Incomplete Kloosterman sums and multiplicative inverses in short intervals
Tim Browning, Alan Haynes
TL;DR
This paper studies the solvability of the equation xy ≡ 1 (mod p) within short intervals, using a mean value theorem for incomplete Kloosterman sums to analyze the distribution of multiplicative inverses.
Contribution
It introduces a new approach leveraging incomplete Kloosterman sums to understand the distribution of solutions in short intervals, advancing previous methods in the field.
Findings
Established conditions for the solvability of xy ≡ 1 in short intervals
Developed a mean value theorem for incomplete Kloosterman sums
Provided bounds on the number of solutions in specified intervals
Abstract
We investigate the solubility of the congruence xy=1 (mod p), where p is a prime and x,y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums.
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Taxonomy
TopicsAnalytic Number Theory Research · advanced mathematical theories · Advanced Mathematical Theories and Applications