**5**

votes

**2**answers

591 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 ...

**3**

votes

**0**answers

518 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 ...

**2**

votes

**1**answer

414 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**

votes

**1**answer

143 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 ...

**2**

votes

**1**answer

245 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} ...

**2**

votes

**0**answers

102 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 ...

**2**

votes

**0**answers

249 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 ...

**2**

votes

**0**answers

136 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 ...

**1**

vote

**1**answer

122 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 ...

**1**

vote

**0**answers

31 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 ...

**1**

vote

**0**answers

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 ...

**1**

vote

**1**answer

330 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 ...

**1**

vote

**0**answers

376 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**

vote

**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 ...

**1**

vote

**0**answers

114 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 ...

**1**

vote

**0**answers

270 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 ...

**0**

votes

**0**answers

129 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**

votes

**0**answers

47 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 ...

**0**

votes

**0**answers

83 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 ...

**0**

votes

**0**answers

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 ...

**0**

votes

**0**answers

110 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 ...

**-2**

votes

**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**

votes

**1**answer

366 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 ...