22 problems

Let $G_2(n)=\sum_{k+m=n}\Lambda(k)\Lambda(m)$ for positive integers $n$, where $\Lambda$ is the von Mangoldt function, and define … Under the Riemann hypothesis, $\Phi_2(s)$ admits…

Let $L(s)=\sum a_n n^{-s}$ be an ordinary Dirichlet series with entire continuation of order at most $1$, and let $\mu(\sigma)$ denote its order function on vertical lines. Refined…

Let $L(s)=\sum a_n n^{-s}$ be an ordinary Dirichlet series with entire continuation of order at most $1$, or meromorphic continuation with finitely many poles. Define the order fun…

Let $S$ be a finite subset of the integers at least $2$, and let $t$ be a real number. Erdős–Ingham conjecture. … The paper's concluding remarks explain that the elementary approac…

Zeta-type relation. The completed function may satisfy a functional equation of the form

Pole-order conjecture. The series $D_{\Q,G}(s)$ should have a meromorphic continuation to

Helson's conjecture. The function $f_\chi$ never has any zeros in its half-plane of convergence.

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.…

Let $N$ be a positive integer, and let $f$ be an arithmetic function with period $N$ such that $f(n)\in\{-1,1\}$ for $n=1,2,\dots,N-1$ and $f(n)=0$ whenever $n\equiv 0\pmod{N}$. Er…

M. Stein's conjecture. One has $\dim_{\mathbb{Q}}V_m=m$ for every positive integer $m$. Moreover, for every $\left(t_1,\ldots,t_{4m}\right)\in V_m$,

Bounded-coefficient Dirichlet-series conjecture. Every function in this class has no poles to the right of the critical line $\operatorname{Re}(z)=1/2$.

Let $0<p\leq2$, and let $\varphi_{1/p}$ be the Dirichlet series defined in the paper by the referenced formula. The space $\mathcal{H}^p$ denotes the corresponding Hardy space of D…

Let $p\leq2$, let $\mathcal{H}^p$ denote the Hardy space of Dirichlet series, and let $\varphi_{1/p}$ be the Dirichlet series defined in the source by equation. Dirichlet-series Ha…

Strict weak-product inclusion conjecture. The inclusion between the standard weak product and its skew counterpart is strict:

Let $f(s)=\sum_{n=1}^{\infty}a_n e^{-\lambda_n s}$ be a Dirichlet series that is meromorphically continuable to the left of its half-plane of absolute convergence. Linnik–Ibragimov…

Let $\zeta(s)$ be a generalized Dirichlet series absolutely convergent on the half-plane $\{z: \operatorname{Re} z\ge 1\}$, and suppose that it has a simple zero at $s=1$. A genera…

Irreducibility conjecture. If $G$ is a finite simple group, then $R_2(G,s)$ is irreducible in $\mathcal{R}$. The examples in the paper show irreducibility for particular finite sim…

Gun–Murty–Rath conjecture. The number

Let $a_1,a_2,a_3,\ldots$ be a sequence of complex numbers, and let $\{b_n\}$ be its partial sum sequence, assumed to be bounded. Choose a hyperreal number system and define…

Average quadratic prime-pair Dirichlet-series conjecture. For $\sigma>1/4$ and $N\to\infty$,

Conditional-convergence conjecture. For an entire $L$-function,

Let $C_n$ be the matrix discussed above, with two large real eigenvalues and $\lfloor \log_2 n\rfloor-1$ remaining small nontrivial eigenvalues. For a small nontrivial eigenvalue,…