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Mean square estimate for relatively short exponential sums involving Fourier coefficients of cusp forms. (English) Zbl 1327.11057
The author estimates the mean square of (relatively) short exponential sums related to the Fourier coefficients of a holomorphic cusp form $$f(z)=\sum_{n=1}^\infty a(n) n^{(\kappa-1)/2} e(nz)$$ of weight $$\kappa$$, namely, $I=\int_{M}^{M+\triangle} w(x) \left|\sum_{x\leq n\leq x+T} a(n) e\left(\frac{h}{k}\, n\right)\right|^2 \; dx.$ In this expression, $$h$$ and $$k$$ are coprime integers with $$0<k\ll M^{1/2-\varepsilon'}$$ for some fixed $$\varepsilon'>0$$, $$\triangle=\min(k^2 M^{1/2+\varepsilon}, M)$$, $$w(x)$$ is a positive pump function on $$[M,M+\triangle]$$ satisfying certain technical conditions, $$T\asymp M^\delta$$ for some $$1/2<\delta<1$$ (“relatively short” exponential sum). She obtains $I=S+O(k^2 M^{1+\varepsilon}+\triangle M^\varepsilon T^{1/2} k)+O\left(\sqrt{|S|(k^2M^{1+\varepsilon}+\triangle M^\varepsilon T^{1/2}k)}\right).$ The quantity $$S$$ is explicit and satisfies $$S\ll \triangle T$$ when $$k\gg TM^{-1/2}$$ and $$S\ll \triangle M^{1/2}k$$ otherwise. The given bounds for $$S$$ are the actual orders of magnitude of that term. When omitting the weight function $$w(x)$$, she gives upper bounds for $$I$$ depending on the size of $$k$$. The proof of the general theorem starts off with a Voronoi-type summation formula and proceeds via careful calculations, in particular, to get cancellations on certain diagonal terms.

##### MSC:
 11L07 Estimates on exponential sums 11F11 Holomorphic modular forms of integral weight 11F30 Fourier coefficients of automorphic forms
##### Keywords:
Fourier coefficients; cusp forms; exponential sums; mean square
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##### References:
  Cramér, H.: Über zwei Sätze von Herrn G. H. Hardy. - Math. Z. 15, 1922, 201–210.  Ernvall-Hytönen, A.-M.: On the mean square of short exponential sums related to cusp forms. - Funct. Approx. Comment. Math. 45:1, 2011, 97–104. · Zbl 1243.11084  Ernvall-Hytönen, A.-M., and K. Karppinen: On short exponential sums involving Fourier coefficients of holomorphic cusp forms. - Int. Math. Res. Not. IMRN 2008:10, Art. ID. rnn022, 1–44, 2008. · Zbl 1247.11106  Heath-Brown, D. R., and K. Tsang: Sign changes of E(t), ∆(x) and P (x). - J. Number Theory 49, 73–83, 1994. · Zbl 0810.11046  Ivić, A.: On the divisor function and the Riemann zeta-function in short intervals. - Ramanujan J. 19:2, 207–224, 2009. · Zbl 1226.11086  Jutila, M.: On the divisor problem for short intervals. - Ann. Univ. Turku. Ser. A I 186, 23–30, 1984. · Zbl 0536.10032  Jutila, M.: Lectures on a method in the theory of exponential sums. - Tata Inst. Fund. Res. Lect. Math. Phys. 80, Tata Inst. of Fundamental Research, Bombay, 1987. · Zbl 0671.10031  Jutila, M.: On exponential sums involving the Ramanujan function. - Proc. Indian Acad. Sci. Math. Sci. 97:1-3, 157–166, 1987. · Zbl 0658.10043  Jutila, M., and Y. Motohashi: Uniform bound for Hecke L-functions. - Acta Math. 195, 61–115, 2005. · Zbl 1098.11034  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. - Proc. Cambridge Philos. Soc. 35, 357–372, 1939. · Zbl 0021.39202  Vesalainen, E. V.: Moments and oscillations of exponential sums related to cusp forms. arXiv:1402.2746, submitted. · Zbl 1423.11087  Wilton, J. R.: A note on Ramanujan’s arithmetical function $$\tau$$(n). - Proc. Cambridge Philos. Soc. 25:2, 121–129, 1929. Received 14 November 2013 ●Revised received 14 July 2013 ●Accepted 26 September 2014 · JFM 55.0709.02
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