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We estimate the sums \[ \sum_{c\leq x} \frac{S(m,n,c,\chi)}{c}, \] where the $S(m,n,c,\chi)$ are Kloosterman sums of half-integral weight on the modular group.
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H. Rademacher · 1973
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On modular forms of half integral weight
G. Shimura · 1973
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S. Niwa · 1975
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On construction of holomorphic cusp forms of half integral weight
T. Shintani · 1975
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The summation formulas for general Kloosterman sums
N. V. Proskurin · 1979
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The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums
N. V. Kuznetsov · 1980
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Asymptotic formulæ in combinatory analysis [Proc. London Math. Soc. (2) 17
G. H. Hardy and S. Ramanujan · 2000
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Arithmetic applications of Kloosterman sums
D. R. Heath-Brown · 2000
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A generalization of the Goldfeld-Sarnak estimate on Selberg’s Kloosterman zeta-function
W. d. A. Pribitkin · 2000
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Ramanujan’s unpublished manuscript on the partition and tau functions with proofs and commentary
B. C. Berndt and K. Ono · 2001
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The subconvexity problem for Artin L L -functions
W. Duke, J. B. Friedlander, and H. Iwaniec · 2002
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Spectral methods of automorphic forms
H. Iwaniec · 2002
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On the Niwa-Shintani theta-kernel lifting of modular forms
B. A. Cipra · 1983
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Sums of Kloosterman sums
D. Goldfeld and P. Sarnak · 1983
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The Selberg trace formula for PSL ( 2 , ℝ ) {\rm PSL}(2,\,{\mathbb{R}}) . Vol. 2
D. A. Hejhal · 1983
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Integral mean value theorems and the Ganelius inequality
I. Knowles · 1984
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Additive number theory and Maass forms
P. Sarnak · 1984
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Introduction to Kloostermania
M. N. Huxley · 1985
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Functoriality for the exterior square of GL 4 {\rm GL}_{4} and the symmetric fourth of GL 2 {\rm GL}_{2}
H. H. Kim · 2003
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On general Kloosterman sums
N. V. Proskurin · 2003
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Numerical computations with the trace formula and the Selberg eigenvalue conjecture
A. R. Booker and A. Strömbergsson · 2007
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Sums of Hecke eigenvalues over values of quadratic polynomials
V. Blomer · 2008
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On Linnik and Selberg’s conjecture about sums of Kloosterman sums
P. Sarnak and J. Tsimerman · 2009
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A generalized Kohnen-Zagier formula for Maass forms
E. M. Baruch and Z. Mao · 2010
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Equidistribution of Heegner points and the partition function
A. Folsom and R. Masri · 2010
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NIST Handbook of Mathematical Functions
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors · 2010
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Sums of Kloosterman sums over arithmetic progressions
S. Ganguly and J. Sengupta · 2012
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Bounds and algorithms for the K K -Bessel function of imaginary order
A. R. Booker, A. Strömbergsson, and H. Then · 2013
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http://dlmf.nist.gov/, Release 1.0.10 of 2015-08-07
NIST Digital Library of Mathematical Functions · 2015
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The L L -functions and modular forms database, Maass forms on Γ 0 ( 6 ) \Gamma_{0}(6)
LMFDB Collaboration · 2015
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