All Questions
128 questions
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 ...
- 7,127
38
votes
4
answers
6k
views
Modular forms and the Riemann Hypothesis
Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions?
What I'm thinking of is this: under the Mellin transform, the Riemann zeta function $...
- 889
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 ...
- 13.1k
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 ...
- 41.4k
19
votes
2
answers
2k
views
Applications of Artin's holomorphy conjecture
I wonder why the Artin conjecture is so important. The only reason I could figure out is that one could use the holomorphy of Artin L-series and Weil's converse theorem to show modularity of two-...
user19475
19
votes
1
answer
1k
views
constants in Gamma factors in functional equation for zeta functions.
Usually the Riemann zeta function $\zeta(s)$ gets multiplied by a "gamma factor" to give a function $\xi(s)$ satisfying a functional equation $\xi(s)=\xi(1-s)$. If I changed this gamma factor by a non-...
- 41.4k
18
votes
1
answer
1k
views
Distinct simple zeros of Dirichlet L-functions
Given a finite set of distinct primitive Dirichlet characters, $\chi_1, \dots, \chi_r$, is it known that the product of the L-functions, $$L(s):=\prod_{i=1}^r L(s,\chi_i),$$ has a simple zero? It's ...
- 1,671
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 $S$...
- 28.7k
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
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 ...
- 1,231
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 ...
- 317
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/...
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 ...
- 253
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
$$\sum_{n&...
- 5,424
14
votes
0
answers
584
views
Moments of derivatives of $L$-functions
I'd like to know why it is important to know the moments of the derivatives of $L$-functions.
The moments of $L$-functions are related to the Lindelöf Hypothesis, but what about the moments of the ...
- 241
13
votes
1
answer
763
views
Least quadratic residue under GRH: an explicit bound
Let $m$ be a positive integer and $\chi$ a primitive character mod $m$. Let $x$ be such that $\chi(p)\ne 1$ for all primes $p<x$. Assume GRH. How can one bound $x$ in terms of $m$ ? I do not need ...
- 1,294
13
votes
0
answers
622
views
No Siegel-Landau zeros for $\mathrm{GL}(n)$
The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact:
There ...
- 17.6k
12
votes
1
answer
2k
views
Artin conjecture on L-functions
Artin conjecture on Artin $L$-functions asserts that the Artin $L$-function $L(\rho,s)$ of a non-trivial irreducible representation $\rho$ of the Galois group $\Gamma$ of a number field admits ...
user95222
11
votes
1
answer
328
views
Critical points of Dirichlet L functions
Let $L(s,\chi)$ denote a Dirichlet $L$-function for a real-valued non-principal
character $\chi$. This has limiting value $L(\infty,\chi) = 1$ and we are interested in how this limit is approached ...
- 1,411
10
votes
2
answers
705
views
Averages over integer points of the sphere
A paper of William Duke proves that integer points on the sphere are equidistributed:
$$ V_n = \{ (x,y,z) \in \mathbb{Z}^2 : x^2 + y^2 + z^2 = n \}. $$
Up to reflections across the $x$, $y$ and $z$ ...
- 22.8k
10
votes
1
answer
1k
views
How does Riemann hypothesis implies estimates?
In Iwaniec, Luo and Sarnak article (precisely (4.23)), it is said that GRH for $L(s, \mathrm{sym}^2(f))$, for a holomorphic cusp newform $f$ of level $N$ and weight $k$, implies
$$\sum_{p \nmid N} \...
- 551
10
votes
1
answer
1k
views
Reference for the odd dihedral case of Artin's conjecture
The example that Matt Emerton cited here prompted me to become interested in how one proves that odd two dimensional dihedral Galois representation are modular. This is the first case of the strong ...
- 7,062
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.
- 20.2k
9
votes
1
answer
521
views
Zeroes of complete L-functions
Hello,
Let $F$ and $G$ be two functions belonging in the Selberg class, $\Lambda_{F}$ and $\Lambda_{G}$ the complete L-functions associated to $F$ and $G$. I would like to know whether this assertion ...
- 195
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,
$$L(1, ...
- 13.3k
9
votes
1
answer
1k
views
Functional equation Dedekind zeta function
I'd like to know to what point is it possible to generalize
this method
for obtaining the functional equation for the Dedekind zeta function $\zeta_K(s)$ of a number field ?
Let $\mathfrak{C}$ be ...
- 3,403
9
votes
1
answer
830
views
Lindelof Hypothesis implying Selberg Eigenvalue Conjecture?
The Generalized Lindelof Hypothesis says that for the $L$-function of an automorphic form we have
$$L(1/2+it)\ll Q(t)^{\epsilon}$$
for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at $...
- 3,804
9
votes
0
answers
399
views
Symmetric Fifth Power Lift of GL(2) Automorphic Form
Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that
$$L(s, \pi, Sym^m)$...
- 3,804
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 ...
- 5,424
8
votes
2
answers
973
views
Easiest way to see that $\zeta_{\mathbb{Z}[i]}(s) = \zeta(s) L(s, \chi)$?
As the question suggests, what is the easiest way to see that$$\zeta_{\mathbb{Z}[i]}(s) = \zeta(s)L(s, \chi)?$$Here, $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ ...
user75660
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')...
- 253
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 ...
- 927
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 ...
8
votes
1
answer
2k
views
Characterizing the newforms s.t. the associated symmetric square $L$-function has a pole
I have a straightforward question. Let $f$ be a holomorphic cusp form of weight $k$, level $N$, and nebentypus $\chi$ that is new in the sense of Atkin-Lehner theory. Write its Fourier expansion at $\...
- 1,217
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
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-...
- 5,424
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 ...
- 12.8k
7
votes
2
answers
478
views
Rankin-Selberg integral for GL(3) form with Odd Maass form on GL(2)
Let $F$ be a Hecke-Maass cusp form for $SL_3(\mathbb Z)$.
Let $u$ be a Hecke-Maass cusp form for $SL_2(\mathbb Z)$.
The following integral
$$\mathcal L(s,F\times u)=\int_{{SL}(2,\mathbb{Z})\...
- 3,804
7
votes
1
answer
1k
views
The Correlation of the Möbius Function and Dirichlet Characters
Let $\chi$ be a Dirichlet character, and define $\phi_\chi (n)$ so that it satisfies $$\sum_{n=1}^\infty \phi_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$
In other words
$$\phi_{\chi}(n)=\sum_{d|...
- 11.4k
7
votes
1
answer
353
views
Numerically double-checking formula with L-values
I'm working with a special case of Ichino's triple product formula, which for classical holomorphic newforms $f$, $g$ ,$h$ of weights $k$, $m-k$, $m$ (and central characters satisfying $\chi_f \chi_g =...
- 131
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 $...
- 353
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 ...
- 26.9k
6
votes
1
answer
678
views
Root number of the Rankin-Selberg convolution of two newforms
Let $p$ and $q$ be two distinct primes. Let $f\in \mathcal{S}_k^{\ast}(pq,\psi)$ be a holomorphic newform of level $pq$, nebentypus $\psi$, and weight $k$, where $\psi = \chi_p \chi_{0(q)}$, with $\...
- 63
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 "...
- 4,015
6
votes
1
answer
713
views
Explicit zero density estimate for Dirichlet $L$-functions
Let's define $N(\alpha,T,\chi)=\sharp\lbrace \rho=\sigma+i\gamma: L(\rho,\chi)=0, \alpha\leq \sigma<1, |\gamma|\leq T\rbrace$ , where $\chi$ is a primitive Dirichlet character. We know, from ...
6
votes
1
answer
1k
views
subconvexity problem for $GL(3) × GL(2)$ $L$-function without involving in symmetric lift
A question in study of subconvexity topic puzzles me for a long time, which mabe a stupid question for many experts. I really wish someone to help me out, and any advice will be highly appreciated.
...
- 177
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
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 ...
- 20.2k
6
votes
0
answers
426
views
Explicit bounds for exceptional zeros and/or $L(1,\chi)$ for real $\chi$
I would like to have an explicit upper bound (that is, one with explicit constants) for a possible real zero $\beta$ for $L(1,\chi)$ for real Dirichlet characters $\chi$. I need such a bound for real ...
- 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=...
- 6,974