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Bounds for Kloosterman sums on .

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This study establishes new power-saving bounds for Kloosterman sums using explicit exponential sum representations. These findings advance the understanding of number theory and have applications beyond Sarnak

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Area of Science:

  • Number Theory
  • Analytic Number Theory
  • Algebraic Number Theory

Background:

  • Kloosterman sums are fundamental objects in number theory.
  • Previous work established bounds for specific cases, but general power-saving bounds were lacking.
  • Weyl elements play a crucial role in the study of automorphic forms and number theoretic sums.

Purpose of the Study:

  • To establish novel power-saving bounds for Kloosterman sums associated with specific Weyl elements.
  • To extend these bounds to a broader class of Kloosterman sums.
  • To apply these results to address open problems, such as Sarnak's density conjecture.

Main Methods:

  • Derivation of explicit representations of Kloosterman sums as exponential sums.
  • Utilizing techniques from analytic number theory to establish bounds.
  • Applying established number theoretic frameworks to analyze Weyl elements.

Main Results:

  • Established power-saving bounds for Kloosterman sums related to the long Weyl element and a Weyl element of order 2.
  • Achieved power-saving bounds for all Kloosterman sums on .
  • Provided an application that surpasses Sarnak's density conjecture for the principal congruence subgroup of prime level.

Conclusions:

  • The established power-saving bounds offer significant improvements in estimating Kloosterman sums.
  • The explicit exponential sum representations are key to achieving these improved bounds.
  • The results have implications for understanding the distribution of number theoretic objects and advancing conjectures in the field.