32 problems

Strict monotonicity conjecture. The function $\boldsymbol D$ is strictly decreasing on $(0,1)$; in particular, $\boldsymbol\rho_0$ is its unique real zero.

A Dirichlet series is a series of the form $\sum_{n\geq1}a_n n^{-s}$, viewed as a function of the complex variable $s$. Linnik–Ibragimov's conjecture. Any reasonable Dirichlet seri…

Let $f\in\mathcal H$ have Dirichlet series expansion $f(s)=\sum_{n=1}^{\infty}a_n n^{-s}$, and let $\chi$ range over the characters considered in the paper. Uniform boundedness con…

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

Infinite upper-rank conjecture. If $R(S)=\infty$, then the Bass stable rank of $\mathscr{H}^\infty_S$ is infinite.

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

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…

For real $s$, let $M_3(s)$ denote the three-dimensional Madelung constant defined by the lattice sum studied in the paper. Zucker's conjecture. $M_3(s)$ may be expressed in terms o…

Local embedding conjecture. The local embedding property should hold only when $p$ is an even integer or $p=\infty$.

Let $P(y)=\sum_{m=0}^n a_m y^m$, and let $k\geq 0$ be the least integer such that $P^{(k)}(1)\neq 0$ after $P(1)=P'(1)=\cdots=P^{(k-1)}(1)=0$. Define … Ki's conjecture concerns the…

Inversion-formula implication. The inversion formula implies the Riemann hypothesis, since it states that the function generated by $1/\zeta(s)$ is $\mu(n)$ for every $s$ in the se…

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

Hardy–Littlewood conjecture. If $\alpha$ is 1-Diophantine but not quadratic irrational, then the line

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…

Non-optimality conjecture. The exponent $(2^\alpha-1)/d$ is not optimal for describing the relevant composition-operator embedding.

Schatten-class symbol conjecture. For every $p>2$ there exists a multiplicative Hankel form $H_\varphi$ in $S_p(\mathscr{H}^2)$ without a bounded symbol.

Erdős's conjecture. One has

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…

Let $M$ be a cancellative monoid with a discrete degree map, let $I_0$ be a subset of $M/\!\!\sim\setminus\{[1]\}$ satisfying $I_0=\min(I_0)$, and let $P_{M,I_0}(\exp(-s))$ and…