All Questions
Tagged with analytic-number-theory l-functions
154 questions
9
votes
1
answer
1k
views
Functional equation Dedekind zeta function
I'd like to know to what point is it possible to generalize
this method
for obtaining the functional equation for the Dedekind zeta function $\zeta_K(s)$ of a number field ?
Let $\mathfrak{C}$ be ...
CommunityBot
- 1
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 ...
- 6,984
0
votes
0
answers
112
views
Explicit formula for k-central numbers
Given a positive integer $ n $ and assuming Goldbach's conjecture, let $r_{0}(n)$ denote the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are primes. Let $k_{0}(n)$ denote 'the ...
- 6,984
14
votes
1
answer
532
views
Bound for $GL(3)$ symmetric square
Let $\pi$ be an automorphic representation of $GL(3)$ over a number field. Let $a_n$ be the coefficients of $L(s, \pi, \mathrm{sym}^2)$. Do we know if
$$\sum_{n>0} \frac{|a_n|}{n^s}$$
and
$$\sum_{n&...
- 5,089
3
votes
2
answers
344
views
Distribution of zeros of real quadratic Dirichlet L-functions in small intervals
Motivation: Some data gathered on least quadratic nonresidues indicate that the zeros of quadratic Dirichlet L-functions are more evenly spaced than that in general Dirichlet L-functions.
Question. ...
- 5,089
6
votes
2
answers
315
views
Functional equation and/or growth estimates for a shifted L function
Consider the $L$-series defined by
$$L_{\alpha,\chi}(s) = \sum_{n\geq 1} \frac{e^{2\pi i \alpha \Omega(n)} \chi(n)}{n^s} = \prod_p \left(1 - \frac{e^{2\pi i \alpha} \chi(p)}{p^s}\right)^{-1}.$$
It ...
- 71
2
votes
1
answer
508
views
Moments of Dirichlet $L$-functions on the critical line
I'm looking for a reference for some questions related to the moments of Dirichlet characters on the critical line,
$$ M_k(T;\chi) = \int_T^{2T} |L(1/2+it,\chi)|^{2k}\,dt, $$
where $\chi$ is a ...
- 1,354
2
votes
0
answers
104
views
The Guinand-Weil explicit formula for Hecke characters
The Guinand-Weil formula for the Riemann zeta function is
\begin{aligned}&\Phi (1)+\Phi (0)-\sum _{\rho }\Phi (\rho )\\={}&\sum _{p,m}{\frac {\log(p)}{p^{m/2}}}{\Big (}F(\log(p^{m}))+F(-\log(p^...
- 9,501
1
vote
1
answer
249
views
What's the motivation for the $3$ appearing in Iwaniec and Kowalski's definition of the analytic conductor?
In their book Analytic Number Theory, Iwaniec and Kowalski, on page 95, define the analytic conductor by the following formula:
$\displaystyle{{\frak{q}}_{\infty}(s)=\prod_{j=1}^{d}\left(\vert s+\...
- 6,984
0
votes
1
answer
178
views
Why does LMFDB refer to L functions having coefficients of type $a_p-a_{p^2}$ instead of just $a_{p^2}$?
Today I was reading LMFDB (the L-functions and Modular Forms DataBase), and I came across something that confused me. When discussing degree 3 L functions on this page, they assert that all the ones ...
- 26
3
votes
0
answers
217
views
Maass--Selberg for any Eisenstein series on higher rank
Does there exist a Maass--Selberg relation for any Langlands Eisensein series on $\mathrm{GL}(n)$? By any I mean an Eisenstein series which is induced from any standard parabolic with any discrete ...
- 1,692
4
votes
0
answers
288
views
Lower bound on symmetric square L-function
In a paper of Soundararajan, equation (16c) states that for any Hecke eigenform $f$ of weight $k$, the symmetric square L-function at $1$ satisfies the bound
$$(\log k)^{-2}\ll L(1, \mathrm{sym}^2f)\...
- 1,890
22
votes
1
answer
1k
views
Hadamard factorization of L-functions
I have already asked this question here in a different form, but really need an answer.
Let $L(s)$ be a "standard" $L$-function, say with Euler product, functional equation, etc...
(Selberg ...
- 105k
8
votes
2
answers
839
views
On the consistency of the definition of the conductor for automorphic forms
Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}_2(F)$, where $F$ is local non-archimedean. The local conductor associated to $\pi$ can be defined in two usual manners:
By its ...
- 37k
3
votes
0
answers
157
views
On the ''generalised'' Chebyshev psi function
Let $\chi$ be a Dirichlet character mod $q$ and $\Lambda(n)$ be the von Mangoldt function. Let $c(\chi)=1$ if $\chi$ is the principal character, and zero otherwise. Let $\Theta_\chi$ be the supremum ...
- 11.1k
0
votes
1
answer
116
views
Logarithms of $L$-functions of irreducible characters of Galois group
We know that the $L$ functions of Dirichlet characters $\chi$ of $(\mathbb Z / m\mathbb Z)^\times$ satisfy the property that $\log L(s, \chi)$ is holomorphic for $\Re(s) \geq 1$ if $\chi$ is a ...
CommunityBot
- 1
4
votes
0
answers
122
views
Small values of $L(\frac{1}{2}, \chi_{-8d})$
In Section 3 of https://arxiv.org/pdf/0708.3990.pdf Soundararajan shows that there exists infinitely many fundamental discriminants $8d$ for which $L(1/2, \chi_{8d})$ is very small.
His argument runs ...
- 4,746
4
votes
2
answers
448
views
Nonvanishing of central L-values of Maass forms
Are there any results on the proportion of nonzero central L-values of Maass cusp forms? More precisely, I am looking for lower bounds for
\begin{equation} \frac{\#\{\phi_j : \, L(1/2, \phi_j) \neq 0,...
- 21
6
votes
1
answer
337
views
"Sub-logarithmic" zero-free regions from Deuring-Heilbronn/Linnik's repulsion theorem
For each $n\in\mathbb{N}$, let:
$\chi_n\pmod{q_n}$ a real non-principal Dirichlet character ($q_1 < q_2 < \cdots$),
$\beta_n$ the largest real zero of $L(s,\chi_n)$,
$\delta_n := (1-\beta_n)\...
- 825
18
votes
0
answers
740
views
Infinite extensions such that every elliptic curve has finite rank
The comments to this answer seem to make the following claim.
Claim. Let $K$ be the maximal abelian extension of $\mathbf Q$ that is unramified away from $p$ (more generally, away from a finite set $S$...
CommunityBot
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8
votes
1
answer
595
views
Does the symmetric square L-function vanish at one?
Take a cuspidal automorphic representation $\pi$ of $GL(3)$ over a number field. My question is quite straightforward and can be related to this one :
Can $L(1, \pi, \mathrm{sym}^2)$ be zero? If ...
- 105k
2
votes
2
answers
284
views
Upper bound of summation $\sum_{m < \frac{1}{2}X} \frac{|a(m_1m_2^2)|}{m_1m_2^2} \log\frac{X}{m}$
I am studying the paper M. Ram Murty, V. Kumar Murty: Mean values of derivatives of modular $L$-series, Ann. of Math. (2) 133 (1991), no. 3, 447-475.
Let $L(s)=\sum_{n=1}^{\infty} \frac{a(m)}{m^s}$ ...
- 105k
15
votes
1
answer
1k
views
How do functional equations for zeta functions arise from the structure of a homology group?
I have read in various disparate sources that certain zeta functions satisfy functional equations as a consequence of some structure on some homology group. Here is an example of a quote in this ...
- 148k
5
votes
2
answers
941
views
$B(\chi), L'(1,\chi)/L(1,\chi),\dotsc$
Let $\chi$ be a primitive Dirichlet character of modulus $q>1$. Write, as is customary, $B(\chi)$ for the constant in the expression
$$\frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} = B(\chi) + \sum_\rho ...
5
votes
0
answers
111
views
Archimedean L-factors for symplectic group
Let $\pi$ be an automorphic representation of $GSp(4)$. Provided a representation $r$ of the Langlands dual group of $GSp(4)$ (namely, the standard or the spinor one), it is possible to define a ...
- 927
15
votes
5
answers
2k
views
$|L'(1,\chi)/L(1,\chi)|$
Let $\chi$ be a primitive Dirichlet character $\mod q$, $q>1$. Is there a neat, simple way to give a good bound on $L'(1,\chi)/L(1,\chi)$?
Assuming no zeroes $s=\sigma+it$ of $L(s,\chi)$ satisfy $\...
- 20.2k
4
votes
1
answer
308
views
Effective, explicit bound on $L'(1,\chi)/L(1,\chi)$
Are there effective, explicit upper bounds on $L'(1,\chi)/L(1,\chi)$ in the literature, valid for all Dirichlet characters $\chi$? Due to the possibility of Siegel zeros, I don't imagine one can do ...
- 20.2k
3
votes
2
answers
316
views
Explicit formula: explicit work with general smoothing?
The following is a literature question, in the sense that I already know how to do what I am asking about, and in fact have already done it; now I'd like to write a brief historical overview as an ...
- 3,486
12
votes
3
answers
2k
views
Are L-functions uniquely determined by their values at negative integers?
Are L-functions uniquely determined by their values at negative integers? In another words, is there a sequence of integers $a_1, a_2, a_3, \cdots$ such that
the corresponding L-function
$$L_{\{a_n\}...
- 3,403
3
votes
0
answers
171
views
Similarity between two $L$-functions (Hasse-Weil $L$-function of twisted ellptic curve and Dirichlet $L$-function)
Let $E$ be an elliptic curve over $\mathbb Q$ with conductor $N$ and $E_d$ be its twisted curve by $d$, where $d$ is a fundamental discriminant with $(d,N)=1$. Let $\chi_d$ be a Dirichlet character ...
- 4,713
5
votes
2
answers
433
views
Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms
Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients.
Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know the ...
- 5,089
2
votes
0
answers
147
views
Well-known estimate for $L(s,\chi)$ for $\sigma=\text{Re}s\geq 1/2$
This is a very short question.
Let $s=\sigma+it$ be a complex number with $\sigma \geq 1/2$.
In the paper 'Jutila, Matti. "On the Mean Value of $L(1/2, \chi)$ FW Real Characters." Analysis 1.2 (...
- 663
13
votes
1
answer
763
views
Least quadratic residue under GRH: an explicit bound
Let $m$ be a positive integer and $\chi$ a primitive character mod $m$. Let $x$ be such that $\chi(p)\ne 1$ for all primes $p<x$. Assume GRH. How can one bound $x$ in terms of $m$ ? I do not need ...
- 63.9k
4
votes
2
answers
566
views
Real non trivial zeros of Dirichlet L-functions
When dealing with the prime number theorem in arithmetic progressions, one cannot exclude the possible presence of a real zero close to $1$ for at most one real character mod $q$. On the other hand, ...
- 5,089
4
votes
0
answers
177
views
How many exceptional conductors are there?
We say that a conductor $q$ is exceptional if there is a primitive quadratic character $\chi$ modulo $q$ such that $L(s,\chi)$ has a real zero $\beta$ such that $\beta > 1-c/\log q$ (where $c$ is ...
- 105k
7
votes
1
answer
564
views
Functional equation for general number fields
When it comes to general number fields beyond $\mathbb{Q}$, the litterature is not so abundant in analytic number theory. For instance over $\mathbb{Q}$, for primitve Dirichlet characters modulo $q$, ...
- 8,374
5
votes
0
answers
384
views
A sum over zeros of L-functions in the paper "Chebyshev's Bias"
Let $\varepsilon>0$ be small and
\begin{align*}
\widetilde{F}_{\varepsilon}(\xi):=\frac{4}{\varepsilon}\sum_{0<\gamma\leq \varepsilon^{-2}}\frac{\sin(\gamma \xi)\sin \frac{\gamma \varepsilon}{2}}...
- 63.9k
0
votes
0
answers
126
views
Summation formula for twisted L-function
Does any expert here know something about the summation formula of the Voronoi type for the sum $$\sum_{n\le X} a_{f}(n)\chi(n) e\left(\frac{an}{c}\right)?$$
Here $f$ is a newform of level $N$, $\chi$...
- 353
5
votes
1
answer
1k
views
What is the analytic conductor of this Hecke L-function?
Following Iwaniek and Kowalski, S5.10, page 130 we consider an angle character $\xi_k$ on the Gaussian integers $\mathbb Z[i]$ defined by
$ \xi_k(\mathfrak a) = \left(\frac{\alpha}{|\alpha|}\right)^k $...
- 1,207
1
vote
0
answers
152
views
A mixed of the Dedekind zeta function and the L-function
I have recently come across the following function, which seems like a "mix" between the Dedekind zeta function and the L-function:
$\sum_I\frac{\chi_k(N(I))}{N(I)^s}$
where $\chi_k(n)$ is the ...
3
votes
2
answers
207
views
What is known about gaps between zeros of L-functions?
In many different settings, it is possible to determine statistics about spacings (pair correlation, small gaps, large gaps, champions, etc.), for instance
prime numbers
Laplacian eigenvalues on a ...
- 6,984
14
votes
2
answers
1k
views
What are zeta functions good for?
I know a couple of answers to the above question:
They can be used for point counting over finite fields/estimating the distribution of primes in characteristic 0.
There are various conjectures/...
- 6,039
11
votes
1
answer
328
views
Critical points of Dirichlet L functions
Let $L(s,\chi)$ denote a Dirichlet $L$-function for a real-valued non-principal
character $\chi$. This has limiting value $L(\infty,\chi) = 1$ and we are interested in how this limit is approached ...
- 13.1k
3
votes
0
answers
139
views
L-functions and Hecke theory for general number fields
I often read about analytic number theory on $\mathbf{Q}$ when it turns to explicit computations, so I wonder how much results generalize as they stand and the use of $\mathbf{Q}$ is merely made for ...
- 31
3
votes
1
answer
266
views
Consistency of the notion of conductor of a representation
The notion of analytic conductor of a generic representation of $\mathrm{GL}(n)$ has been defined by Iwaniec and Sarnak, and since then is at the heart of many works in analytic number theory and used ...
- 8,374
6
votes
1
answer
246
views
The L-function of Q(-1/2) and the "number of prime $p\equiv 3$ divisors" function
In the framework of classical motives, there is no such thing as a motive $\mathbb Q(-\tfrac 12)$, i.e. a tensor root of $\mathbb Q(-1)$. There is one, however, in a more general setting of "...
- 12.8k
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=...
- 105k
6
votes
0
answers
230
views
Relation between Gross-Koblitz and Chowla-Selberg formulas
The Chowla-Selberg formula relates the eta function with values of the gamma function at rational numbers. The eta function appears, at least in the proofs I have seen, related to values of $L$-...
- 3,107
3
votes
0
answers
540
views
Questions about the exceptional zeros of Dirichlet $L$-functions
I have couple questions regarding the exceptional zeros of Dirichlet $L$-functions. We have the following result:
There is a constant $c_1 > 0$ such that $L(\sigma, \chi) \not = 0$ whenever
$$
\...
- 3,625
14
votes
1
answer
1k
views
Is the adjoint L-function on GL(m) holomorphic?
Let $\pi$ be an automorphic representation on $\mathrm{GL}(m)/\mathbb{Q}$.
Define $$L(s,\pi,\mathrm{Ad}):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an $L$-function with Euler product of ...
- 105k