Asymptotic bounds for special values of shifted convolution Dirichlet series

Hoffstein and Hulse defined the shifted convolution series of two cusp forms by "shifting" the usual Rankin-Selberg convolution L-series by a parameter h. We use the theory of harmonic Maass forms to study the behavior in h-aspect of certain values of these series and prove a polynomial bound as h approaches infinity. Our method relies on a result of Mertens and Ono, who showed that these values are Fourier coefficients of mixed mock modular forms.

[1]  Zachary A. Kent,et al.  Special L -values and periods of weakly holomorphic modular forms , 2014 .

[2]  Michael H. Mertens,et al.  Special values of shifted convolution Dirichlet series , 2014, 1406.0770.

[3]  Michael H. Mertens.  Eichler–Selberg type identities for mixed mock modular forms , 2014, 1404.5491.

[4]  G. Harcos.  Twisted Hilbert modular L-functions and spectral theory , 2014, 1402.1332.

[5]  P. Maga.  Shifted convolution sums and Burgess type subconvexity over number fields , 2013, 1312.0553.

[6]  Ö. Imamoḡlu,et al.  Holomorphic projections and Ramanujan’s mock theta functions , 2013, Proceedings of the National Academy of Sciences.

[7]  Zachary A. Kent,et al.  Eichler–Shimura theory for mock modular forms , 2013 .

[8]  Byungchan Kim,et al.  Eichler integrals and harmonic weak Maass forms , 2012, 1210.3783.

[9]  J. Hoffstein,et al.  Multiple Dirichlet Series and Shifted Convolutions , 2011, 1110.4868.

[10]  G. Harcos,et al.  Twisted L-Functions Over Number Fields and Hilbert’s Eleventh Problem , 2009, 0904.2429.

[11]  G. Harcos,et al.  The spectral decomposition of shifted convolution sums , 2007, math/0703246.

[12]  G. Harcos,et al.  A Burgess-like subconvex bound for twisted L-functions , 2007 .

[13]  J. Bruinier,et al.  On two geometric theta lifts , 2002, math/0212286.

[14]  G. Harcos.  An additive problem in the Fourier coefficients of cusp forms , 2001, math/0101096.

[15]  D. Zagier,et al.  Period functions for Maass wave forms. I. , 2001, math/0101270.

[16]  R. Borcherds.  Automorphic forms with singularities on Grassmannians , 1996, alg-geom/9609022.

[17]  Don Zagier,et al.  Heegner points and derivatives ofL-series , 1986 .

[18]  J. Sturm.  Projections of $C^\infty$ automorphic forms , 1980 .

[19]  K. Ono.  Unearthing the visions of a master: harmonic Maass forms and number theory , 2008 .

[20]  G. Harcos,et al.  Hybrid bounds for twisted L-functions , 2008 .

[21]  J. Sturm.  PROJECTIONS OF C ° ° AUTOMORPHIC FORMS , 2007 .

[22]  H. Iwaniec,et al.  Bounds for automorphic L-functions , 2005 .

[23]  V. Blomer.  Shifted convolution sums and subconvexity bounds for automorphic L-functions , 2004 .

[24]  Pham Do Tuan,et al.  On the estimation of Fourier coefficients. , 1969 .