Asymptotische Formeln für die Fourierkoeffizienten ganzer Modulformen

On sums of $\mathit{SL}(3,\mathbb{Z})$ Kloosterman sums

“…with J w,µ given by the integrals (14), (15), (28), (29), and (30), J I (µ) given by (17), and C(µ) replaced with C * (µ).…”

“…The classical Kloosterman sums originated in 1926 in the context of applying the circle method to counting representations of integers by the four-term quadratic form ax 2 + by 2 + cz 2 + dt 2 [16] where c c ≡ 1 (mod c), cc ≡ 1 (mod c ). In 1927, Kloosterman [15] used these sums to estimate Fourier coefficients of modular forms, as did Rademacher in 1937 [23]. Optimal estimates for individual Kloosterman sums were obtained in 1948 by André Weil [32]: |S(a, b, c)| ≤ d(c) (a, b, c) √ c, where d(c) is the number of positive divisors of c and (a, b, c) is the greatest common divisor.…”

The role of the Ramanujan conjecture in analytic number theory

“…This refined circle method approach turned out to be fruitful in the theory of holomorphic cusp forms. Soon after Hecke proved his bound λ f (n) = O(n 1/2 )-the simplest approximation towards the Ramanujan conjecture-from quite general principles, Kloosterman [79] was able to improve this to λ f (n) = O ε (n 3/8+ε ). Estermann [47] and Salié [118] then completely clarified this connection between non-trivial bounds for Kloosterman sums and bounds for Fourier coefficients of holomorphic cusp forms of integer weight k 2.…”

Generalized Kloosterman sums and the Fourier coefficients of cusp forms

“…In this final section we improve upon the Fourier coefficient estimate of § 1.5 and simultaneously obtain a Fourier coefficient estimate for cusp forms on congruence subgroups of G(y/2) and G{y/3). This is accomplished by modifying the classical Kloosterman version of the circle method [5].…”

“…By Lemma 3.5 it suffices to prove c" = 0{nr'2-xl4ln3/2no_x,2{n)) for F G C°{rm{N'), -r, vty with Fourier expansion (3)(4)(5)(6) FXz) = Z cne{nz/N'y/hT). where tí* is uniquely determined by mh'h'* = 1 (mod qN*), 0 < /'* < qN* and, for tí* I', qN* < h'* < 2qN*.…”