Sums with hyperbolic conditions

by Leo Schäfer
Doctoral thesis
Date of Examination:2025-07-14
Date of issue:2025-12-18
Advisor:Prof. Dr. Jörg Brüdern
Referee:Prof. Dr. Jörg Brüdern
Referee:Prof. Dr. Damaris Schindler
Persistent Address: http://dx.doi.org/10.53846/goediss-11715

 

 

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Abstract

English

We develop a generalization of the hyperbola method, a technique used to obtain asymptotic estimates for summations involving a product of variables as a constraint, given that similar sums over rectangular regions can be evaluated. Compared to previous variations, our approach widens the range of admissible main terms and requires milder assumptions on the error terms. We then make use of the hyperbola method to work on a specific counting problem. Using the Hardy-Littlewood circle method, we compute an asymptotic formula where our previous results are used to evaluate or bound the exponential sums that arise.
Keywords: Analytic number theory; Hardy-Littlewood circle method; Hyperbola method; Diophantine equations
 
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