Correlated sums of \(r(n)\). (English) Zbl 0933.11045

Chamizo, Fernando

J. Math. Soc. Japan 51, No. 1, 237-252 (1999).

The author considers the asymptotic formula \[ S(N,m) := \sum_{n\leq N}r(n)r(n+m) = 8\left|2^{k+1}-3\right|\sigma(\frac{m}{2^k})\frac{N}{m} + E(N,m),\tag{1} \] where \(r(n)\) is the number of representation of \(n\) as a sum of two integer squares, \(m\) is a natural number which is not necessarily fixed, \(2^k||n\), \(\sigma(n)\) is the sum of divisors of \(n\), and \(E(N,m)\) is the error term in (1).
The method of proof consists of using the spectral theory of automorphic forms, similarly as was done by Y. Motohashi [Ann. Sci. Ec. Norm. Supér., IV. Sér. 27, 529–572 (1994; Zbl 0819.11038)], who obtained precise results for the analogous problem of the evaluation of \(D(N,m) := \sum_{n\leq N}d(n)d(n+m)\), where \(d(n)\) is the number of divisors of \(n\). The author first connects \(S(N,m)\) and Hecke operators in an elementary way. Then he provides the spectral expansion of \(S(N,m)\) in terms of non-holomorphic modular forms, valid both for even and odd \(m\). In the ensuing identity the Selberg–Harish-Chandra transform appears, and the proof depends on the so-called “pretrace” formula, which appears distinctly simpler than Kuznetsov’s famous trace formula, which was used by Motohashi (op. cit.) in dealing with \(D(N,m)\).
Estimates of the error term function \(E(N,m)\) are made to depend on bounds for \(U_j(T) := \sum_{T<\kappa_j\leq 2T}|u_j(z)|^4\), where \(\{u_j(z)\} (\operatorname{Im} z > 0)\) is the set of Hecke cusp forms of \(\Gamma = \mathrm{PSL}_2(\mathbb C)\) or \(\Gamma = \Gamma_0/\{\pm\mathrm{Id}\}\) with respective eigenvalues \(\{\lambda_j = \frac{1}{4} + \kappa_j^2\}\). Two conditional bounds for \(E(N,m)\) are derived, assuming plausible conjectures on the order of \(U_j(T)\). Unconditionally the author proves \[ E(N,m) \ll_\varepsilon N^\varepsilon\min\left(N^{23}m^{5/42}, N^{17/23} + N^{1/2}m^{47/196}\right) \qquad(m \leq N), \]
\[ E(N,m) \ll_\varepsilon N^{7/24}m^{11/24+\varepsilon}\qquad(N \leq m), \] as well as \[ \sum_{M<m\leq 2M}\alpha_mE(N,m) \ll_\varepsilon ||\alpha||_2(N^{2/3+\varepsilon}M^{1/2} + N^{1/3}m^{5/6+\varepsilon}), \] where \(\alpha = \{\alpha_m\} \in \mathbb C\) is arbitrary and \(M,N > 1\).

Reviewer: Aleksandar Ivić (Beograd)

MSC:

11N37	Asymptotic results on arithmetic functions
11N75	Applications of automorphic functions and forms to multiplicative problems
11F72	Spectral theory; trace formulas (e.g., that of Selberg)
11E25	Sums of squares and representations by other particular quadratic forms

Keywords:

asymptotic formulas; pretrace formula; Hecke operators; non-holomorphic cusp form; sums of squares

Citations:

Zbl 0819.11038