Duke Mathematical Journal

A sieve method for shifted convolution sums

Roman Holowinsky

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Abstract

We study the average size of shifted convolution summation terms related to the problem of quantum unique ergodicity (QUE) on ${\rm SL}_2 (\mathbbm{Z})\backslash \mathbbm{H}$. Establishing an upper-bound sieve method for handling such sums, we achieve an unconditional result that suggests that the average size of the summation terms should be sufficient in application to quantum unique ergodicity. In other words, cancellations among the summation terms, although welcomed, may not be required. Furthermore, the sieve method may be applied to shifted sums of other multiplicative functions with similar results under suitable conditions

Article information

Source
Duke Math. J. Volume 146, Number 3 (2009), 401-448.

Dates
First available in Project Euclid: 14 January 2009

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1231947434

Digital Object Identifier
doi:10.1215/00127094-2009-002

Mathematical Reviews number (MathSciNet)
MR2484279

Zentralblatt MATH identifier
05505895

Subjects
Primary: 11N36: Applications of sieve methods 11F30: Fourier coefficients of automorphic forms
Secondary: 11M99: None of the above, but in this section

Citation

Holowinsky, Roman. A sieve method for shifted convolution sums. Duke Math. J. 146 (2009), no. 3, 401--448. doi:10.1215/00127094-2009-002. http://projecteuclid.org/euclid.dmj/1231947434.


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