46 problems

Let $\mathfrak{L}_f(s)=\sum_{n=1}^{\infty}f(n)n^{-s}$ be a member of the Selberg class $\mathcal{S}$, with coefficients $f(n)$. Ramanujan's conjecture. One has…

Let $\mathcal S$ be the Selberg class, viewed as a multiplicative semigroup, and call $F\in\mathcal S$ primitive if it is not a product of non-trivial functions in $\mathcal S$. Ev…

Let $F$ and $G$ be primitive functions in the Selberg class, let $t(m;F)$ denote the ordinates of the non-trivial zeros of $F$, and define … where $w(u)=4/(4+u^2)$. Murty–Perelli c…

Let $\mathcal S$ be the Selberg class. Let $F,G\in\mathcal S$ be primitive functions with Dirichlet coefficients $a(n)$ and $b(n)$, respectively, and define $\delta_{F,G}=1$ if…

Let $k\geq2$ and $2\leq d_1<\cdots<d_k$ be fixed integers. For each $j$, let $\mathfrak{L}_{f_j}(s)\in\mathcal{S}$ have degree $d_j\geq2$, and let $\ell_j\geq1$ be fixed; put…

Let $k\geq2$ and $2\leq d_1<\cdots<d_k$ be fixed integers. For each $j$, let $\mathfrak{L}_{f_j}(s)\in\mathcal{S}_{\mathrm{real}}$ have degree $d_j\geq2$, and let $E_{f_j}(x)$ be i…

Let $k\geq2$ and $2\leq d_1<\cdots<d_k$ be fixed integers. For each $j$, let $\mathfrak{L}_{f_j}(s)\in\mathcal{S}$ have degree $d_j$, and let $\ell_j\geq1$ be fixed; put…

Let $\ell\geq2$ be fixed. Let $\mathfrak{L}_f(s)\in\mathcal{S}$ have degree $d\geq2$, and let $\mathfrak{L}(s)\in\mathcal{S}$ satisfy … for some $\eta>0$, where $P$ has degree $m$.…

Let $\ell\geq2$ be a fixed integer. Let $\mathfrak{L}_f(s)\in\mathcal{S}$ have degree $d\geq2$, and let $E_f(x)$ denote its error term in the associated summatory asymptotic formul…

Let $\mathcal{S}$ denote the Selberg class. For $\mathfrak{L}_f(s)\in\mathcal{S}$, write its degree as $d=2\sum_j\lambda_j$, where the $\lambda_j$ occur in its functional equation.…

Kaczorowski's degree-equals-degree conjecture. For every $L\in\mathcal{S}_{\mathrm{poly}}$, one has

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Let $L(s)$ be an $L$-function in the Selberg class, and let $d(L)$ denote its degree. Integer-degree conjecture. The degree is a non-negative integer: … The conjecture is known whe…

Let $\mathcal{L}\in\mathcal{S}$, and let $a(p)$ denote its prime coefficients. Let $\widehat{\mathcal{C}_{\mathcal{L}}^{P_1}}(x)$ and $\widehat{\mathcal{C}_{\mathcal{L}}^{P_2}}$ sa…

Let $\mathcal{L}$ belong to the subclass $\mathcal{S}'$ of the Selberg class, and let $N_\mathcal{L}(\sigma,T)$ denote the number of zeros…

Let $F(s)$ be an $L$-function in the Selberg class $\mathcal S$. It has a polynomial Euler product if, for $\operatorname{Re}(s)>1$, … where the $\gamma_i(p)$ are complex numbers a…

Let $\mathcal S$ denote the Selberg class, and let $\mathcal S^*$ be the subclass consisting of functions satisfying the Selberg-class axioms together with a polynomial Euler produ…

Generalized Selberg conjecture A.

Let $\Gamma \subset \operatorname{SL}(2,\mathbb{R})$ be a discrete, co-finite group with cusps, let $\phi$ be a Maaß form for $\Gamma$, and let … be its associated $L$-function. A…

Let $\mathcal{L}\in\mathcal{S}$, where $\mathcal{S}$ is the Selberg class, and let … be its degree. The degree is well-defined, and $\mathrm{d}_{1}=0$ while $\mathrm{d}_{\zeta}=1$;…

Let $\mathcal{S}$ be the Selberg class of functions satisfying the standard Selberg-class axioms, and let $\mathcal{SP}$ be the subclass whose members have a polynomial Euler produ…

Let ${{\mathfrak A}}^{\#}_d$ denote the set of functions $F(s)$ in the extended Selberg class ${{\mathfrak A}}^{\#}$ whose degree is $d_F=d$, where $d_F$ is the degree determined b…

Let ${\mathcal S}^\sharp$ be the extended Selberg class, let $F\in{\mathcal S}^\sharp$ have degree $d$, and let its functional equation have conductor, root number, and $H$-invaria…

Let ${\mathcal S}$ be the Selberg class of $L$-functions, and let a primitive function mean a primitive element of ${\mathcal S}$. Automorphic $L$-functions include those attached…