All Questions
128 questions
3
votes
1
answer
276
views
Positivity of partial Dirichlet series for a quadratic character?
Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet ...
- 105k
1
vote
0
answers
111
views
Whether or not the root number of GL$_3\times$GL$_2$ $L$-function $L(s, F \otimes g)$ contains the coefficients $\lambda_g(n)$ of $g$?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ and $q$ be two distinct primes. Let $$\Gamma_0(p)= \left\{ g\in \GL_3(\mathbb{Z}):g \equiv \left(\begin{matrix} \ast &\ast&\...
- 41
8
votes
1
answer
356
views
Average bounds on Rankin-Selberg coefficients for modular forms
Let $f$ be a cuspidal Hecke newform of weight $k$ and level $N$, and denote by $a_f(n)$ its $n$-th Fourier coefficient. The newform $f$ is normalized so that $a_f(1) = 1$. As a consequence of Rankin-...
CommunityBot
- 1
1
vote
0
answers
68
views
Connection between a special integral transform and averages of L-functions
Let $\Gamma = \operatorname{SL}_2(\mathbb{Z})$ and $\mathcal{H}$ be the upper half-plane. For $A>1$, define the truncated Eisenstein series $E_A(z,s)$ as $$E_A(z,s) = \begin{cases} E(z,s), & \...
- 63.9k
4
votes
1
answer
299
views
Non-vanishing of archimedean integral representations
Let $\psi$ denote a non-trivial additive character of $\mathbb{R}$ and $n$ be a positive integer. Let $(\pi,V)$ and $(\pi',V')$ be two irreducible generic Casselman-Wallach representations of $G_n=\...
- 8,374
6
votes
1
answer
642
views
Generalizations of Hamburger's Theorem
(Despite the name, the theorem in question is not a joke nor is it a statement about a delicious food).
An old theorem of Hans Hamburger from 1921, as stated in Marvin Knopp's paper "On Dirichlet ...
CommunityBot
- 1
1
vote
0
answers
133
views
Automorphy of the twisted representation
The Artin reciprocity says that if
$$
\chi: \operatorname{Gal}(K/\mathbb Q) \to \mathbb C
$$
is a 1-dimensional representation of a finite Galois extension $K/ \mathbb Q$, then it corresponds to a ...
3
votes
1
answer
178
views
The lower bound for the automorphic $L$-function $L(s,\pi)$ at the edge of the critical strip $\Re s=1$
Let $\pi$ be any automorphic Maass form on $\text{GL}_m$ of level $N$, say. Assume that the associated $L$-function $L(s,\pi)$ satisfies some good conditions; for example, it satisfies the functional ...
- 177
5
votes
1
answer
162
views
A question on hybrid subconvexity for individual L-functions
Sorry to disturb. I have a question need some explanations from the experts on the MO-website.
As usual, we let $L(f,s)$ be the corresponding $L$-function associated to the newform $f$ on $SL_2(\...
- 4,713
2
votes
1
answer
587
views
Bounds for Dirichlet L-functions
Let $L$ denote a Dirichlet L-function attached to the primitive character $\chi$. What are the best known bounds for $L(\sigma+it, \chi)$?
PS: For $L=\zeta$ and $0\leq\sigma\leq 1$, i'm aware of a ...
- 105k
4
votes
1
answer
247
views
Do Artin L functions have polynomial growth in the critical strip?
Given an irreducible representation $\rho$ of the Galois group $G$ of a number field $K$ over $\mathbb{Q}$, we have the associated Artin $L$ function which we denote by $L(s, \rho)$. By Brauer ...
- 5,089
4
votes
1
answer
245
views
Conditional convergence of Artin $L$-functions
Let $k$ be a number field and $V$ a non-trivial irreducible Artin representation over $k$. Consider the associated Artin $L$-function with corresponding Euler product decomposition $L(V,s)= \prod_v ...
- 105k
8
votes
1
answer
401
views
Is $\frac{1}{L(1+it)}$ unbounded?
Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding L-function. Is $$\frac{1}{L(1+it, \chi)}$$ unbounded for $t \in \mathbb{R}$? I'm aware that this is true if $L=\zeta$, but I'm ...
- 105k
1
vote
1
answer
258
views
Classification of L functions and Dirichlet series by poles
I am interested in the connection between particular Dirichlet series' abscissa of convergence and the poles of L-functions.
Let $D(z) = \sum_{n=1}^\infty\frac{a_n}{n^z}$ be a Dirichlet series ...
- 105k
9
votes
1
answer
533
views
Effective bound of $L(1,\chi)$
Let $d$ be a fundamental discriminant and let $\chi$ be the associated primitive real character of modulus $\vert d \vert$. Assuming GRH, Littlewood proved that as $\vert d \vert$ grows large,
$$L(1, ...
0
votes
1
answer
179
views
A question about the setup of zero density estimates for Dirichlet $L$-functions
For $L(s,\chi)= \sum_{n \geq 1}\frac{\chi(n)}{n^s}$, where $s = \sigma + it$, we define the function $N(\sigma, T, \chi)$ which counts the zeros $\rho = \beta + i\gamma$ for which $L(\rho, \chi) =0$ ...
- 105k
2
votes
0
answers
141
views
Analyticity of unramifed part of Rankin-Selberg $L$-functions on $\Re(s)=1$
I have only a little knowledge about automorphic representations and $L$-functions. Now I am reading the textbook of Goldfeld and Hundley on automorphic representations, and also planning to read the ...
- 663
2
votes
1
answer
264
views
'$\times$' or '$\otimes$' when writing $L$-functions?
Recently, I came across the Langlands correspondence theorem, there is the following line:
$$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$
where $\sigma$ and $\tau$ are ...
- 105k
15
votes
1
answer
738
views
Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$
Euler proved
$$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$
where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
- 105k
1
vote
0
answers
101
views
Motivation behind a result of Munshi on nonvanishing of L-functions in families of elliptic curves
In this article in Compositio (2011), Munshi proves a mean value result for
$$ \sum_{d} r(d) \Lambda^{(l)}(1/2,f,\chi_d) F(d/Y),$$
where here $f$ is a primitive holomorphic form of level $q$ with ...
- 1,354
4
votes
1
answer
229
views
Abscissa of convergence of the $\tau$ Dirichlet series
Define the $\tau$ Dirichlet series $L$ by
$$L(s)=\sum_{n=1}^\infty \frac{\tau (n)}{n^s}$$
where $\tau$ is defined by
$$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$
where $|q|\lt 1$....
- 105k
3
votes
1
answer
223
views
Non-Schwartz test functions for the explicit formula for L-functions
The statements of the explicit formula for L-functions that I am aware of require the test function to be a Schwartz function (see, e.g., equation (4.11) in Section 4 of Low lying zeros of families of ...
- 5,089
1
vote
1
answer
214
views
Bound on Von Mangoldt for automorphic L-functions
Following the notation in Iwaniec+Kowalski, let $L(f,s)$ be an L-function. Denote
$$\frac{L'}{L}(f,s)=\sum_{n\ge1} \Lambda_f(n)n^{-s} $$
In terms of the local roots of the Euler product:
$$ \Lambda_f(...
- 6,367
0
votes
0
answers
101
views
Relating the multiplicative Fourier transform and the derived characteristic polynomial
(Tuesday, Sept 5:) For a number field $Fˣ$ and a number ring $Oˣ$ it is common to define:
$Z(f,χ) = ʃ_{Fˣ} f(x) χ(x) dˣ x$
$g(ω,ψ) = ʃ_{Oˣ} ω(x) ψ(x) dˣ x$
where $dˣx$ is the multiplicative Haar ...
CommunityBot
- 1
7
votes
1
answer
1k
views
The Correlation of the Möbius Function and Dirichlet Characters
Let $\chi$ be a Dirichlet character, and define $\phi_\chi (n)$ so that it satisfies $$\sum_{n=1}^\infty \phi_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$
In other words
$$\phi_{\chi}(n)=\sum_{d|...
- 105k
4
votes
2
answers
559
views
Explicit formula for Artin L-functions
The classical explicit formula for the Riemann Zeta function states that
$$
\psi(x)=x-\sum_{\rho} \frac{x^{\rho}}{\rho}+O(1),
$$
where $\psi(x)=\sum_{n \leq x} \Lambda(n)$ and the sum is over all non-...
- 1,570
2
votes
1
answer
388
views
What are the best known upper bounds for $\frac{1}{L(s, \chi)}$?
Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding Dirichlet L-function. What are the best known bounds for $\frac{1}{L(s, \chi)}$ in the half-plane of convergence?
I'm aware of ...
- 8,374
8
votes
1
answer
643
views
Absolute convergence of Rankin–Selberg series
Let $\pi$ and $\pi'$ be two general automorphic representations on $\operatorname{GL}(n)$ and $\operatorname{GL}(n')$ over $\mathbb{Q}$.
I heard that the Rankin-Selberg $L$-function $L(s,\pi\times\pi')...
- 105k
4
votes
0
answers
509
views
Ramanujan's conjecture on modular forms and Riemann hypothesis
I just watched Kannan Soundararajan's talk on the distributions of valus of zeta and $L$-functions at virtual ICM 2022. In his talk, he introduced a theorem on Ramanujan's ternary form $\phi_{1}: x^{2}...
- 8,374
4
votes
0
answers
450
views
Question about a paper by Franca and LeClair in analytic number theory
I am reading an article "Transcendental equations satisfied by the individual
zeros of Riemann $\zeta$, Dirichlet and modular
L-functions" by G. Franca and A. LeClair (2015) see here. The ...
- 41
9
votes
3
answers
658
views
Vinogradov-Korobov for Dirichlet L-functions?
Where can one find a Vinogradov-Korobov zero-free region for Dirichlet L-functions? It has to be in a standard reference, but I'm having a non-trivial time finding it.
- 358
2
votes
2
answers
308
views
Reference for zero sum estimates of Dirichlet L functions
Let $\chi$ be a primitive character mod $p$ (prime) and $\rho = \beta + i \gamma$ be a non-trivial zero of $L(s, \chi)$.
I am reading a paper by Ihara and Murty where they use following estimate :
$\...
- 4,195
2
votes
1
answer
757
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/...
- 105k
7
votes
3
answers
826
views
Analytic equivalents for primes in arithmetic progressions
By way of context: it is known that the prime number theorem $\pi(x) \sim x/\log x$ is (nontrivially) equivalent to the statement that $\zeta(s)$ does not vanish on the line $\Re s=1$.
I would like ...
- 50.6k
1
vote
0
answers
203
views
Large values of $L(1,\chi)$ for quadratic Dirichlet characters $\chi$
Granville and Soundararajan, in "Upper Bounds for $L(1, \chi)$", first paragraph,
say it is known that there exist quadratic Dirichlet characters $\chi$ for which $L(1, \chi)$ is about $\log\...
- 105k
12
votes
1
answer
2k
views
Artin conjecture on L-functions
Artin conjecture on Artin $L$-functions asserts that the Artin $L$-function $L(\rho,s)$ of a non-trivial irreducible representation $\rho$ of the Galois group $\Gamma$ of a number field admits ...
- 105k
2
votes
2
answers
263
views
Sign of the special value at s=0 of Hecke L-functions
Let $L/K$ be an abelian extension of number fields with Galois group $G$ and let $\chi : G \to \{\pm 1\}$ denote a real linear character of $G$. Denote $L(\chi,s)$ the Artin L-function associated to $\...
- 617
6
votes
1
answer
943
views
Voronoi formula for the symmetric $L$-function with level $N $
Sorry to disturb. Does any experts here know something upon the Voronoi type for the symmetric $L$-functions$$\sum_{n\le X} A_F(1,n)e\left ( \frac{an}{c}\right)=?$$
Here $F$ is a symmetric-lift of a $...
- 571
8
votes
1
answer
310
views
Explicit estimates for $N(T,\chi)$ (not $N(T,\chi)+N(T,\overline{\chi})$)
Let $N(T,\chi)$ denote the number of zeros of $L(s,\chi)$ with imaginary part between $0$ and $T$, with any zero with imaginary part equal to $T$ or to $0$ (not that the latter kind really exists) ...
- 20.2k
0
votes
1
answer
195
views
Are Li's numbers $\lambda_n$ absolutely convergent for $n>1$?
Li's numbers $\{\lambda_n\}$ are defined as $$\lambda_n=\frac{1}{(n-1)!}\frac{d^n}{ds^n} [s^{n-1}\log\xi(s)]_{s=1} $$ for all positive integers $n$.
Also $\lambda_n$ is given as a sum over the non ...
- 7,024
20
votes
3
answers
2k
views
Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field.
Let $M$ be the splitting field of
x^8 + 3*x^7 + 13*x^6 + 17*x^5 + 45*x^4 + 37*x^3 + 11*x^2 + 112*x + 108
over the rationals. If I've understood some tables ...
- 25.4k
1
vote
1
answer
329
views
Behaviour of a certain $L$ function at $s=1$
I was going through this paper. Corollary 7.3.4 says the $L$-function $L(s,\pi, \rm{sym}^4)$ is holomorphic except possibly at $s=0,1$ and gives a necessary and sufficient condition for it to have a ...
- 5,089
4
votes
1
answer
318
views
Watson's triple product for automorphic forms shifted by Maass rising operators
Let $\phi_i$ be a holomorphic Hecke eigencusp form of weight $k_i$ for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$, or a Maass cusp form (we then say that $k_i=0$). We assume they are normalised such that $\...
- 37k
67
votes
17
answers
12k
views
Shortest/Most elegant proof for $L(1,\chi)\neq 0$
Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$?
Background: The non-vanishing ...
CommunityBot
- 1
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 ...
- 6,974
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 ...
- 6,974
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