All Questions
Tagged with analytic-number-theory l-functions
154 questions
3
votes
2
answers
337
views
Poles of Rankin-Selberg $L(s,\pi\times\tilde \pi)$?
Let $F$ be a number field and let $\pi$ be a cuspidal automorphic representation of $GL_n(\mathbb{A_F})$.
Do we know that $L(s,\pi\times \tilde\pi)$ has a simple pole at $s=1$?
Do we know that $\...
- 3,804
3
votes
1
answer
654
views
Order of vanishing of Artin $L$-functions at $s=1$
Let $E/F$ be a finite Galois extension of number fields with Galois group $G$. Let $S$ be a finite set of places of $F$ containing the infinite places. For $\chi$ an irreducible complex character of $...
- 1,661
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
3
votes
1
answer
242
views
Divisibility of Dirichlet L-functions
Let $k$ be an even integer and $p$ a prime number such that $p-1|k$.
Suppose that $p$ does not divide $L(1-k,\chi)$ and $L(1-k,\psi)$, where $\chi$ and $\psi$ are quadratic characters.
Can we deduce ...
- 91
4
votes
2
answers
768
views
Relation of these two Dirichlet $L$-functions
Let $\chi$ and $\psi$ be two quadratic Dirichlet characters and let $L(s,\chi)$ and $L(s,\psi)$ their associated Dirichlet $L$-functions.
Is there a realtion between these two Dirichlet $L$-functions:...
- 1,257
4
votes
1
answer
391
views
Non-vanishing of L-function of modular form
There is a theorem by Langlands and Shalika (link) that the L-function of a cuspidal automorphic representation does not vanish on the line $\mathrm{Re}( s)=1$ (in their normalization which might be ...
- 7,377
1
vote
1
answer
74
views
Asymptotic for zeroes of $L(s,\chi)$ in a disk $|s|<R$
In 'Remarks on Weil's quadratic functional..' p.191 Bombieri claims any given $L$-function $L(s,\chi)$ has at least
$$\big(\frac{1}{\pi}+o(1)\big)R\log R$$
zeroes in a disk $|s|<R$. Is there a ...
- 3,799
3
votes
1
answer
367
views
Weil Conjectures Analog for Multivariate Zeta Functions
We know that the Riemann zeta function can be generalized to multivariate zeta functions.
Is there a multivariate analog of the Weil conjectures?
- 13.9k
2
votes
0
answers
270
views
On a sequence of L-functions having same zeros in critical strip and GRH
I had an idea on GRH involving a sequence of L-functions having same zeros, then at one step I need a bound on these function and I wonder if this bound is in fact not as hard as GRH itself ?
Let's ...
- 1,199
1
vote
0
answers
154
views
Behavior of partial Euler product in the critical strip (with Dirichlet Character)
Consider a primitive Dirichlet Character $\chi$ (non principal) and the partial Euler product attached to the L-function $L(\chi,s)$ ($p_i$ are the prime numbers) :
$$P(\chi,N)=\prod_{i=1}^{N} \frac{...
- 1,199
13
votes
0
answers
622
views
No Siegel-Landau zeros for $\mathrm{GL}(n)$
The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact:
There ...
- 17.6k
9
votes
1
answer
533
views
Effective bound of $L(1,\chi)$
Let $d$ be a fundamental discriminant and let $\chi$ be the associated primitive real character of modulus $\vert d \vert$. Assuming GRH, Littlewood proved that as $\vert d \vert$ grows large,
$$L(1, ...
- 13.3k
5
votes
0
answers
195
views
Moments of completed L-functions?
This is a follow up question to this one.
It seems that results on moments of L-functions, that is, estimates for integrals of the form
$$\int^{T}_1|\zeta(\sigma+it)|^{2k}dt$$
are typically for the ...
- 3,799
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
3
votes
1
answer
259
views
Question about mean square estimate for sums of Dirichlet coefficients of Symmetric Power $L$-functions
I have a question related to Coefficients of Symmetric power $L$-functions and I would be grateful if you could answer it.
Let $\lambda_{Sym^rf}(n)$ be the $n$th Dirichlet coefficient of $L(Sym^rf,s).$...
- 1,257
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
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
2
votes
1
answer
131
views
Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$, does $L_c(s, \chi)$ necessarily equal $1$?
Consider an analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, let $c \...
user75660
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
5
votes
1
answer
687
views
Subconvexity bound for Hecke $L$-functions in the $s$-aspect
Let $L(s,\chi)$ be the $L$-function of a non-trivial Hecke character of a general number field $K$, so that $L(s,\chi)$ which has no pole or zero at $s=1$.
I am looking for a reference for upper ...
- 3,062
5
votes
1
answer
290
views
Bounding a Sum of Adjoint L-Function Values
Fix integers $k\geq2$ and $N>1$, and let $S(k,N)$ denote the normalized new Hecke eigenforms in $S_k(\Gamma_1(N))$. [If it makes my question easier to answer, feel free to replace this with $\...
- 1,422
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
5
votes
1
answer
324
views
Symmetry type of non-cohomological automorphic forms
By Katz-Sarnak philosophy a family of $L$-functions would have a symmetry type which would reflect the statistics of $L$-functions, such as low lying zeros and moments. Shin-Templier's paper on Sato-...
- 809
3
votes
0
answers
163
views
Oscillatory integral moments of $L(\frac{1}{2} + it, f \times f)$
Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding
$$
\int_0^T L\left(\tfrac{1}{2} + it, f \...
- 1,570
9
votes
1
answer
830
views
Lindelof Hypothesis implying Selberg Eigenvalue Conjecture?
The Generalized Lindelof Hypothesis says that for the $L$-function of an automorphic form we have
$$L(1/2+it)\ll Q(t)^{\epsilon}$$
for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at $...
- 3,804
3
votes
1
answer
204
views
Lower bound of first moment of $L$-function on $\mathrm{GL}(3)$
Let $\pi$ be an automorphic form on $\mathrm{GL}(3,\mathbb{A}_{\mathbb{Q}})$.
Do we know any case that
\[\int_0^{T} \left|L(\frac{1}{2} + it, \pi)\right| dt \gg T\]
holds unconditionally?
I know the ...
- 3,804
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
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
$$L(s, \pi, Sym^m)$...
- 3,804
14
votes
1
answer
1k
views
Is the adjoint L-function on GL(m) holomorphic?
Let $\pi$ be an automorphic representation on $\mathrm{GL}(m)/\mathbb{Q}$.
Define $$L(s,\pi,\mathrm{Ad}):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an $L$-function with Euler product of ...
- 253
8
votes
1
answer
643
views
Absolute convergence of Rankin–Selberg series
Let $\pi$ and $\pi'$ be two general automorphic representations on $\operatorname{GL}(n)$ and $\operatorname{GL}(n')$ over $\mathbb{Q}$.
I heard that the Rankin-Selberg $L$-function $L(s,\pi\times\pi')...
- 253
3
votes
1
answer
224
views
Are there L-functions of degree 1 that aren't Hecke L-functions?
I don't know of any examples and I don't know of any results which prohibit them
- 31
5
votes
1
answer
842
views
Generalization of Watson's triple product
In Watson's thesis (page 51) we can find his beautiful triple product formula. My question is that does there exist a generalization of this formula? By generalization, I mean:
If $\phi_n$'s are ...
- 1,692
5
votes
0
answers
384
views
A sum over zeros of L-functions in the paper "Chebyshev's Bias"
Let $\varepsilon>0$ be small and
\begin{align*}
\widetilde{F}_{\varepsilon}(\xi):=\frac{4}{\varepsilon}\sum_{0<\gamma\leq \varepsilon^{-2}}\frac{\sin(\gamma \xi)\sin \frac{\gamma \varepsilon}{2}}...
- 2,719
1
vote
1
answer
273
views
Does $L(-1+it,f)\ll_f \log^c q(f)t$ hold ture?
Let $f$ be a holomorphic or Maass cusp form for $SL(2,Z)$. Define $L(s,f)=\sum_{n\ge 1}\frac{a_f(n)}{n^s}$, for $\Im s$ sufficiently large.
Then
$$L(-1+it,f)\ll_f \log^c q(f)t$$
holds, for some ...
- 177
3
votes
1
answer
426
views
On link between Riemann hypothesis and partial GRH
Is there a way to show that if the Riemann hypothesis holds for Dirichlet L-function associated to primitive Dirichlet character (excluding trivial character $\chi(1)$ which could be qualified of ...
- 1,199
6
votes
1
answer
1k
views
subconvexity problem for $GL(3) × GL(2)$ $L$-function without involving in symmetric lift
A question in study of subconvexity topic puzzles me for a long time, which mabe a stupid question for many experts. I really wish someone to help me out, and any advice will be highly appreciated.
...
- 177
2
votes
2
answers
491
views
Summation of certain series
Suppose $f(n)$ is a periodic function with period $q$. Now from this paper we get that if $\displaystyle\sum_{n=1}^{q}f(n)=0$ then $\displaystyle\sum_{n=1}^{\infty}\frac{f(n)}{n}=-\frac{1}{q}\...
- 1,692
5
votes
0
answers
394
views
a generalization of a formula of Shimura
Let $\phi$ be a $GL(2)$ automorphic form with Fourier coefficients $a(n)$ and $a(1)=1$.
Obviously we have $L(s,\phi)=\sum \frac{a(n)}{n^s}$.
Shimura have the following formula
$L(s, Ad\; \phi)=\zeta(...
- 585
1
vote
1
answer
324
views
Off critical line zeros for half integer weight $L$-functions
Let $f(z) = \sum_{n=1}^\infty A(n)n^{\frac{k-1}{2}}e(nz)$ be a modular form of weight $k$ for a half integer $k$. Put
$$L(s,f) = \sum_{n=1}^\infty \frac{A(n)}{n^s} $$
to be the $L$-function.
Further ...
0
votes
1
answer
207
views
Can GRH for complex primitive Dirichlet characters fail with a single non-trivial zero off the critical line?
Can GRH for complex primitive Dirichlet character fail with a
single non-trivial zero off the critical line?
For real characters this is impossible because the non-trivial zeros are in quadruples.
...
- 25.4k
3
votes
0
answers
240
views
Is this extension of the Selberg class trivial?
I came across the following modification of the Selberg class in some of my work (see below), and while I've moved on in some sense -- I submitted the paper in question -- I can't get it off of my ...
- 1,671
12
votes
1
answer
969
views
Montgomery's pair correlation function without RH?
In the theory of the Riemann zeta function, Montgomery's Pair correlation function is defined as
$$
F(\alpha) = \frac{1}{N(T)}
\sum_{T < \gamma, \gamma' < 2T} T^{i \alpha (\gamma - \gamma')} \...
- 123
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
$$\phi_{\chi}(n)=\sum_{d|...
- 11.4k
18
votes
1
answer
1k
views
Distinct simple zeros of Dirichlet L-functions
Given a finite set of distinct primitive Dirichlet characters, $\chi_1, \dots, \chi_r$, is it known that the product of the L-functions, $$L(s):=\prod_{i=1}^r L(s,\chi_i),$$ has a simple zero? It's ...
- 1,671
2
votes
2
answers
606
views
Axioms for zeta functions
The Selberg class is an axiomatization of arithmetically significant zeta functions (a.k.a. L-functions) by a few analytic properties (functional equation etc.) However there do exist other zeta ...
- 333
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
-1
votes
1
answer
304
views
Consequences of the degree conjecture
the title is quite explicit: I would like to know the consequences of the degree conjecture for the Selberg class.
Thank you in advance.
- 6,974
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
1
vote
1
answer
365
views
Selberg's orthonormality conjecture and density
Let $F$ and $G$ be two primitive functions of the Selberg class, and let $\mathbb{A}$ be the set of values taken by the function which maps a prime number $p$ to $a_{p}(F)\overline{a_{p}(G)}$. $\...
- 6,974
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