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On exponential sums involving the Ramanujan function. (English) Zbl 0658.10043
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 R. A. Rankin’s result in [Math. Proc. Camb. Philos. Soc. 35, 357-372 (1939; Zbl 0021.39202)] that $$\sum_{n\leq x}\tau^ 2(n)=x^{12}+O(x^{12- 2/5}).$$ 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 $$x^ 6 \log x$$. 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.
Reviewer: M.Sheingorn

##### MSC:
 11L40 Estimates on character sums 11F11 Holomorphic modular forms of integral weight
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##### References:
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