What question did this study set out to answer?

The central aim is to examine the power moments of functions counting representations of integers by ideals.

March 10, 2026Open Access

Generalized divisor problem associated with the number of integral ideals

Q: What does this research mean for the field?

The function τ_k,K(n) exhibits specific statistical properties, including established asymptotic behavior for representations of n as a product of norms of k ideals in non-normal cubic fields. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

NGNaveen K. GodaraIndian Institute of Technology Madras AJAnuj JakharIndian Institute of Technology Madras RJRenu JoshiTechnische Universität Braunschweig

Key Points

The central aim is to examine the power moments of functions counting representations of integers by ideals.
Analyzed the function τ k , K ( n ) to count representations of n by norms of k ideals.
Studied statistical properties, including mean and variance of τ k , K ( n ).
Derived asymptotic formulae for positive integral coefficient mixed forms.
Identified specific power moments of τ k , K ( n ).
Established statistical properties, including calculated mean and variance.
Formulated asymptotic expressions for fixed mixed forms with positive integral coefficients.

Abstract

In this article, we study the power moments of the function τ k , K ( n ) , which counts the number of representations of n as a product of norms of k ideals in the ring of integers of a non-normal cubic field K . Additionally, we analyze the statistical properties of τ k , K ( n ) , including its mean and variance. Furthermore, we establish asymptotic formulae for cases where n is represented by a fixed mixed form with positive integral coefficients.

KI fragen

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