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Sums of Fourier coefficients of a Maass form for SL3(ℤ) twisted by exponential functions
-
Xiumin Ren
and
Yangbo Ye
Published/Copyright:
December 2, 2011
Abstract.
Let f be a Maass cusp form for SL3(ℤ) with Fourier coefficients
. We consider the sum
, where
. A bound better than
is proved to be valid for certain transcendental numbers 
. This
bound improves Miller's result (2006) for these α. For α close
to a rational number 
with 
,
the smooth sum is
further proved to decay rapidly. This extends a result of
Booker (2000 and 2005) on a smooth sum of
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.
Received: 2011-07-07
Revised: 2011-11-24
Published Online: 2011-12-02
Published in Print: 2014-01-01
© 2014 by Walter de Gruyter Berlin Boston
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Articles in the same Issue
- Masthead
- Pseudovarieties generated by Brauer type monoids
- On some arithmetic properties of Siegel functions (II)
- Perfect vector alignment of the vorticity and the vortex stretching is one forever
- Frobenius groups of automorphisms and their fixed points
- Schubert calculus and the Hopf algebra structures of exceptional Lie groups
- The sovability of norm, bilinear and quadratic equations over finite fields via spectra of graphs
- Periodicity in the stable representation theory of crystallographic groups
- Sums of Fourier coefficients of a Maass form for SL3(ℤ) twisted by exponential functions
- The rational classification of links of codimension > 2
- On the classifying space for proper actions of groups with cyclic torsion
- Turán determinants of Bessel functions
