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91 views

### 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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96 views

### dimension of a scheme and degree of an L-function [closed]

I try to understand correctly the notion of scheme, as Serre in the second volume of his Oeuvres defines zeta and L-functions in this context. What seems interesting to me is that he states a theorem ...

**2**

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**1**answer

164 views

### 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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137 views

### Does the proportion of non trivial zeroes of given real part increase on [0,1/2] for all L-functions?

Here what I call an L-function is either an element of the Selberg class or an automorphic L-function. For such a function $F$, $x\in [0,1/2]$ and $T\gt 0$, let's define $\delta_{F,T}(x)$ such that ...

**0**

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49 views

### Arguments of Dirichlet coefficients of prime index of primitive elements of the Selberg class

Let $F$ and $G$ be two primitive elements of the Selberg class such that for $\Re(s)>1$, $\displaystyle{F(s)=\sum_{n>0}\dfrac{a_{F}(n)}{n^s}}$ and ...

**3**

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**1**answer

168 views

### 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 ...

**-3**

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**1**answer

374 views

### 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 ...

**2**

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**1**answer

249 views

### 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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84 views

### 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 ...

**2**

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39 views

### 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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41 views

### 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 ...

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112 views

### Invariants of an L-function

For any element $F$ of the Selberg class, one can define invariants of $F$ such as the degree, the conductor, the h- invariants, etc. My question is: these any known set of invariants such that a ...

**-2**

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**1**answer

162 views

### 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 ...

**3**

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259 views

### Automorphisms of the L-function associated to an elliptic $\mathbb{Q}$-curve

Edited after Noam Elkies' comment: From what I understand (very few actually), there exist elliptic curves defined over some number fields $\mathbb{K}$ Galois over $\mathbb{Q}$ which are isogenous to ...

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**2**answers

595 views

### 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 ...

**2**

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**1**answer

339 views

### “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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**1**answer

123 views

### 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 ...

**3**

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**0**answers

143 views

### 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 ...

**2**

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**0**answers

380 views

### 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 ...

**1**

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**0**answers

158 views

### 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 ...

**2**

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**1**answer

422 views

### 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 ...

**2**

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119 views

### 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 ...

**2**

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272 views

### 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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112 views

### Selberg's orthonormality conjecture and permutations

Let $(F_n)_{n>0}$ be an enumeration of the primitive elements of the Selberg class. Let $F_i$ and $F_j$ be such primitive L-functions. What is known about the group $G(i,j)$ of permutations ...

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525 views

### 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 ...