My question is twofold:
Does anyone know if there's a connection between the Monster Group and the Riemann Zeta function?
If there is a known connection, then I would be curious to know when and if the number 163 arises in the function, or in other Zeta or Eta functions.
Disclaimer: I am a ley mathematics enthusiast, not a mathematician. I can promise no degree of rigor in this post.
Background: 163 is my favorite number, and I have a question about it. It's the largest Heegner number. The Heegner primes, and especially 163, are really interesting because of their connection to the J Invariant. The J invariant, is in turn connected via "monstrous moonshine" to the "classification of the finite simple groups", which is widely considered to be the largest mathematical project ever conducted. Group theory is the study of symmetry. A $1 million Clay Mathematics Institute prize will be awarded to the first person to prove the existence of a property of the Riemann Zeta function which is related to at least one of its symmetries. Often touted as the "holy grail of mathematics," the Riemann Hypothesis asks for a proof of the unchanging uniformity of the Real part of the function's non-trivial zeros: do they always remain at +1/2, regardless of how enormous the absolute value of the imaginary part might be? We already know that the critical line is an axis of symmetry (see https://en.wikipedia.org/wiki/Riemann_zeta_function#Other_results).
As far as I know there is no rigorous connection between the Monster Group and the Riemann Zeta function, but I believe such a connection exists. The number 163 and the other Heegner numbers seem likely to play some role in any proof of such a connection. Such a proof would not prove the Riemann Hypothesis, but any new links that can be drawn between the hypothesis and other areas of mathematics may prove useful in the eventual construction of a rigorous proof.
The so-called class number problem for the ideal class group includes the classification of the imaginary quadratic number fields with class number $1$. The "last" one in this list is $\Bbb Q(\sqrt{-p})$ for the prime $p=163$. Dirichlet's class number formula connects the class numbers $h(D)$, where $D$ is the discriminant, with $L$-functions $L(s,\chi)$, a generalization of $\zeta(s)$. Deuring proved, that if the classical Riemann hypothesis is false, then the class number $h(D)$ satisfies $h(D) > 2$ for $-D$ sufficiently large. Heibronn proved that if the generalized Riemann hypothesis is false then $h(D)\to \infty$ if $D\to -\infty$.
This is not yet a direct link why $p=163$ arises in RH, but it is true that Gauss listed the following related problems in his book Disquisitiones Arithmeticae together:
The class number satisfies $h(D)\to \infty$ if $D\to -\infty$.
There are exactly $9$ imaginary quadratic fields with class number one, namely $\Bbb Q(\sqrt{-d})$ with $$ d= 1,2,3,7,11,19,43,67,163 $$
There are infinitely many real quadratic fields with class number one.