All Questions
Tagged with analytic-number-theory l-functions
154 questions
10
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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
$$\sum_{p \nmid N} \...
- 105k
1
vote
1
answer
324
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Off critical line zeros for half integer weight $L$-functions
Let $f(z) = \sum_{n=1}^\infty A(n)n^{\frac{k-1}{2}}e(nz)$ be a modular form of weight $k$ for a half integer $k$. Put
$$L(s,f) = \sum_{n=1}^\infty \frac{A(n)}{n^s} $$
to be the $L$-function.
Further ...
7
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2
answers
478
views
Rankin-Selberg integral for GL(3) form with Odd Maass form on GL(2)
Let $F$ be a Hecke-Maass cusp form for $SL_3(\mathbb Z)$.
Let $u$ be a Hecke-Maass cusp form for $SL_2(\mathbb Z)$.
The following integral
$$\mathcal L(s,F\times u)=\int_{{SL}(2,\mathbb{Z})\...
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6
votes
1
answer
713
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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 ...
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38
votes
4
answers
6k
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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 $...
CommunityBot
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0
votes
1
answer
207
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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.
...
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6
votes
1
answer
678
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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 $\...
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19
votes
2
answers
2k
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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-...
- 17.6k
8
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1
answer
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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 $\...
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14
votes
0
answers
584
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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 ...
1
vote
1
answer
365
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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 $a_{p}(F)\overline{a_{p}(G)}$. $\...
- 17.6k
1
vote
1
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253
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infinitely many non-trivial zeros for $L(s,\chi)$ using Hadamard product for $\xi(s,\chi)$
Here is the definition of $\xi(s,\chi)$:
$\xi(s,\chi)= \left(\frac{s(s-1)}{2} \right)^{1_{\chi=1}} (q/\pi)^{\frac{s+a}{2}} \Gamma \left( \frac{s+a}{2} \right) L(s,\chi)$
Here is the definition of ...
- 105k
5
votes
1
answer
1k
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Subconvexity bounds and zero-free regions
When I see results in analytic number theory, I often have trouble seeing how they relate and their relative strength. Here is a specific question that should help me (and hopefully others too).
Here ...
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1
vote
0
answers
138
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Analytic continuation of an Integral involving product of L-functions
Let $L_i(s)$ be some $L$-functions. (I am interested in the case when $L_1, L_2$ are two different Hecke $L$-function associated to the same number field.) Let
$$F(s)=\int_{\Re w=\sigma} \frac{1}{w} ...
- 105k
1
vote
0
answers
325
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Extension of Heath-Brown's fourth moment $\sum_ \chi \lvert L (s, \chi) \rvert^4$ to complex characters
Let $L(s, \chi)$ denote the Dirichlet $L$-function associated to the character $\chi$.
In his paper A Mean Value Estimate for Real Character Sums, Heath-Brown proves a mean value bound for $L(s,\chi)$...
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3
votes
2
answers
337
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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 $\...
- 17.6k
5
votes
1
answer
687
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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 ...
- 105k
10
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1
answer
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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 ...
CommunityBot
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2
votes
2
answers
606
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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 ...
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3
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1
answer
654
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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 $...
- 17.6k
4
votes
2
answers
769
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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:...
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3
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1
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242
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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 ...
- 17.6k
4
votes
1
answer
391
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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 ...
- 17.6k
3
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1
answer
224
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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
- 17.6k
-1
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1
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304
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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.
- 17.6k
1
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1
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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$$
zeroes in a disk $|s|<R$. Is there a ...
3
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1
answer
367
views
Weil Conjectures Analog for Multivariate Zeta Functions
We know that the Riemann zeta function can be generalized to multivariate zeta functions.
Is there a multivariate analog of the Weil conjectures?
- 17.6k
2
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0
answers
270
views
On a sequence of L-functions having same zeros in critical strip and GRH
I had an idea on GRH involving a sequence of L-functions having same zeros, then at one step I need a bound on these function and I wonder if this bound is in fact not as hard as GRH itself ?
Let's ...
CommunityBot
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1
vote
0
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154
views
Behavior of partial Euler product in the critical strip (with Dirichlet Character)
Consider a primitive Dirichlet Character $\chi$ (non principal) and the partial Euler product attached to the L-function $L(\chi,s)$ ($p_i$ are the prime numbers) :
$$P(\chi,N)=\prod_{i=1}^{N} \frac{...
CommunityBot
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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
830
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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 $...
- 17.6k
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 ...
CommunityBot
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7
votes
1
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353
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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 =...
CommunityBot
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3
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1
answer
259
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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 $L(Sym^rf,s).$...
- 105k
8
votes
2
answers
973
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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 $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ ...
CommunityBot
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5
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1
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290
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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 $\...
CommunityBot
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6
votes
0
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426
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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 ...
- 105k
2
votes
1
answer
131
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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 \...
- 14.6k
5
votes
1
answer
324
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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-...
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10
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2
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705
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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 $z$ ...
- 105k
3
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0
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163
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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
5
votes
1
answer
842
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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
3
votes
1
answer
204
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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 ...
- 177
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
$$L(s, \pi, Sym^m)$...
- 3,804
6
votes
1
answer
1k
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subconvexity problem for $GL(3) × GL(2)$ $L$-function without involving in symmetric lift
A question in study of subconvexity topic puzzles me for a long time, which mabe a stupid question for many experts. I really wish someone to help me out, and any advice will be highly appreciated.
...
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1
vote
1
answer
273
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Does $L(-1+it,f)\ll_f \log^c q(f)t$ hold ture?
Let $f$ be a holomorphic or Maass cusp form for $SL(2,Z)$. Define $L(s,f)=\sum_{n\ge 1}\frac{a_f(n)}{n^s}$, for $\Im s$ sufficiently large.
Then
$$L(-1+it,f)\ll_f \log^c q(f)t$$
holds, for some ...
- 105k
3
votes
1
answer
426
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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 ...
- 11.2k
2
votes
2
answers
491
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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
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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
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