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Results tagged with l-functions
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user 13625
Questions about generalizations of the Riemann Zeta function of arithmetic interest whose definition relies on meromorphic continuation of special kinds of Dirichlet series, such as Dirichlet L-functions, Artin L-functions, elements of the Selberg class, automorphic L-functions, Shimizu L-functions, p-adic L-functions, etc.
-1
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1
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237
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Does unique factorization for automorphic L-functions imply a weakened form of Ramanujan con...
Selberg orthonormality conjecture for automorphic L-functions was proven under Ramanujan conjecture, and SOC itself implies unique factorization for those L-functions.
My question is: does the uniq …
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0
votes
0
answers
131
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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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4
votes
3
answers
841
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what is exactly the difference between the Selberg class and the set of Artin L-functions?
The question is in the title: from what I read in the answer to another question, Artin L-functions are conjecturally cuspidal automorphic L-functions for some algebraic group that can be transfered t …
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4
votes
1
answer
291
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References on Erdos conjecture on arithmetic progressions
Erdos conjectured that any set $ A $ of positive integers such that $ \sum_{n\in A}\dfrac{1}{n} $ diverges contains arbitrary long arithmetic progressions. The celebrated Green-Tao theorem is a spec …
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3
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0
answers
369
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Is an automorphic form of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ determined by its...
To an automorphic representation $\pi$ of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ one can associate its L-function $s\mapsto L_{\pi}(s)$.
Is the map $\pi\mapsto L_{\pi}$ bijective?
Edit March …
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-2
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1
answer
207
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Which properties of L-functions can be proven assuming they are objects of a symmetric bimon...
The title says it all : assuming all L-functions are objects of a symmetric bimonoidal category $ (\mathcal{C},\oplus,\otimes,s\mapsto 1,\zeta) $ , where $ \oplus $ stands for the usual product, wh …
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2
votes
1
answer
242
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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 d …
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1
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What is known about gaps between zeros of L-functions?
Typing "gaps between zeros of Selberg class" on Google gives as the fifth research result a survey by Caroline Turnage-Butterbaugh that may be helpful.
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3
votes
0
answers
206
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Are quantities involved in Generalized Ramanujan Conjecture eigenvalues of some unitary oper...
If I'm not mistaken, every automorphic L-function $L(s,\pi)$ verifies $\displaystyle{L(s,\pi)=\prod_{p}L_{p}(s,\pi_{p})}$ where $L_{p}(s,\pi_{p})=\displaystyle{\prod_{j=1}^{m}\big(1-\frac{\alpha_{\pi} …
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2
votes
1
answer
181
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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 $\gamma …
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5
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1
answer
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Does the existence of a Landau-Siegel zero imply the existence of a complex zero off the cri...
The question is in the title: can a Landau-Siegel zero be the only zero off the critical line for a Dirichlet L-function or does its existence imply the existence of a complex non trivial zero in the …
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1
vote
1
answer
183
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Has a universality theorem been proved for the Davenport-Heilbronn L function?
The question is in the title: has a universality theorem in the sense of Voronin been proved for the Davenport-Heilbronn function, or do we expect such a theorem to hold true only for L functions that …
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0
votes
1
answer
221
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Under Ramanujan conjecture, is primitivity equivalent to cuspidality and irreducibility?
Lemma 4.2 in M. Ram Murty, Selberg conjectures and Artin L-functions(1994), states that under Ramanujan conjecture, an irreducible cuspidal automorphic representation of $\operatorname{GL}_{n}(\mathbb …
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0
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1
answer
567
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degree of an isobaric sum
I'm trying to understand a few things about automorphic L-functions. In page 5 of http://arxiv.org/pdf/1401.0390.pdf, the author mentions the isobaric sum decomposition $\pi=n_{1}\pi_{1}\boxplus\cdots …
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3
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0
answers
273
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Have there been recent developments of Booker's approach to L-functions as distributions?
Andrew Booker introduced a framework to study L-functions through distributions in https://arxiv.org/abs/1308.3067v2. This allowed him and others to get new results about zeros of automorphic L-functi …
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