28 votes
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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}, ...
Dan Romik's user avatar
  • 2,480
23 votes
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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 ...
Will Sawin's user avatar
  • 135k
22 votes
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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 ...
Lucia's user avatar
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22 votes
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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 ...
Jack Sprat's user avatar
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)...
Desiderius Severus's user avatar
20 votes
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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 ...
Will Sawin's user avatar
  • 135k
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 ...
KConrad's user avatar
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19 votes
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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 ...
Terry Tao's user avatar
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17 votes
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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 ...
Will Sawin's user avatar
  • 135k
17 votes
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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 ...
David Loeffler's user avatar
16 votes
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Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$

Consider the Dirichlet series $$F(s)=\sum_{n=1}^\infty\frac{f(n)}{n^s},$$ where $f(n)$ is the completely multiplicative function which satisfies $f(2)=-1$ and $f(p)=(-1)^{(p-1)/2}$ for odd primes $p$. ...
GH from MO's user avatar
  • 97.8k
15 votes
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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 ...
Joël's user avatar
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15 votes
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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}...
Peter Humphries's user avatar
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 ...
GH from MO's user avatar
  • 97.8k
14 votes
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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 ...
David Loeffler's user avatar
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, ...
KConrad's user avatar
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14 votes
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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 $\mod q$, ...
Lucia's user avatar
  • 43.3k
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 ...
Joe Silverman's user avatar
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 ...
Peter Humphries's user avatar
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 ...
David Loeffler's user avatar
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 = ...
Peter Humphries's user avatar
13 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}{\...
Peter Humphries's user avatar
13 votes
Accepted

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 ...
Olivier's user avatar
  • 10.2k
13 votes

Does the number of roots of the modular form associated to an elliptic curve, on the positive imaginary axis, equal the analytic rank?

Let $Z$ be the number of zeroes of $y\mapsto f(iy)$ for $0<y<\infty$ and let $r$ be the analytic rank of $E$. Then $Z\geqslant r$ and $Z\equiv r \pmod{2}$ as I will explain below. I don't know ...
Chris Wuthrich's user avatar
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 $\pi$,...
Peter Humphries's user avatar
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 ...
GH from MO's user avatar
  • 97.8k
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 $T$)....
Terry Tao's user avatar
  • 108k
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, ...
Lior Silberman's user avatar
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 ...
Olivier Ramaré's user avatar
11 votes
Accepted

$|L'(1,\chi)/L(1,\chi)|$

Suppose that $\chi(-1)=1$ and that all non-trivial zeros $\beta+i\gamma$of $L(s,\chi)$ with $|\gamma|\le 1/2$ are on the critical line $\beta=1/2$. Recall the Hadamard factorization formula (see ...
Lucia's user avatar
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