12 problems

Let $\alpha$ be an algebraic irrational parameter, and let the Hurwitz zeta function be $\zeta(s,\alpha)$. Gonek's universality conjecture. Universality also holds for the Hurwitz…

Let $\nu$ be the Cantor measure used in the paper and let $\widetilde\nu$ be its pushforward to $(0,1)$. Let $\zeta(s,\alpha)$ denote the Hurwitz zeta function. Cantor-weighted Hur…

Equivalent Chowla–Milnor formulation. One has

For a real number $x$ with $0<x\leqslant 1$, let the Hurwitz zeta function be … For $x=1$, write $\zeta(s)=\zeta(s,1)$. Chowla–Chowla conjecture. For any prime number $p$, the…

Let $\alpha=a/q$ be rational with $\gcd(a,q)=1$ and $1\leq a\leq q$, and let the Hurwitz zeta function be … For $k>0$, Sahay's conjecture. There is a constant $c_k(\alpha)$ such th…

Let $S\subset\mathbb{C}$ be Borel, and let $\zeta(s,\alpha)$ denote the Hurwitz zeta function for a shift parameter $0<\alpha<1$. Gaussian value-distribution conjecture. For large…

Let $k\in\mathbb{N}$ and let $0<\alpha\leqslant 1$ be irrational. Write $\zeta(s,\alpha)$ for the Hurwitz zeta function and let $d$ denote the algebraic degree of an algebraic numb…

Let $s=\sigma+it$ be a complex variable, let $\zeta(s,\alpha)$ be the Hurwitz zeta-function, and let … Let $\alpha\in\mathcal{A}$, let $K$ be a compact subset of $D$ with connected…

Let $p$ be a prime, let $k>1$ be an integer, and let the Hurwitz zeta function be … for $0<x\le1$. Milnor's conjecture. The real numbers … are linearly independent over…

Let $q$ and $k$ be positive integers greater than one. Consider the Hurwitz zeta values $\zeta(k,a/q)$ for integers $a$ satisfying $1\le a\le q$ and $(a,q)=1$. Milnor's conjecture.…

Hurwitz zeta moment conjecture. For every fixed real number $k>0$, there is a constant $c_k(\alpha)$ such that

Let $\alpha$ be a parameter with $0<\alpha<1/2$ or $1/2<\alpha<1$, and let Theorem 1 denote the universality statement for the Hurwitz zeta-function in the preceding theorem. Hurwi…