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<p>Various results about a divisor function with Diophantine properties are obtained, including a simple asymptotic formula for its sum and a Voronoï-type formula. The proofs rely on analytic properties of certain Dirichlet series that are expressed in terms of Hecke’s zeta functions with Grössencharaktere associated to a real quadratic number field. Also used are new estimates and asymptotics for the standard hypergeometric function that are uniform in parameters, which are of independent interest.</p>

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A Diophantine divisor problem and Hecke zeta functions

  • 期刊论文

  • 发表日期：

    2026年6月29日

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  • Olga Balkanova,
  • William Duke,
  • Dmitry Frolenkov

摘要

Various results about a divisor function with Diophantine properties are obtained, including a simple asymptotic formula for its sum and a Voronoï-type formula. The proofs rely on analytic properties of certain Dirichlet series that are expressed in terms of Hecke’s zeta functions with Grössencharaktere associated to a real quadratic number field. Also used are new estimates and asymptotics for the standard hypergeometric function that are uniform in parameters, which are of independent interest.