Sums of Fourier coefficients of a Maass form for SL3(Z) twisted by exponential functions

作者：Ren Xiumin^*; Ye Yangbo

来源：Forum Mathematicum, 2014, 26(1): 221-238.

DOI：10.1515/FORM.2011.157

• Maass form
• SL3(Z)
• oscillation of Maass form
• Voronoi summation formula

Let f be a Maass cusp form for SL3(Z) with Fourier coefficients A(f)(m, n). We consider the sum Sigma(n > 0) A(f)(m, n)phi(n/X)e(alpha n), where phi is an element of C-c(infinity)(0, infinity). A bound better than O-m,O-f,O-epsilon(X3/4+epsilon) is proved to be valid for certain transcendental numbers alpha is an element of R. This bound improves Miller's result (2006) for these alpha. For alpha close to a rational number a/q with q(3) << X1-epsilon/m, the smooth sum Sigma(n > 0) A(f)(m, n)phi(n/X)e(alpha n) is further proved to decay rapidly. This extends a result of Booker (2000 and 2005) on a smooth sum of A(f)(m, n) without the exponential function. These bounds manifest a strange vibration nature of Maass cusp forms. The main techniques include rational approximation of real numbers, a Voronoi summation formula for SL3(Z), and its asymptotic expansion.

• 出版日期2014-1
• 单位山东大学