All Questions
15 questions
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
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
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
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
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
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
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
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
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
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
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
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 ...
- 20.2k
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 ...
- 5,424