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Convolutions of Fourier coefficients of cusp forms. (English) Zbl 1006.11019

The author proves the interesting bounds \[ \begin{aligned} T(N;f)&=\sum_{n=1}^Nt(n)t(n+f)\ll N^{2/3+\varepsilon}, \qquad(1\leq f\ll N^{2/3}),\\ T(N)&=\sum_{n=1}^{N-1}t(n)t(N-n)\ll N^{1/2+\theta+\varepsilon}. \end{aligned} \] Here as usual \(\rho(n)=\rho(1)t(n)\), where \(\rho(n)\) is the \(n\)-th Fourier coefficient of a nonholomorphic cusp form (the Maass wave form) for the full modular group. The constant \(\theta\) satisfies \(t(n)\ll n^{\theta +\varepsilon}\) (\(\theta\leq 5/28\) is known; \(\theta=0\) is the Ramanujan-Petersson conjecture). The sums above with \(t(n)\) replaced by \(d(n)\), the number of divisors of \(n\), are a classical problem in analytic number theory. For the error terms in the corresponding problem one has precisely the same error terms as those obtained above by the author. There are, of course, deep reasons for this analogy, which is explained in detail. Any improvements of the exponents above probably would have to come from completely new ideas. One may obtain results analogous to these also for the case of holomorphic cusp forms but the Maass wave forms are somewhat more difficult to treat.
The author’s proof is based on several powerful techniques from modern analytic number theory. It briefly consists of the following. Instead of the sums \(T(N;f)\) and \(T(N)\) the author works with certain smoothed sums. He applies his variant of the circle method to these sums, and singles out the relevant expression involving exponential sums. These sums are then transformed by using the automorphy of Maass wave forms. At this stage sums of Kloosterman sums emerge, that are transformed by the Kuznetsov trace formulas. The resulting expression is then estimated by the spectral large sieve inequality, yielding finally the above displayed result.

MSC:

11F30 Fourier coefficients of automorphic forms
11N37 Asymptotic results on arithmetic functions
11P55 Applications of the Hardy-Littlewood method
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
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