Let \(\zeta (s)\) be the Riemann zeta function, so \(\zeta^3 (s)= \sum_{n\geq 1} \tau_3 (n) n^{-s}\), where \(\tau_3 (n)\) is the number of factorizations \(n=klm\). Let \(f(z)\) be a holomorphic cusp form of weight \(k\) for the modular group, with Fourier expansion \(f(z)= \sum_{n\geq 1} a(n) n^{(k- 1)/2} \exp(2\pi i nz)\). The author studies the shifted convolution of \(\zeta^3\) with \(f\). That is, let \(\Phi (f, s)= \sum_{n\geq 1} \tau_3 (n) a(n-1) n^{-s}\). This sum converges absolutely for \(\operatorname{Re}(s)> 1\). The author proves that \(\Phi (f, s)\) has analytic continuation to \(\operatorname{Re}(s)> 23/24\). He also shows that \[ \Psi (f, x)= \sum_{n\leq x} \tau_3 (n) a(rn- 1)\ll x^{71/ 72+ \varepsilon}, \] uniformly in \(0< r< x^{1/ 24}\), where the implied constant depends on \(f\) and \(\varepsilon\); the weaker bound \(x^{1+ \varepsilon}\) follows directly from Deligne’s bound for \(a(n)\).
The proofs of these theorems are obtained by studying \(\sum g(n) \tau_3 (n) a(rn- 1)\) in place of \(\Psi (f, x)\), where \(g\) is a compactly-supported \(C^\infty\) function which is identically 1 on \([x/2, x]\) and has some additional properties. Following the work of Duke, Friedlander, and Iwaniec, the estimation of this sum may be obtained provided one controls a certain sum of Kloosterman sums times an appropriate smoothing function. Using the Poisson summation formula and Cauchy’s inequality, the problem is further reduced to the estimation of a certain three-variable Kloosterman-like sum. The author establishes such an estimate by applying the work of A. Adolphson and S. Sperber on exponential sums and Newton polyhedra [Ann. Math. (2) 130, 367–406 (1989; Zbl 0723.14017)], which is based on Deligne’s work on the Weil conjectures, along with techniques of Bombieri and Hooley to deal with some of the degenerate cases.