Abstract
In this paper we apply some simple approaches to improve a recent result of Luo on the shifted convolution sum of the Fourier coefficients of a cusp form and a theta series. A mean square formula is established in order to explore the right order of magnitude of this shifted convolution sum.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Luo’s proof seems to only give that \(g_{\ell }\) is the smallest integer strictly greater than \((\ell +1)/2\). This means \(\vartheta _{3}=\tfrac{1}{8}\) and \(\vartheta _{5}=\tfrac{1}{5}\) instead of \(\vartheta _{3}=\tfrac{1}{6}\) and \(\vartheta _{5}=\tfrac{1}{4}\).
The symbols \(F(x) = \Omega _{+}(G(x))\), \(F(x) = \Omega _{-}(G(x))\) and \(F(x) = \Omega (G(x))\) (as \(x\rightarrow \infty \)) mean that \(\displaystyle \limsup _{x\rightarrow \infty } F(x)/G(x)>0\), \(\displaystyle \limsup _{x\rightarrow \infty } \,(-F(x))/G(x)>0\) and \(\displaystyle \limsup _{x\rightarrow \infty } |F(x)|/G(x)>0\), respectively.
References
Bateman, P.T.: On the representations of a number as the sum of three squares. Trans. Am. Math. Soc. 71, 70–101 (1951)
Deshouillers, J.-M., Iwaniec, H.: An additive divisor problem. J. Lond. Math. Soc. 26(2), 1–14 (1982)
Duke, W., Iwaniec, H.: Bilinear forms in the Fourier coefficients of half-integral weight cusp forms and sums over primes. Math. Ann. 286, 783–802 (1990)
Hafner, J.L., Ivić, A.: On sums of Fourier coefficients of cusp forms. Enseign. Math. 35(3–4), 375–382 (1989)
Harcos, G., Michel, P.: The subconvexity problem for Rankin–Selberg \(L\)-functions and equidistribution of Heegner points. II. Invent. Math. 163, 581–655 (2006)
Hardy, G.H.: Representation of a number as the sum of squares. Trans. Am. Math. Soc. 17, 255–284 (1920)
Heath-Brown, D.R.: The twelfth power moment of the Riemann zeta-function. Quart. J. Math. Oxford 29, 443–462 (1978)
Holowinsky, R.: A sieve method for shifted convolution sums. Duke Math. J. 146(3), 401–448 (2009)
Ingham, A.E.: Some asymptotic formulae in the theory of numbers. J. Lond. Math. Soc. 2, 202–208 (1927)
Ivić, A., Motohashi, Y.: On some estimates involving the binary additive divisor problem. Quart. J. Math. Oxford 46, 471–483 (1995)
Iwaniec, H.: Topics in Classical Automorphic Forms. Graduate Studies in Mathematics, vol. 17. American Mathematical Society, Providence, RI (1997)
Iwaniec, H.: Spectral Methods of Automorphic Forms, 2nd edn. American Mathematical Society, Providence, RI (2002)
Jutila, M.: A method in the theory of exponential sums. Tata Institute of Fundamental Research, Bombay Lectures No. 80. Springer-Verlag, Berlin-Heidelberg-New York (1987)
Jutila, M.: On exponential sums involving the Ramanujan function. Proc. Indian Acad. Sci. (Math. Sci.) 97(1–3), 157–166 (1987)
Jutila, M.: The Additive Divisor Problem and Exponential Sums. Advances in Number Theory. Oxford University Press, New York (1993)
Jutila, M.: The additive divisor problem and its analogs for Fourier coefficients of cusp forms. I. Math. Z. 223(3), 435–461 (1996)
Jutila, M.: The additive divisor problem and its analogs for Fourier coefficients of cusp forms. II. Math. Z. 225, 625–637 (1997)
Lü, G.: The average value of Fourier coefficients of cusp forms in arithmetic progressions. J. Number Theory 129, 477–487 (2009)
Luo, W.: Shifted convolution of cusp-forms with \(\theta \)-series. Abh. Math. Semin. Univ. Hambg. 81, 45–53 (2011)
Meurman, T.: On the Binary Additive Divisor Problem, Number Theory (Turku, 1999), pp. 223–246. de Gruyter, Berlin (2001)
Motohashi, Y.: The binary additive divisor problem. Ann. Sci. Ec. Norm. Supér. 27, 529–572 (1994)
Smith, R.A.: Fourier coefficients of modular forms over arithmetic progressions. I, II. With remarks by M. R. Murty. C. R. Math. Rep. Acad. Sci. Canada 15(2–3), 91–98 (1993). 85-90
Walfisz, A.: Über Gitterpunkte in mehrdimensional Ellipsoiden, VIII. Trudy Tbil. Math. Inst. 5, 1–65 (1938)
Wu, J.: Power sums of Hecke eigenvalues and applications. Acta Arith. 137, 333–344 (2009)
Acknowledgments
The authors thank the referee for his remarks and careful reading. In particular, he indicated: (a) Concerning the foot note on p. 2, the results in [19] are in fact correct as stated since one can just simply split the \(n\)-sum as \(\sum _{n\leqslant X^A} + \sum _{n>X^A}\) for sufficiently large \(A>0\), and bound the tail using instead \(p=g_l+1\) in the notation of [19]. (b) In view of the simple identity \(\lambda _f(mn) = \sum _{d\mid (m, n)} \mu (d) \lambda _f(m/d)\lambda _f(n/d)\), Lemma 3.1 follows trivially, where \(\mu (n)\) is the Möbius function.
Author information
Authors and Affiliations
Corresponding author
Additional information
Guangshi Lü is supported in part by the National Natural Science Foundation of China (11031004, 11171182), NCET (NCET-10-0548) and Shandong Province Natural Science Foundation for distinguished Young Scholars (JQ201102). The first two authors are supported in part by IRT1264. This work was completed when Jie Wu visited China University of Mining and Technology in 2013. He would like to thank the institute for the pleasant working conditions. Wenguang Zhai is supported by the National Key Basic Research Program of China (Grant No. 2013CB834201), the Natural Science Foundation of China (Grant No. 11171344) and the Natural Science Foundation of Beijing (Grant No. 1112010).
Rights and permissions
About this article
Cite this article
Lü, G., Wu, J. & Zhai, W. Shifted convolution of cusp-forms with \(\theta \)-series. Ramanujan J 40, 115–133 (2016). https://doi.org/10.1007/s11139-015-9678-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-015-9678-8