A Sieve Method for Shifted Convolution Sums

Details A Sieve Method for Shifted Convolution Sums A Sieve Method for Shifted Convolution Sums

2008-09-09

arxiv.org/abs/0809.1669v1

Mathematics

Number Theory

To appear in Duke Math. J. See also arXiv:0809.1640 for analogous results

Scientific paper

We study the average size of shifted convolution summation terms related to the problem of Quantum Unique Ergodicity on ${\rm SL}_2 (\mathbbm{Z})\backslash \mathbbm{H}$. Establishing an upper-bound sieve method for handling such sums, we achieve an unconditional result which 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.

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