# Questions tagged [selberg-class]

Questions about Selberg class and the related conjectures such as the analogue of Riemann Hypothesis, Selberg's orthonormality conjecture, degree conjecture, general converse conjecture that says the Selberg class exactly consists of automorphic L-functions, etc.

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### Bounds for the logarithmic derivative in the Selberg Class

Let $F \in \mathcal{S}$, where $\mathcal{S}$ is the set of $L-$functions in the Selberg Class. Are there established upper and lower bounds for $$\left|\frac{F^{'}(s)}{F(s)}\right|,$$ where $s = \...

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### Selberg class definition and Riemann hypothesis

Looking at the Selberg class definition on Wikipedia, under "Comment on definition", there is this paragraph:
"The condition that the real part of $\mu_i$ be non-negative is because ...

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### Does the Ramanujan-Petersson condition correspond to a Fourier type property?

The Ramanujan-Petersson is one of the requirements used in Selberg's class of L-functions, and as such is a necessary condition for the Riemann Hypothesis to hold. The general converse conjecture ...

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### Does Rankin-Selberg convolution preserve primitivity?

Call $L$-function any element of an L-rig (see Are there infinitely many L-rigs? for a definition). Suppose $F$ and $G$ are two primitive L-functions. Is $F\otimes G$ itself primitive? If yes, does ...

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### Are there natural Dirichlet series whose completions have poles in the region of absolute convergence?

The Selberg class of $L$-functions are Dirichlet series
$$ L(s, f) = \sum_{n \geq 1} \frac{a(n)}{n^s}, $$
satisfying certain properties that can be abbreviated as analyticity, a Ramanujan conjecture, ...

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### Does this particular L-series built from L-functions of prime degree define an L-function?

Throughout this question, I call 'L-function' any automorphic L-function belonging to the Selberg class.
Suppose $ (F_i)_{(i>0)} $ is a sequence of L-functions with $ F_i $ of degree $ p_i $ ...

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### Known degrees of L-functions F and G whose Rankin-Selberg convolution is an L-function

Calling '$L$-function' any automorphic $L$-function belonging to the Selberg class, what are the known $L$-functions $L(s,F)$ and $L(s,G)$ of respective degrees $d$ and $d'$ such that the Rankin-...

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### Asymptotic number of zeros for Dirichlet series with functional equation

I think the usual proof for the asymptotic number of zeros of the Riemann zeta function
$$N(T) = \#\left\{\rho : \ \zeta(\rho)=0, \begin{array}{l}\scriptstyle Im(\rho)\ \in\ [0,T]\\ \scriptstyle Re(\...

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### Consequences of Langlands functoriality conjecture

I would like to know whether Langlands' functoriality conjecture implies that the Selberg class coincides with the class of automorphic L-functions and, if so, whether this class is closed under ...

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### gamma-factor of a primitive element of the Selberg class

Suppose $F$ is a primitive element of the Selberg class and $\displaystyle{\prod_{j=1}^{r}\Gamma(\lambda_{j}s+\mu_{j})}$ with $r>1$ the product of Gamma functions appearing in the gamma factor $\...

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### Assymptotics of a Selberg type integral

Does any one know some references/ ideas on how to study the assymptotics as $N$ goes to $\infty$ of the following Selberg type integral
$$\int _{\mathbb R^N} e^{-|x|^2}\ \prod_{1\le i<j\le N} \...

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### seminar about the strong multiplicity one for the Selberg class

Very recently, a seminar took place in Seoul with Haseo Ki as an invited speaker to talk about the strong multiplicity one theorem for the whole Selberg class that he did manage to prove. I would like ...

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### Rankin-Selberg convolution and product of degrees

As I'm kinda obsessed with the Selberg class and because of the general converse conjecture, I'm still trying to get a rough idea of what automorphic representations and their L-functions as well as ...

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### Tensor product of two elements of the Selberg class

Maybe too easy a question for most members of this site, but suppose whenever $F$ and $G$ belong to the Selberg class, then so does $F\otimes G$ where the considered tensor product of $F$ and $G$ is ...

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### Is SOC known to imply the Grand Riemann Hypothesis? [closed]

I'm currently working on a conditional proof of the Grand Riemann Hypothesis, which is based on the assumption that every field automorphism of $\mathbb{C}$ that commutes with an element of the ...

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### Automorphic L-functions over $GL_n( \mathbb{Q} )$

A paper by Kaczorowski & Perelli arXiv:1207.2312 dealing with the elements of the Selber class with degree two suggests that $S_2$ coincides with the automorphic l-functions over $GL_2( \mathbb{Q} ...

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### Has universality been definitely established for the whole Selberg class?

I juste googled to get some insight about universality for l-functions belonging to the Selberg class, but it seems that the proof requires the validity of the prime number theorem for the considered ...

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### Automorphisms of $\mathbb{C}$, Selberg class and surjectivity

Let $F$ be an element of the Selberg class and $\sigma$ be a field automorphism of $\mathbb{C}$ such that $\sigma\circ F=F\circ\sigma$. Let $Fix_{\sigma}$ be the set of all complex numbers $z$ such ...

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### Would countability conjecture and degree conjecture imply unique factorization for the Selberg class?

I already asked this question on MSE but didn't get any comment or answer, so I ask it here.
Assuming countability conjecture as stated in http://www.mat.unimi.it/users/molteni/research/papers-pdf/4-...

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### Can the isometry group of the set of zeros of an L-function $F$ be used to make $F$ automorphic?

I'm still trying to understand the notion of automorphic (L-)function. Due to my lack of knowledge of the subject, this question may appear pretty vague and therefore may not be suitable for MO. I ...

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### On extended Riemann Hypothesis and coefficients of Selberg Class L-functions

There is the conjecture that Selberg Class L-functions satisfy RH.
So that an L-function needs to have its coefficient multiplicatives (plus other conditions: functional equation,...) in order to ...

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### "good" automorphisms of Galois classes of L functions

This question is a follow-up to Galois classes of L-functions. My goal here is to make things clearer.
Definition 1
Let $A$ be a subclass of the Selberg class containing $s\mapsto 1$, closed under ...

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### On properties of coefficients of Selberg Class L-function

The coefficient of Selberg Class L-function satisfy:
$a_n <M_{\epsilon} n^{\epsilon}$ (for any $\epsilon >0$) and the $a_n$ are multiplicative.
So I would like to know if it can be shown that ...

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### Strong automorphisms of the Selberg class

Following Automorphisms of the Selberg class, I define strong automorphisms of the Selberg class by adding as an hypothesis that every invariant of $F$ (i.e all the $H$-invariants, conductor and root ...

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### A possible application of representation theory to Galois classes of L-functions

I define the notion of a Galois class of L-function as follows:
$A$ is a Galois class of L-functions if and only if the following conditions simultaneously hold true:
1) $A$ is a subset of the ...

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### Would a proof of both (G)RH and Montgomery's pair correlation conjecture imply SOC?

Would a proof of both (G)RH and Montgomery's pair correlation conjecture imply SOC?
It seems, judging by the abstract of a 2002 paper of Ram Murty and a possibly Romanian co-author published on www....

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### Would a proof of Ramanujan Conjecture together with other known results about automorphic L-functions imply the Grand Riemann Hypothesis?

The question is in the title: if I'm not mistaken, what misses to prove that all automorphic L-functions belong to the Selberg class is a proof of the Ramanujan Conjecture. But the Selberg class is ...

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### Is a certain group related to a primitive L function isomorphic to $Gal(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$ for some $\ell$?

I define the notion of "Galois class of L functions" in the following way:
$A$ is a Galois class of L functions if and only if the follwing three conditions hold simultaneously:
1) every element ...

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### Automorphisms of S and representations

EDIT July 22nd 2013: I add further details in bolded sentences:
Assuming Selberg's orthonormality conjecture and following Automorphisms of the Selberg class, I define automorphisms of the Selberg ...

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### Galois classes of L-functions

Around one month ago, I posted on math.stackexchange a draft I wrote in which I define the notion of Galois class of L-functions: see https://math.stackexchange.com/questions/280876/definition-of-a-...

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### Consequences of the degree conjecture

the title is quite explicit: I would like to know the consequences of the degree conjecture for the Selberg class.
Thank you in advance.

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### Selberg's orthonormality conjecture and density

Let $F$ and $G$ be two primitive functions of the Selberg class, and let $\mathbb{A}$ be the set of values taken by the function which maps a prime number $p$ to $a_{p}(F)\overline{a_{p}(G)}$. $\...