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Results tagged with automorphic-forms
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An automorphic form is a well-behaved function from a topological group $G$ to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup $\Gamma \subset G$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.
5
votes
Accepted
Fourier expansion at inequivalent cusps
This can be done numerically. The $n$-th Fourier coefficient around the cusp is given by an integral of the form $$\int_0^{1} f|_{\sigma_2} (z) e(-nz) dx,$$ where $\sigma_2$ is a scaling matrix for t …
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11
votes
Accepted
Do we care about multiple zeta functions?
It may help clarify things to work out a specific example, although the OP may know this. In case $n=3$, the double Dirichlet series evaluates as
$$ \sum_{m,n=1}^{\infty} \frac{A_F(m,n)}{m^{w} n^s} = …
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11
votes
what is the equivalent of the Euler constant for higher dimensional lattices
I work out the case $d=2$ below. I didn't check everything carefully, so hopefully there are no errors.
Up to homothety, any lattice is equivalent to one generated by the complex numbers $1,z$ with …
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4
votes
Characterizing the real analytic Eisenstein series
The properties (1)-(5) do not characterize $E(z,s)$. The issue is that there's no enough control on it as a function of $s$. For an example, let $$F(z,s) = e^{s(1-s)} E(z,s).$$
Then $F(z,s)$ satisfi …
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9
votes
Accepted
Lower bound of Hecke eigenvalues of Maass form
If $x$ is large enough, then Rankin-Selberg theory will show that $S(x) \gg x^{1-\varepsilon}$. However, if $x$ is not large enough, then it is unknown how to obtain a lower bound for $S(x)$. In par …
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5
votes
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Product of two cuspforms is not a cuspform
I don't think it's possible to find a "nice" (say, smooth) function $f \in L_2(\Gamma \backslash \mathbb{H})$ such that $(1) \int_0^{1} f(x+iy) dx = 0$ for all $y > 0$ and $\lim_{y\rightarrow \infty} …
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9
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What is the relation of the Kuznetsov-Bruggeman trace formula and the Selberg trace formula?
Sorry to give a reference to my own paper, but perhaps what you are looking for is contained in section 2 of this paper; see also Theorem 1.3. The basic idea is that the Selberg and Kuznetsov trace f …
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