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 ${f(z)}$ and ${g(z)}$ are weight ${k}$ holomorphic cusp forms on ${\text{GL}_2}$ 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 ${n}$ coefficients of a cusp form ${f}$ by $$ S_f(n) := \sum_{m \leq n} a(m). \tag{1}$$
We consider upper bounds for the second moment of ${S_f(n)}$.
The famous Ramanujan-Petersson conjecture gives us that ${a(n)\ll n^{\frac{k-1}{2} + \epsilon}}$. So one might assume ${S_f(X) \ll X^{\frac{k-1}{2} + 1 + \epsilon}}$. 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 ${B(x)}$ 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 ${C}$ 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 ${B(X)}$. 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 ${D}$.
The question of better understanding ${B(X)}$ is analogous to understanding the error term in the circle problem or divisor problem. In our paper, we introduce the Dirichlet series $$D(s, S_f \times S_g) := \sum_{n \geq 1} \frac{S_f(n) \overline{S_g(n)}}{n^{s + k - 1}}$$ and $$D(s, S_f \times \overline{S_g}) := \sum_{n \geq 1} \frac{S_f(n)S_g(n)}{n^{s + k - 1}}$$ 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 ${S_f(n)}$.
Our primary result is the meromorphic continuation of ${D(s, S_f \times S_g)}$. As a first application, we prove a smoothed generalization to~(3).
Suppose either that ${f = g}$ is a Hecke eigenform or that ${f}$ and ${g}$ 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 ${\theta}$ 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 ${f = g}$, 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 ${0 \leq \alpha < k}$.
In the path to proving these results, we explicitly demonstrate remarkable cancellation between Rankin-Selberg convolution ${L}$-functions ${L(s, f\times g)}$ 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 ${D(s, S_f \times S_g)}$ 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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