Quaternary quadratic forms with prime discriminant
Jeremy Rouse, Katherine Thompson

TL;DR
This paper establishes explicit lower bounds for the number of representations of positive integers by positive-definite quaternary quadratic forms with prime discriminant, linking it to bounds on the Petersson norm of associated theta series.
Contribution
It provides a novel explicit lower bound on representations and connects it to bounds on the Petersson norm, depending on the smallest non-represented integer by the dual form.
Findings
Explicit lower bounds on representation counts for prime discriminant forms.
Upper bounds on Petersson norms related to non-represented integers.
Non-trivial bounds on the sum of integers not represented by the form.
Abstract
Let be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer by . This problem is connected with deriving an upper bound on the Petersson norm of the cuspidal part of the theta series of . We derive an upper bound on that depends on the smallest positive integer not represented by the dual form . In addition, we give a non-trivial upper bound on the sum of the integers excepted by .
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Taxonomy
TopicsAdvanced Mathematical Identities · Analytic Number Theory Research · Advanced Algebra and Geometry
