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That is must there either be no non-trivial zeros off the critical line or infinitely many? I'm sure that no one believes otherwise, but I've never seen a theorem in the literature addressing this. …

Consider an extension of the Riemann zeta function $\zeta(s)$ where $s$ now runs over a non-standard enlargement of the complex plane. Observe that if $s=\sigma + it$ with $\sigma>1$ real and finit …

Following Arias-De-Reyna, the x-ray of an analytic function $f$ means markings on the complex plane, with one color showing the real locus of $f$ and another color the purely imaginary locus. Suppo …

Even though the title of this question pretty much captures what I'd like to know, I'll add two side questions: 1) Is it difficult to get a handle on the totality of functions that arise if one drop …

As in the famous Euler product identity, the primes occur on only one side of the following: $\sum_n \ln(n) x^n = \sum_p ln(p)(\sum_k\frac{x^{p^k}}{1-x^{p^k}})\ .$ My basic question: Does this ident …

Various questions on MO concerning the "surprise" occurrence of the gamma function in the functional equation of the Riemann zeta function got me wondering whether the Gamma function alone suffice for …

Hurwitz's automorphisms theorem bounds the order of the automorphism group of a negatively curved Riemann in terms of the genus. Now suppose a finite group $G$ acts faithfully on a Riemann surface …

Suppose one has function $f(z)$ analytic in the unit disk. Suppose closed loop $L$ lies in the disk except for one point $P$ on the boundary. Then the Cauchy integral theorem generally does not appl …

The following series evaluation $\sum \frac{n^2-1}{(n^2+1)^2}=\frac{1}{2}(1-\frac{\pi^2}{\sinh(\pi)^2})$ seems attractive to me, and has a proof related to the evaluation of $\zeta(2)$. Does this i …

Pedagogically speaking, I see two problems with defining $\Gamma(z)$ (at least for real $z$) by the limit $$\Gamma(z)=\lim_{m\to\infty}\frac{m! m^z}{\prod_{i=0}^m (z+i)}$$ as compared with the formula …

The totality of all holomorphic functions on the unit disk forms some sort of infinite-dimensional complex manifold, where the coefficients of the Taylor expansion might serve as coordinates for the s …

Fix a radius $r \leq 1$. I'm interested in any necessary conditions, or any sufficient conditions, for a subset $S$ of $B(0,r)$, the origin-centered open disk of radius $r$, for $S$ to be the set of …