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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.
8
votes
1
answer
1k
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Would Elliott-Halberstam conjecture follow from GRH?
The Wikipedia article about Elliott-Halberstam (EH for short) conjecture says that the so-called Bombieri-Vinogradov theorem, which is a weaker form of EH conjecture, is in some sense an averaged form …
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5
votes
1
answer
2k
views
Does $ M(x)=O(\sqrt{x}) $ if and only if the De Bruijn-Newman constant is negative?
The Riemann hypothesis is equivalent to the assertion that the De Bruijn-Newman constant $ \Lambda $ , as defined in https://www.sciencedirect.com/science/article/pii/S0001870809001133/pdf?md5=d2b0cb …
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5
votes
1
answer
2k
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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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4
votes
1
answer
291
views
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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4
votes
References for general Hasse-Weil zeta function
This recent preprint may be of interest for you, as the authors first consider L-functions and then find back the algebraic variety they come from.
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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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4
votes
3
answers
841
views
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
245
views
Is the imaginary part of $t\mapsto\zeta(1/2+it)$ close to the derivative of its real part?
Plotting $t\mapsto\zeta(1/2+it)$ on Wolfram alpha, it seems that the maxima of its real part are close to the zeros of its imaginary part, while the maxima of the latter seem close to the inflection p …
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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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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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3
votes
1
answer
1k
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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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3
votes
L-functions of Calabi-Yau varieties
Sorry for answering my own question, but it may be useful to some people. It seems, judging by http://arxiv.org/pdf/1301.2225v2.pdf, that the "right" notion of L-function for a Calabi-Yau variety is n …
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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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2
votes
1
answer
242
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 d …
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