# Tag Info

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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}, ...
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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 ...
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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 ...
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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 ...
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### 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)...

### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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### 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 ...

### 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 ...
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### 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 $\pi$,...
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### 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}{\...
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### 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 ...
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### 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 $T$)....