The variance of the number of sums of two squares in Fq[T] in short intervals
Consider the number of integers in a short interval that can be represented as a sum of two squares. What is an estimate for the variance of these counts over random short intervals? We resolve a function field variant of this problem in the large q limit, finding a connection to the z-measures firs...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
| Published: |
Johns Hopkins University Press
2021
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| Summary: | Consider the number of integers in a short interval that can be represented as a sum
of two squares. What is an estimate for the variance of these counts over random short
intervals? We resolve a function field variant of this problem in the large q limit, finding
a connection to the z-measures first investigated in the context of harmonic analysis
on the infinite symmetric group. A similar connection to z-measures is established
for sums over short intervals of the divisor functions dz(n). We use these results to
make conjectures in the setting of the integers which match very well with numerically
produced data. Our proofs depend on equidistribution results of N. Katz and W.
Sawin. |
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