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Slides from a Talk at the Dartmouth Number Theory Seminar

I recently gave a talk at the Dartmouth Number Theory Seminar (thank you Edgar for inviting me and to Edgar, Naomi, and John for being such good hosts). In this talk, I described the recent successes we’ve had on working with variants of the Gauss Circle Problem.

The story began when (with Tom Hulse, Chan Ieong Kuan, and Alex Walker — and with helpful input from Mehmet Kiral, Jeff Hoffstein, and others) we introduced and studied the Dirichlet series
$$\begin{equation}
\sum_{n \geq 1} \frac{S(n)^2}{n^s}, \notag
\end{equation}$$
where $S(n)$ is a sum of the first $n$ Fourier coefficients of an automorphic form on GL(2)$. We’ve done this successfully with a variety of automorphic forms, leading to new results for averages, short-interval averages, sign changes, and mean-square estimates of the error for several classical problems. Many of these papers and results have been discussed in other places on this site.

Ultimately, the problem becomes acquiring sufficiently detailed understandings of the spectral behavior of various forms (or more correctly, the behavior of the spectral expansion of a Poincare series against various forms).
We are continuing to research and study a variety of problems through this general approach.

The slides for this talk are available here.

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Paper:Short-Interval Averages of Sums of Fourier Coefficients of Cusp Forms

This is joint work with Thomas Hulse, Chan Ieong Kuan, and Alex Walker, and is a sequel to our previous paper.

We have just uploaded a paper to the arXiv on estimating the average size of sums of Fourier coefficients of cusp forms over short intervals. (And by “just” I mean before the holidays). This is the second in a trio of papers that we will be uploading and submitting in the near future.

Suppose $latex {f(z)}$ is a weight $latex {k}$ holomorphic cusp form on $\text{GL}_2$ with Fourier expansion

$$f(z) = \sum_{n \geq 1} a(n) e(nz).$$

Denote the sum of the first $n$ coefficients of $f$ by $$S_f(n) := \sum_{m \leq n} a(m). \tag{1}$$
We consider upper bounds for the second moment of $latex {S_f(n)}$ over short intervals.

In our earlier work, we mentioned the conjectured bound $$ S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{4} + \epsilon}, \tag{2}$$
which we call the “Classical Conjecture.” There has been some minor progress towards the classical conjecture in recent years, but ignoring subpolynomial bounds the best known result is of shape $$ S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{3}}. \tag{3}$$

One can also consider how $latex {S_f(n)}$ behaves on average. Chandrasekharan and Narasimhan [CN] proved that the Classical Conjecture is true on average by showing that $$ \sum_{n \leq X} \lvert S_f(n) \rvert^2 = CX^{k- 1 + \frac{3}{2}} + B(X), \tag{4}$$
where $latex {B(x)}$ is an error term. Later, Jutila [Ju] improved this result to show that the Classical Conjecture is true on average over short intervals of length $latex {X^{\frac{3}{4} + \epsilon}}$ around $latex {X}$ by showing $$ X^{-(\frac{3}{4} + \epsilon)}\sum_{\lvert n – X \rvert < X^{3/4} + \epsilon} \lvert S_f(n) \rvert^2 \ll X^{\frac{k-1}{2} + \frac{1}{4}}. \tag{5}$$
In fact, Jutila proved a much more complicated set of bounds, but this bound can be read off from his work.

In our previous paper, we introduced the Dirichlet series $$ D(s, S_f \times S_f) := \sum_{n \geq 1} \frac{S_f(n) \overline{S_f(n)}}{n^{s + k – 1}} \tag{6}$$
and provided its meromorphic continuation In this paper, we use the analytic properties of $latex {D(s, S_f \times S_f)}$ to prove a short-intervals result that improves upon the results of Jutila and Chandrasekharan and Narasimhan. In short, we show the Classical Conjecture holds on average over short intervals of width $latex {X^{\frac{2}{3}} (\log X)^{\frac{2}{3}}}$. More formally, we prove the following.

Theorem 1 Suppose either that $latex {f}$ is a Hecke eigenform or that $latex {f}$ has real coefficients. Then \begin{equation*} \frac{1}{X^{\frac{2}{3}} (\log X)^{\frac{2}{3}}} \sum_{\lvert n – X \rvert < X^{\frac{2}{3}} (\log X)^{\frac{2}{3}}} \lvert S_f(n) \rvert^2 \ll X^{\frac{k-1}{2} + \frac{1}{4}}. \end{equation*}

We actually prove an ever so slightly stronger statement. Suppose $latex {y}$ is the solution to $latex {y (\log y)^2 = X}$. Then we prove that the Classical Conjecture holds on average over intervals of width $latex {X/y}$ around $latex {X}$.

We also demonstrate improved bounds for short-interval estimates of width as low as $latex {X^\frac{1}{2}}$.

There are two major obstructions to improving our result. Firstly, we morally use the convexity result in the $latex {t}$-aspect for the size of $latex {L(\frac{1}{2} + it, f\times f)}$. If we insert the bound from the Lindel\”{o}f Hypothesis into our methodology, the corresponding bounds are consistent with the Classical Conjecture.

Secondly, we struggle with bounds for the spectral component $$ \sum_j \rho_j(1) \langle \lvert f \rvert^2 y^k, \mu_j \rangle \frac{\Gamma(s – \frac{3}{2} – it_j) \Gamma(s – \frac{3}{2} + it_j)}{\Gamma(s-1) \Gamma(s + k – 1)} L(s – \frac{3}{2}, \mu_j) V(X, s) \tag{7}$$
where $latex {\mu_j}$ are a basis of Maass forms and $latex {V(X,s)}$ is a term of rapid decay. For our analysis, we end up bounding by absolute values and are unable to understand cancellation from spin. An argument successfully capturing some sort of stationary phase could significantly improve our bound.

Supposing these two obstructions were handled, the limit of our methodology would be to show the Classical Conjecture in short-intervals of width $latex {X^{\frac{1}{2}}}$ around $latex {X}$. This would lead to better bounds on individual $latex {S_f(X)}$ as well, but requires significant improvement.

For more details and specific references, see the paper on the arXiv.

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Paper: The Second Moments of Sums of Fourier Coefficients of Cusp Forms

This is joint work with Thomas Hulse, Chan Ieong Kuan, and Alex Walker.

We have just uploaded a paper to the arXiv on the second moment of sums of Fourier coefficients of cusp forms. This is the first in a trio of papers that we will be uploading and submitting in the near future.

Suppose $latex {f(z)}$ and $latex {g(z)}$ are weight $latex {k}$ holomorphic cusp forms on $latex {\text{GL}_2}$ with Fourier expansions

$$\begin{align} f(z) &= \sum_{n \geq 1} a(n) e(nz) \\
g(z) &= \sum_{n \geq 1} b(n) e(nz). \end{align}$$

Denote the sum of the first $latex {n}$ coefficients of a cusp form $latex {f}$ by $$ S_f(n) := \sum_{m \leq n} a(m). \tag{1}$$

We consider upper bounds for the second moment of $latex {S_f(n)}$.

The famous Ramanujan-Petersson conjecture gives us that $latex {a(n)\ll n^{\frac{k-1}{2} + \epsilon}}$. So one might assume $latex {S_f(X) \ll X^{\frac{k-1}{2} + 1 + \epsilon}}$. However, we expect the better bound $$ S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{4} + \epsilon}, \tag{2}$$

which we refer to as the “Classical Conjecture,” echoing Hafner and Ivić [HI].

Chandrasekharan and Narasimhan [CN] proved that the Classical Conjecture is true on average by showing that $$ \sum_{n \leq X} \lvert S_f(n) \rvert^2 = CX^{k- 1 + \frac{3}{2}} + B(X), \tag{3}$$

where $latex {B(x)}$ is an error term, $$ B(X) = \begin{cases} O(X^{k}\log^2(X)) \ \Omega\left(X^{k – \frac{1}{4}}\frac{(\log \log \log X)^3}{\log X}\right), \end{cases} \tag{4}$$

and $latex {C}$ is the constant, $$ C = \frac{1}{(4k + 2)\pi^2} \sum_{n \geq 1}\frac{\lvert a(n) \rvert^2}{n^{k + \frac{1}{2}}}. \tag{5}$$

A application of the Cauchy-Schwarz inequality to~(3) leads to the on-average statement that $$ \frac{1}{X} \sum_{n \leq X} |S_f(n)| \ll X^{\frac{k-1}{2} + \frac{1}{4}}. \tag{6}$$

From this, [HI] were able to show in some cases that $$ S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{3}}. \tag{7}$$

Better lower bounds are known for $latex {B(X)}$. In the same work [HI] improved the lower bound of [CN] for full-integral weight forms of level one and showed that $$ B(X) = \Omega\left(X^{k – \frac{1}{4}}\exp\left(D \tfrac{(\log \log x )^{1/4}}{(\log \log \log x)^{3/4}}\right)\right), \tag{8}$$

for a particular constant $latex {D}$.

The question of better understanding $latex {B(X)}$ is analogous to understanding the error term in the circle problem or divisor problem. In our paper, we introduce the Dirichlet series $$D(s, S_f \times S_g) := \sum_{n \geq 1} \frac{S_f(n) \overline{S_g(n)}}{n^{s + k – 1}}$$

D(s, S_f \times \overline{S_g}) &:= \sum_{n \geq 1} \frac{S_f(n)S_g(n)}{n^{s + k – 1}} and provide their meromorphic continuations. From our review of the literature, these Dirichlet series and their meromorphic continuations are new and provide new approaches to the classical problems related to $latex {S_f(n)}$.

Our primary result is the meromorphic continuation of $latex {D(s, S_f \times S_g)}$. As a first application, we prove a smoothed generalization to~(3).

Theorem 1 Suppose either that $latex {f = g}$ is a Hecke eigenform or that $latex {f}$ and $latex {g}$ have real coefficients. \begin{equation*} \frac{1}{X} \sum_{n \geq 1}\frac{S_f(n)\overline{S_g(n)}}{n^{k – 1}}e^{-n/X} = CX^{\frac{1}{2}} + O_{f,g,\epsilon}(X^{-\frac{1}{2} + \theta + \epsilon}) \end{equation*} where \begin{equation*} C = \frac{\Gamma(\tfrac{3}{2}) }{4\pi^2} \frac{L(\frac{3}{2}, f\times g)}{\zeta(3)}= \frac{\Gamma(\tfrac{3}{2})}{4\pi ^2} \sum_{n \geq 1} \frac{a(n)\overline{b(n)}}{n^{k + \frac{1}{2}}}, \end{equation*} and $latex {\theta}$ denotes progress towards Selberg’s Eigenvalue Conjecture. Similarly, \begin{equation*} \frac{1}{X} \sum_{n \geq 1}\frac{S_f(n)S_g(n)}{n^{k – 1}}e^{-n/X} = C’X^{\frac{1}{2}} + O_{f,g,\epsilon}(X^{-\frac{1}{2} + \theta + \epsilon}), \end{equation*} where \begin{equation*} C’ = \frac{\Gamma(\tfrac{3}{2})}{4\pi^2} \frac{L(\frac{3}{2}, f\times \overline{g})}{\zeta(3)} = \frac{\Gamma(\tfrac{3}{2})}{4\pi ^2} \sum_{n \geq 1} \frac{a(n)b(n)}{n^{k + \frac{1}{2}}}.\end{equation*}

We have a complete meromorphic continuation, and it would not be hard to give additional terms in the asymptotic. But the next terms come from zeroes of the zeta function and are complicated to nail down exactly.

Choosing $latex {f = g}$, we recover a proof of the Classical Conjecture on Average. More interestingly, we show that the secondary growth terms do not arise from a pole, nor are there prescribed polar reasons for growth. The secondary growth in the classical result comes from choosing a sharp cutoff instead of the nicely behaving and natural smooth cutoffs.

We prove analogous results for sums of normalized Fourier coefficients $$ S_f^\alpha(n) := \sum_{m \leq n} \frac{a(m)}{m^\alpha} \tag{9}$$

for $latex {0 \leq \alpha < k}$.

In the path to proving these results, we explicitly demonstrate remarkable cancellation between Rankin-Selberg convolution $latex {L}$-functions $latex {L(s, f\times g)}$ and shifted convolution sums $$ Z(s, 0; f,g) := \sum_{n, h} \frac{a(n)\overline{b(n-h)}}{n^{s + k – 1}}. \tag{10}$$

Comparing our results and methodologies with the main results of [CN] guarantees similar cancellation for general level and general weight, including half-integral weight forms.

We provide additional applications of the meromorphic continuation of $latex {D(s, S_f \times S_g)}$ in forthcoming works, which will be uploaded to the arXiv and described briefly here soon.

For exact references, see the paper.

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Back to Work

It is January, and tomorrow studies at Brown resume for the spring. In addition to my course load, it is about time for me to figure out what area to specialize in. So I expect to reenter the math blogosphere!

 

Here’s to a new year.

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