A SHIFTED CONVOLUTION SUM OF AND THE FOURIER COEFFICIENTS OF HECKE–MAASS FORMS II

Let $d_{3}(n)$ be the divisor function of order three. Let $g$ be a Hecke–Maass form for $\unicode[STIX]{x1D6E4}$ with $\unicode[STIX]{x1D6E5}g=(1/4+t^{2})g$. Suppose that $\unicode[STIX]{x1D706}_{g}(n)$ is the $n$th Hecke eigenvalue of $g$. Using the Voronoi summation formula for $\unicode[STIX]{x1D706}_{g}(n)$ and the Kuznetsov trace formula, we estimate a shifted convolution sum of $d_{3}(n)$ and $\unicode[STIX]{x1D706}_{g}(n)$ and show that $$\begin{eqnarray}\mathop{\sum }_{n\leq x}d_{3}(n)\unicode[STIX]{x1D706}_{g}(n-1)\ll _{t,\unicode[STIX]{x1D700}}x^{8/9+\unicode[STIX]{x1D700}}.\end{eqnarray}$$ This corrects and improves the result of the author [‘Shifted convolution sum of $d_{3}$ and the Fourier coefficients of Hecke–Maass forms’, Bull. Aust. Math. Soc.92 (2015), 195–204].

Shifted convolution sums for higher rank groups

In this paper, we study some shifted convolution sums for higher rank groups. In particular, we establish an asymptotic formula for a {\mathrm{GL}(4)\times\mathrm{GL}(2)} shifted convolution sum\sum_{n\leq x}\lvert\lambda_{f}(n)\rvert^{2}r_{l}(n+b),where {\lambda_{f}(n)} are normalized Fourier coefficients of a Hecke holomorphic cusp form and {r_{l}(n)} denotes the number of representations of n by the quadratic form {x_{1}^{2}+\cdots+x_{l}^{2}}.

An elementary property of correlations

International audience We study the "shift-Ramanujan expansion" to obtain a formulae for the shifted convolution sum $C_{f,g} (N,a)$ of general functions f, g satisfying Ramanujan Conjecture; here, the shift-Ramanujan expansion is with respect to a shift factor a > 0. Assuming Delange Hypothesis for the correlation, we get the "Ramanujan exact explicit formula", a kind of finite shift-Ramanujan expansion. A noteworthy case is when f = g = Λ, the von Mangoldt function; so $C_{\Lamda, \Lambda} (N, 2k)$, for natural k, corresponds to 2k-twin primes; under the assumption of Delange Hypothesis, we easily obtain the proof of Hardy-Littlewood Conjecture for this case.

A shifted convolution sum for \mathrm{GL}(3) × \mathrm{GL}(2)

In this paper, we estimate the shifted convolution sum \sum_{n\geqslant 1}\lambda_{1}(1,n)\lambda_{2}(n+h)V\Big{(}\frac{n}{X}\Big{)}, where V is a smooth function with support in {[1,2]} , {1\leqslant|h|\leqslant X} , and {\lambda_{1}(1,n)} and {\lambda_{2}(n)} are the n-th Fourier coefficients of {\mathrm{SL}(3,\mathbf{Z})} and {\mathrm{SL}(2,\mathbf{Z})} Hecke–Maass cusp forms, respectively. We prove an upper bound {O(X^{\frac{21}{22}+\varepsilon})} , updating a recent result of Munshi.

SHIFTED CONVOLUTION SUM OF AND THE FOURIER COEFFICIENT OF HECKE–MAASS FORMS

Let $\{{\it\phi}_{j}(z):j\geq 1\}$ be an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue $1/4+t_{j}^{2}$. Let ${\it\lambda}_{j}(n)$ be the $n$th Fourier coefficient of ${\it\phi}_{j}$ and $d_{3}(n)$ the divisor function of order three. In this paper, by the circle method and the Voronoi summation formula, the average value of the shifted convolution sum for $d_{3}(n)$ and ${\it\lambda}_{j}(n)$ is considered, leading to the estimate $$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\leq X}d_{3}(n){\it\lambda}_{j}(n-1)\ll X^{29/30+{\it\varepsilon}}, & & \displaystyle \nonumber\end{eqnarray}$$ where the implied constant depends only on $t_{j}$ and ${\it\varepsilon}$.