On exponential sums involving the Ramanujan function
Let \(\tau\) (n) be the Ramanujan \(\tau\)-function, i.e. \(x\prod^{\infty}_{m=1}(1-x^ m)^{24}=\sum^{\infty}_{n=1}\tau (n)x^ n.\) This paper establishes the following estimate: \[ (*)\quad \sum_{n\leq x}\tau (n) e(n\alpha)\quad \ll \quad x^ 6. \] Here \(e(x)=e^{2\pi ix}\) and the implied constant is independent of \(\alpha\in [0,1].\) This result is best possible, as may be easily seen from \textit{R. A. Rankin}'s result in [Math. Proc. Camb. Philos. Soc. 35, 357-372 (1939; Zbl 0021.39202)] that \(\sum_{n\leq x}\tau^ 2(n)=x^{12}+O(x^{12- 2/5}).\) \textit{J. R. Wilton} [Math. Proc. Camb. Philos. Soc. 25, 121-129 (1929; JFM 55.0709.02)] showed an estimate like (*) with \(x^ 6\) replaced by \(x^ 6 \log x\). \textit{L. A. Parson} and the reviewer [Mathematika 29, 270-277 (1982; Zbl 0512.10029)] suggested that (*) might well be true and proved a simple version with \(\alpha =p/q\), an exponent less than 6, but the constant approaching \(\infty\) with q. Such a result is also given in the author's paper [``Lectures on a method in the theory of exponential sums'' (Lectures on Mathematics and Physics, Vol. 80, Bombay 1987)]. The proofs in the present paper are similar to those given by the author in [J. Reine Angew. Math. 355, 173-190 (1985; Zbl 0542.10032)] for the corresponding exponential sums involving the divisor function d(n). At a crucial stage in the estimates, a very beautiful approximate functional equation for certain differences of the basic exponential sums is given. The result (*) also holds, mutatis mutandis, for Fourier coefficients of any (analytic) cusp form.
scientific article; zbMATH DE number 4075192