I am happy to announce that my frequent collaborators, Alex Walker and Chan
Kuan, and I have just posted a preprint to the
arxiv called *Sums of cusp form coefficients
along quadratic sequences*.

Our primary result is the following.

Let $f(z) = \sum_{n \geq 1} a(n) q^n = \sum_{n \geq 1} A(n) n^{\frac{k-1}{2}} q^n$ denote a holomorphic cusp form of weight $k \geq 2$ on $\Gamma_0(N)$, possibly with nontrivial nebentypus. For any $h > 0$ and any $\epsilon > 0$, we have that \begin{equation} \sum_{n^2 + h \leq X^2} A(n^2 + h) = c_{f, h} X + O_{f, h, \epsilon}(X^{\eta(k) + \epsilon}), \end{equation} where \begin{equation*} \eta(k) = 1 - \frac{1}{k + 3 - \sqrt{k(k-2)}} \approx \frac{3}{4} + \frac{1}{32k - 44} + O(1/k^2). \end{equation*}

The constant $c_{f, h}$ above is typically $0$, but we weren't the first to notice this.

## Context

We approach this by studying the Dirichlet series \begin{equation*} D_h(s) := \sum_{n \geq 1} \frac{r_1(n) a(n+h)}{(n + h)^{s}}, \end{equation*} where \begin{equation*} r_1(n) = \begin{cases} 1 & n = 0 \\ 2 & n = m^2, m \neq 0 \\ 0 & \text{else} \end{cases} \end{equation*} is the number of ways of writing $n$ as a (sum of exactly $1$) square.

This approach isn't new. In his 1984 paper *Additive number theory and Maass
forms*, Peter Sarnak suggests that one could relate the series
\begin{equation*}
\sum_{n \geq 1} \frac{d(n^2 + h)}{(n^2 + h)^2}
\end{equation*}
to a Petersson inner product involving a theta function, a weight $0$
Eisenstein series, and a half-integral weight Poincaré series. This inner
product can be understood spectrally, hence the spectrum of the half-integral
weight hyperbolic Laplacian (and half-integral weight Maass forms) reflect the
behavior of the ordinary-seeming partial sums
\begin{equation*}
\sum_{n \leq X} d(n^2 + h).
\end{equation*}

A broader class of sums was studied by Blomer.Blomer. *Sums of Hecke
eigenvalues over values of quadratic polynomials*. **IMRN**, 2008.

Let $q(x) \in \mathbb{Z}[x]$ denote any monic quadratic polynomial. Then Blomer showed that \begin{equation*} \sum_{n \leq X} A(q(n)) = c_{f, q}X + O_{f, q, \epsilon}(X^{\frac{6}{7} + \epsilon}). \end{equation*} Blomer already noted that the main term typically doesn't occur.

More recently, Templier and TsimermanTemplier and Tsimerman.
*Non-split sums of coefficients of $\mathrm{GL}(2)$-automorphic forms*.
**Israel J. Math** 2013

showed that $D_h(s)$ has polynomial growth in vertical strips and has reasonable polar behavior. This allows them to show that \begin{equation*} \sum_{n \geq 0} A(n^2 + h) g\big( (n^2 + h)/X \big) = c_{f, h, g} X + O_\epsilon(X^{\frac{1}{2} + \Theta + \epsilon}), \end{equation*} where $\Theta$ is a term that is probably $0$ coming from a contribution from potentially exceptional eigenvalues of the Laplacian and the Selberg Eigenvalue Conjecture, and where $g$ is a smooth function of sufficient decay.

Templier and Tsimerman approach their result in two different ways: one studies the Dirichlet series $D_h(s)$ with the same initial steps as outlined by Sarnak. The second way is more representation theoretic and allows greater flexibility in the permitted forms.

## Placing our techniques in context

Broadly, our approach begins in the same way as Templier and Tsimerman — we study $D_h(s)$ through a Petersson inner product involving half-integral weight Poincaré series. The great challenge is to understand the discrete spectrum and half-integral weight Maass forms, and we deviate from Templier and Tsimerman sharply in our treatment of the discrete spectrum.

For each eigenvalue $\lambda_j$ there is an associated type $\frac{1}{2} +
it_j$ and form
\begin{equation*}
\mu_j(z) = \sum_{n \neq 0} \rho_j(n) W_{\mathrm{sgn}(n) \frac{k}{2}, it_j}(4\pi \lvert n \rvert y) e(nx),
\end{equation*}
where $W$ is a Whittaker function and the coefficients $\rho_j(n)$ are very
mysterious. We average Maass forms in long averages over the eigenvalues and
types (indexed by $j$) *and* long averages over coefficients $n$. We base the
former approach average on Blomer's work above, and for the latter we improve
on (a part of) the seminar work of Duke, Friedlander, and
Iwaniec.Duke, Friedlander, Iwaniec. *The subconvexity problem for
Artin $L$-functions*. **Inventiones**. 2002.

For this, it is necessary to establish certain uniform bounds for Whittaker functions.

To apply our bounds for the discrete spectrum, Maass forms, and Whittaker functions, we use that $f$ is holomorphic in an essential way. We decompose $f$ into a sum of finitely many holomorphic Poincaré series. This is done by Blomer as well. But in contrast, we study the resulting shifted convolutions whereas Blomer recollects terms into Kloosterman and Salié type sums.

Ultimately we conclude with a standard contour shifting argument.

## Additional remarks

### Continuous vs discrete spectra

The quality of our bound mirrors the quality of our understanding of the discrete spectrum. This is interesting in that the continuous and discrete spectra are typically of a similar calibre of size and difficulty.

But here we are examining a half-integral weight object into half-integral weight spectra. The continuous spectrum comes from Eisenstein series, and it turns out that the coefficients of real-analytic half-integral weight Eisenstein series are (essentially) Dirichlet $L$-functions — and these are relatively easy to understand.

Existing bounds for half-integral weight Maass forms are much weaker than corresponding bounds for full-integral weight Maass forms.

In principle, there is also a residual spectrum to consider here (in contrast to weight $0$ spectral expansions). But in practice this is perfectly handled by in the work of Templier and Tsimerman and presents no further difficulty.

### Two too-brief summaries

One too-brief-to-be-correct summary of this paper is that *by restricting to a
smaller class of quadratic polynomials than Blomer, it is possible to prove a
stronger result*. In reality, we restrict to a class of quadratic polynomials
that allows the corresponding Dirichlet series to be easily recognized as a
Petersson inner product involving a standard theta function and a half-integral
weight Poincaré series.

Another too-brief-to-be-correct summary of this paper is that *examining
Whittaker functions and Bessel functions even closer reveals that they control
all of multiplicative number theory*. Actually, this might be
correct.As a corollary, I guess all multiplicative number theory is
controlled by monodromy?

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