zbMATH — the first resource for mathematics

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

11L40 Estimates on character sums
11F11 Holomorphic modular forms of integral weight
Full Text: DOI
[1] Apostol T M, Modular functions and Dirichlet series in number theory, in:Graduate texts in mathematics (New York, Heidelberg, Berlin: Springer Verlag) (1976)
[2] Deligne P, La conjecture de Weil,Inst. Hautes √Čtudes Sci. Publ. Math. 53 (1974) 273–307 · Zbl 0287.14001 · doi:10.1007/BF02684373
[3] Jutila M, On exponential sums involving the divisor function.J. Reine Angew. Math. 355 (1985) 173–190 · Zbl 0542.10032
[4] Jutila M, Lectures on a method in the theory of exponential sums, in:Lectures on mathematics and physics (Bombay: Tata Institute of Fundamental Research) Vol. 80 (1987) · Zbl 0671.10031
[5] Jutila M, Mean value estimates for exponential sums,Journ. Arithm. Ulm (1987) (to appear)
[6] Parson L A and Sheingorn M, Exponential sums connected with Ramanujan’s function \(\tau\)(n),Mathematika 29 (1982) 270–277 · Zbl 0512.10029 · doi:10.1112/S0025579300012365
[7] Rankin R A, Contributions to the theory of Ramanujan’s function \(\tau\)(n) and similar arithmetical functions II. The order of Fourier coefficients of integral modular forms,Math. Proc. Cambridge Philos. Soc. 35 (1939) 357–372 · Zbl 0021.39202 · doi:10.1017/S0305004100021101
[8] Wilton J R, A note on Ramanujan’s function \(\tau\)(n),Math. Proc. Cambridge Philos. Soc. 25(1929) 121–129 · JFM 55.0709.02 · doi:10.1017/S0305004100018636
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.