472 research outputs found
Exponential Sums Related to Maass Forms
We estimate short exponential sums weighted by the Fourier coefficients of a
Maass form. This requires working out a certain transformation formula for
non-linear exponential sums, which is of independent interest. We also discuss
how the results depend on the growth of the Fourier coefficients in question.
As a byproduct of these considerations, we can slightly extend the range of
validity of a short exponential sum estimate for holomorphic cusp forms.
The short estimates allow us to reduce smoothing errors. In particular, we
prove an analogue of an approximate functional equation previously proven for
holomorphic cusp form coefficients.
As an application of these, we remove the logarithm from the classical upper
bound for long linear sums weighted by Fourier coefficients of Maass forms, the
resulting estimate being the best possible. This also involves improving the
upper bounds for long linear sums with rational additive twists, the gains
again allowed by the estimates for the short sums. Finally, we shall use the
approximate functional equation to bound somewhat longer short exponential
sums.Comment: 58 p
An additive problem in the Fourier coefficients of cusp forms
We establish an estimate on sums of shifted products of Fourier coefficients
coming from holomorphic or Maass cusp forms of arbitrary level and nebentypus.
These sums are analogous to the binary additive divisor sum which has been
studied extensively. As an application we derive, extending work of Duke,
Friedlander and Iwaniec, a subconvex estimate on the critical line for
L-functions associated to character twists of these cusp forms.Comment: 16 pages, LaTeX2e; v2: lots of changes, Theorem 2 is new, notation
changed to standard one, abstract and further references added; v3: minor
changes, some restriction imposed in Theorem 2, additional references; v4:
introduction revised, references added, typos corrected; v5: final, revised
version incorporating suggestions by the referee (e.g. Section 5 was added
The shifted convolution of generalized divisor functions
We prove an asymptotic formula for the shifted convolution of the divisor
functions and with , which is uniform in the shift
parameter and which has a power-saving error term, improving results obtained
previously by Fouvry and Tenenbaum and, more recently, by Drappeau
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