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 ${f(z)}$ is a weight ${k}$ 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 ${S_f(n)}$ 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 ${S_f(n)}$ 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 ${B(x)}$ 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 ${X^{\frac{3}{4} + \epsilon}}$ around ${X}$ 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 ${D(s, S_f \times S_f)}$ 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 ${X^{\frac{2}{3}} (\log X)^{\frac{2}{3}}}$. More formally, we prove the following.
Suppose either that ${f}$ is a Hecke eigenform or that ${f}$ 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 ${y}$ is the solution to ${y (\log y)^2 = X}$. Then we prove that the Classical Conjecture holds on average over intervals of width ${X/y}$ around ${X}$.
We also demonstrate improved bounds for short-interval estimates of width as low as ${X^\frac{1}{2}}$.
There are two major obstructions to improving our result. Firstly, we morally use the convexity result in the ${t}$-aspect for the size of ${L(\frac{1}{2} + it, f\times f)}$. If we insert the bound from the Lindelöf 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 ${\mu_j}$ are a basis of Maass forms and ${V(X,s)}$ 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 ${X^{\frac{1}{2}}}$ around ${X}$. This would lead to better bounds on individual ${S_f(X)}$ as well, but requires significant improvement.
For more details and specific references, see the paper on the arXiv.
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