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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.
0
votes
0
answers
441
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Spiegel Vermutung: no Siegel zeros iff GRH is equivalent to Goldbach's conjecture
I apologize for using German language in the title, but this question came to my mind after watching the French movie "le théorème de Marguerite" in which the protagonist gets an insight about Goldbac …
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0
votes
1
answer
198
views
Spacings of Satake parameters under Ramanujan conjecture
I would like to know if, under Ramanujan conjecture, the following three distributions are known or conjectured to match:
the distribution of spacings between Satake parameters of an L-function $F$ a …
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2
votes
1
answer
747
views
Does asymptotic Goldbach imply GRH?
It seems to me that a proof of $\alpha_{n}=o(n)$ where the quantity $\alpha_{n}$ is defined in About Goldbach's conjecture together with the main result of https://kyushu-u.pure.elsevier.com/en/public …
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3
votes
Selberg class definition and Riemann hypothesis
The only reference I managed to find is page 116 of Value Distribution of $L$-Functions, by Jörn Steuding (Springer, 2007).
If we assume the existence of an arithmetic
subgroup of $\mathsf{SL}_2(\mat …
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3
votes
0
answers
273
views
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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4
votes
0
answers
295
views
Automorphisms of the ring of periods
The set of periods $\mathcal{P}$ introduced by Kontsevich and Zagier forms a ring, see for example https://en.m.wikipedia.org/wiki/Period_(algebraic_geometry).
Moreover J. Wan introduced in 2011 in ht …
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5
votes
1
answer
2k
views
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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0
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Accepted
Structure of the automorphism group of an L-rig
This an answer following an argument from Wojowu: as we require the equality $\epsilon_{g(\varphi),s}=\epsilon_{\varphi,s}$ to hold for all $(g,\varphi,s)$, and thus for all $s$, this means that $g(\v …
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-3
votes
1
answer
199
views
Structure of the automorphism group of an L-rig
This question is a follow-up to Are there infinitely many L-rigs? which is already pretty convoluted.
Define the $\varphi$-evaluation morphism at a complex number $s$ as $\epsilon_{\varphi,s}:F\mapsto …
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0
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0
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286
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Is $\operatorname{Aut}(\mathcal{M})$ a fundamental group in Grothendieck's sense?
This question is a follow-up to Are there infinitely many L-rigs? and to Is an automorphic form of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ determined by its L-function?.
I copy paste a deepl …
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-4
votes
Functional equation and/or growth estimates for a shifted L function
This preprint by Kaczorowski and Perelli may contain the pieces of information you're looking for: https://arxiv.org/abs/1911.10497
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1
vote
1
answer
243
views
What's the motivation for the $3$ appearing in Iwaniec and Kowalski's definition of the anal...
In their book Analytic Number Theory, Iwaniec and Kowalski, on page 95, define the analytic conductor by the following formula:
$\displaystyle{{\frak{q}}_{\infty}(s)=\prod_{j=1}^{d}\left(\vert s+\kapp …
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3
votes
1
answer
1k
views
Are there infinitely many L-rigs?
$\DeclareMathOperator{\Q}{\mathbb{Q}}$Call "L-rig" any class $\mathcal{L}$ of L-functions of automorphic representations of $\operatorname{GL}_{n}(\mathbb{A}_{\Q})$ for some $n$ belonging to the Selbe …
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0
votes
1
answer
221
views
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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3
votes
0
answers
369
views
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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