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Math-4181 · mathematics · Number theory

Selberg's orthogonality conjecture

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

The Selberg class $\mathcal S$ consists of functions \[F(s)=\sum_{n\ge1}a_F(n)n^{-s},\qquad \operatorname{Re}s>1,\] satisfying the following conditions. The coefficient $a_F(1)=1$ and, for every $\varepsilon>0$, $a_F(n)=O_{F,\varepsilon}(n^\varepsilon)$, so the series is absolutely convergent for $\operatorname{Re}s>1$. For some integer $m\ge0$, $(s-1)^mF(s)$ is entire of finite order (its absolute value is at most $\exp(C(1+|s|)^A)$ for suitable positive $A,C$). There are $Q>0$, an integer $r\ge0$, $\alpha_j>0$, $\mu_j\in\mathbb C$ with $\operatorname{Re}\mu_j\ge0$, and $\omega\in\mathbb C$ with $|\omega|=1$ such that \[\Phi_F(s)=Q^s\prod_{j=1}^r\Gamma(\alpha_j s+\mu_j)F(s),\qquad \Phi_F(s)=\omega\overline{\Phi_F(1-\overline s)}.\] Here $\Gamma$ is Euler's gamma function, defined initially by $\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}\,dt$ for $\operatorname{Re}z>0$ and then continued meromorphically. Finally, \[F(s)=\prod_{p\text{ prime}}\exp\left(\sum_{k\ge1}b_F(p^k)p^{-ks}\right),\qquad \operatorname{Re}s>1,\] where $b_F(p^k)=O_F(p^{k\theta})$ for some $\theta<\tfrac12$. A nonconstant $F\in\mathcal S$ is primitive if $F=F_1F_2$ with $F_1,F_2\in\mathcal S$ implies $F_1=1$ or $F_2=1$. All $O(1)$ terms below are bounded as $x\to\infty$ and may depend on the indicated functions.

Statement recorded in the supplied source.
For every $F\in\mathcal S\setminus\{1\}$, is there a positive integer $n_F$ such that \[\sum_{p\le x}\frac{|a_F(p)|^2}{p}=n_F\log\log x+O_F(1),\] with $n_F=1$ whenever $F$ is primitive? Moreover, for every pair of distinct primitive $F,G\in\mathcal S$, is it true that \[\sum_{p\le x}\frac{a_F(p)\overline{a_G(p)}}p=O_{F,G}(1)?\] [S2]

Short English statement

Do the coefficients at primes of primitive Selberg-class functions have logarithmic squared norm with coefficient one and bounded cross correlations, with integral squared-norm coefficients for general nonconstant members?

Sources

Catalogue entry

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