Fantastic question. Here is one attempt. Not clear if there is a better way. (I remark first in passing that the partial zeta sums here are almost periodic functions.) I would divide up the primes $p$ to those at or below $x$ and those above, taking $x$ to be as large as computationally possible, and divide up the sum of the $e^{-i a \log p}/p$ into the two corresponding parts. The high-$p$ part should act like noise and should largely cancel, but "cannot" be controlled in any case. On the low-$p$ part I would produce a record of many "early" values of $a$ for which the low-$p$ sum is large. As a first pass, each of these early values of $a$ could then be used to produce high-$a$ candidates by adding to it large multiples of the product of the early periodicities. In this almost-modular approach, the early zeta sum would remain large (indeed unchanged) and the later zeta sum would almost always be small except for the fact that the periodicities are not commensurate. What is desirable then is that the angles associated with the terms of the early zeta sum should keep "pointing" as much as possible in the same directions as before. Candidates for $a$ could then be sought among values for which the histogram of the angle differences was concentrated around $0$, and this histogram itself could be weighted towards the higher-impact lower values of $p$. Admittedly this still involves much computation. One possible upshot is that if one were indeed reduced to trial and error search, the searching could be substantially restricted to the early-$p$ sum