28
votes
Accepted
Are L-functions uniquely determined by their values at negative integers?
Are L-functions uniquely determined by their values at negative integers?
No. The rescaled Riemann zeta function
$$
\zeta(2s) = \sum_{m=1}^\infty \frac{1}{m^{2s}} =
\sum_{n=1}^\infty \frac{a_n}{n^s},
...
- 2,376
23
votes
Accepted
Why do we care about the eigenvalues of the Frobenius map?
The Riemann hypothesis is very important for the relationship between the cohomology and combinatorics of the variety.
First, the Riemann hypothesis lets us read off the Betti numbers from the point ...
- 129k
22
votes
Accepted
Least quadratic residue under GRH: an explicit bound
See the work of Lamzouri, Li, and Soundararajan (I link the arXiv version; the paper appeared in Math. Comp.). Assuming that $\chi$ is a primitive quadratic character (as the title suggests) then ...
- 43k
22
votes
Accepted
The modularity theorem as a special case of the Bloch-Kato conjecture
That is not what the link says. To quote (emphasis mine):
... in which this conjecture was reduced to a special instance of the Bloch-Kato conjecture for the symmetric square motive of an elliptic ...
- 244
21
votes
Accepted
Relation between Schanuel's theorem and class number equation
A spirit. A general approach that may interest you is the following: counting laws can be obtained by studying suitable generating functions, via Tauberian arguments: the rightmost pole (resp. residue)...
- 5,324
20
votes
Why do we care about the eigenvalues of the Frobenius map?
Here are a few different uses of knowing how large the eigenvalues are as complex numbers.
Application 1: Bounding exponential sums. Many classical exponential sums can be interpreted essentially as a ...
- 48.5k
19
votes
Accepted
How do functional equations for zeta functions arise from the structure of a homology group?
There are examples that don't come from number theory, although it's not much simpler. Specifically, the Lefschetz zeta function.
Let $X$ be a compact manifold of dimension $d$ and let $f: X\to X$ be ...
- 129k
19
votes
Accepted
Does the existence of a Landau-Siegel zero imply the existence of a complex zero off the critical line?
A result of Sarnak and Zaharescu, stated
in the contrapositive, implies the existence of a complex zero off the critical line for at least one Dirichlet L-function if one has a sufficiently strong ...
- 104k
16
votes
Accepted
Euler factors of L-function at bad primes
1) True.
2) True.
3) True.
4) They are equal in the case of "tame ramification", e.g. if the degree of $K(\mu_p)$ over $K$, $K$ the coefficient field of the motive, is greater than $d$.
One just ...
- 129k
16
votes
Accepted
What is the Perrin-Riou logarithm (or regulator)?
I am sure I've already written an expository account of this somewhere, but I looked over the lecture and seminar notes on my webpage and couldn't find it, so I'll write one here instead.
Suppose we ...
- 35.1k
15
votes
Accepted
Is there a "Langlands philosophy" reason for the fact that the L-function of the Jacobi theta function is (almost) the Riemann zeta function?
Good question. I don't understand fully what's happening, but here is an idea.
Let $f=\sum a_n q^n$ be a modular form of weight $k+1/2$, nebentypus $\chi$. Assuming $k \geq 1$, the Shimura ...
- 25.5k
15
votes
Accepted
Definitions of $\pi_1 \times \pi_2, \pi_1 \boxplus \pi_2, \pi_1 \boxtimes \pi_2$
The right person to answer this question is probably Dinakar Ramakrishnan (a good working reference is his paper "Modularity of the Rankin–Selberg $L$-series, and multiplicity one for $\mathrm{SL}...
- 7,959
14
votes
Accepted
Does $ M(x)=O(\sqrt{x}) $ if and only if the De Bruijn-Newman constant is negative?
The de Bruijn-Newman constant is nonnegative, as proved in this brand new preprint by Rodgers and Tao. It is also conjectured, but not proven yet, that $M(x)=O(\sqrt{x})$ is false, in which case it is ...
- 92.7k
14
votes
Accepted
BSD conjecture for rank 1 elliptic curves
The following theorem is due to Chris Skinner, in this 2014 paper.
Let E/Q be an elliptic curve such that rank E(Q) = 1 and the
Tate-Shafarevich group Sha(E / Q) is finite, and some other ...
- 35.1k
14
votes
Are L-functions uniquely determined by their values at negative integers?
Your restriction that all $a_n$ are integers is too restrictive to define $L$-functions in general (even most Dirichlet $L$-functions are not like that), and you left out the Euler product. Anyway, ...
- 48.5k
14
votes
Accepted
Hadamard factorization of L-functions
I believe you are correct and $b$ is zero, although I find it inexplicable why this is not better known (certainly I didn't know it before). Let's stick to a primitive Dirichlet character $\mod q$, ...
- 43k
14
votes
Why do we care about the eigenvalues of the Frobenius map?
All of the other answers give good reasons why the eigenvalues and their absolute values are important, but it should be noted that the eigenvalues can be used to give an exact point count via the ...
- 45.1k
13
votes
First formulation of the Dedekind and Hasse-Weil conjectures
Regarding the first of these conjectures, I believe it was first explicitly stated (in the more general setting of a relative extension $K/k$) in Artin's 1923 paper [Über die Zetafunktionen gewisser ...
- 13.6k
13
votes
Accepted
"Extra Euler factors" in one definition of the L-function of a twist of a modular form
Let me translate this into a problem purely about automorphic forms:
Take a newform $f \in \mathcal{S}_k^{\ast}(q,\chi)$, and a primitive Dirichlet character $\psi$ modulo $q'$. Then there exists a ...
- 7,959
13
votes
What kind of non-cuspidal automorphic representation are not isobaric sums?
EDIT. A colleague wrote to me to point out that my original answer to this question was actually completely wrong: I had confused "isobaric" representations with "pure" representations (which are not ...
- 35.1k
13
votes
How does Riemann hypothesis implies estimates?
We have that
\[-\frac{L'}{L}(s,\mathrm{sym}^2 f) = \sum_{n = 1}^{\infty} \frac{\Lambda_{\mathrm{sym}^2 f}(n)}{n^s},\]
where $\Lambda_{\mathrm{sym}^2 f}(n)$ is equal to $\lambda_f(p^2) \log p$ if $n = ...
- 7,959
13
votes
Some questions on the $p$-adic properties of special $L$-values
1) What generalizations of the Kummer congruences are known?
This is somewhat imprecise as a question and in particular, I would dispute a little your assertion that
This is probably the same ...
- 9,915
12
votes
Intuition for Zagier's theorem for $\zeta_K(2)$
There is a K-theory reformulation of Zagier's theorem which explains both the appearance of the integrals $A(x)$ as well as the appearance of hyperbolic geometry. I will try to give a sketch what is ...
- 17k
12
votes
Accepted
On the consistency of the definition of the conductor for automorphic forms
These definitions are consistent, though it's not immediate.
The conductor quantifies the extent to which $\pi$ is ramified. As an aside, I prefer to write $c(\pi)$ for the conductor exponent of $\pi$,...
- 7,959
12
votes
Accepted
Special values of adjoint $L$-functions of automorphic representations of $\mathrm{GSp}(4)$ as Petersson norms
For $\mathrm{GL}_2$, the relationship between the Petersson norm of a newform $f$ and its adjoint $L$-function is roughly a statement of the form
\[\frac{|a_f(1)|^2}{\langle f, f\rangle} = \frac{c_f}{\...
- 7,959
12
votes
Accepted
Does asymptotic Goldbach imply GRH?
Granville (2008) proved that a sufficiently strong average error term for the Goldbach-Hardy-Littlewood conjecture is equivalent to RH, and a refinement of it is equivalent to GRH. See MR2357316 and ...
- 92.7k
11
votes
Accepted
Subconvexity bounds and zero-free regions
Let's just discuss the $t$-aspect, i.e. bounds for the zeta function and its zeroes.
Let $T$ be a large ordinate, and let $H$ be a medium-sized quantity (much larger than $1$, but much less than $T$)....
- 104k
11
votes
Accepted
infinitely many non-trivial zeros for $L(s,\chi)$ using Hadamard product for $\xi(s,\chi)$
The existence of the Hadamard product by itself doesn't show the function has any zeroes (it might just be an exponential!) -- you need a little more.
The idea is to study the logarithmic derivative, ...
- 2,705
11
votes
$|L'(1,\chi)/L(1,\chi)|$
You may use the local method of Landau with some bounds for L(s,chi) (expressing L'/L in terms of the local zeros, the approximation being controlled by an upper bound for |L(s,chi)| in a slightly ...
- 321
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