Fourier coefficients of Eisenstein series

Question

Several questions/reading reference requests for the following topics.

1. I require some point-wise bounds on the absolute value of the Eisenstein series $E_{\mathfrak{a}}(\sigma_{\mathfrak{b}}z,1/2 + it)$ for the congruence subgroup $\Gamma_0(q)$, when $|t|\le 1$.

2. Is there any place in the literature where someone constructs an explicit bound on the Fourier coefficients of the Eisenstein series? For example for the constant term, $\phi_{\mathfrak{a},\mathfrak{b}}(s)$ where $$E_{\mathfrak{a}}(\sigma_{\mathfrak{b}}z, s) = \delta_{{\mathfrak{a}} = {\mathfrak{b}}}y^{s} + \phi_{{\mathfrak{a}},{\mathfrak{b},0}}(s)y^{1 - s} + \sum_{m\neq0}a_{{\mathfrak{a}},{\mathfrak{b}},m}(1/2 + it,y)e(mx)...$$ I am particularly interested in upper bounds on $\phi_{\mathfrak{a},\mathfrak{a}}(1/2 + it)$ when $|t|\le 1$. I know that it is at most 1 in absolute value, however I need something that is strictly less than 1 when $q$ is large, say.

3. I found some information on a paper by Deshouillers and Iwaniec where they compute $\phi_{\mathfrak{a},\infty,m}(s)$ explicitly, but without much detail I find it hard to deduce how their construction generalizes. Is there any place in the literature you would suggest a student to search for in order to decipher this text?

4. Perhaps this is somewhat of a duplicate of #2, however, I found a lot of material in the literature on upper bounds on the Fourier coefficients of cusp forms, but couldn't find anything on those of the Eisenstein series. Is there any obvious way to bound the coefficients $\phi_{\mathfrak{a},\mathfrak{b},m}(s)$ in terms of $m$ and $s$? I surely am missing out on something.

Any answer to any one of the questions would be much appreciated. Thanks in advance!

Answer 1

2. I don't think much is explicitly written down in the literature anywhere. With that being said... the bounds should be at least as strong as those for Maass cusp forms.

3. You want the Bruhat decomposition. See Theorem 2.7 of Iwaniec's book "Spectral methods of automorphic forms". I do something similar in my paper on lower bounds for $L(1,\chi)$ via Eisenstein series.

4. See below.

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The long answer is that this question is relatively easy to answer when $q$ is squarefree, because then every cusp is of the form $1/v$ for some $v \mid q$ with the crucial fact that $(v,\frac{q}{v}) = 1$. If your cusp is of this form, then the Bruhat decomposition isn't too hard and you can write things out explicitly.

A somewhat better approach is to use the theory of Eisenstein newforms (see Matt Young's recent paper on explicit calculations for Eisenstein series). The benefit of this approach is that you use a different basis of Eisenstein series for which the Fourier coefficients are completely explicit off the bat.

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Anyway, it's hard for me to answer this question without knowing more about your applications and exactly what bounds you need, so you're welcome to email me for more specific discussions.

Answer 2

My recent preprint, Eisenstein series and an asymptotic for the $K$-Bessel function, gives some bounds on both the Fourier coefficients and the Eisenstein series but for large |t|. I also give some references to recent papers that may be helpful.