Two-parameter Identities for Divisor Sums in Algebraic Number Fields

TL;DR

This paper extends Ramanujan's divisor sum identities to algebraic number fields, introducing two-parameter Riesz sum identities involving ideal counting functions and providing bounds for these sums.

Contribution

It introduces novel two-parameter Riesz sum identities for divisor functions in algebraic number fields, connecting classical identities to algebraic number theory.

Findings

Established Riesz sum identities with an extra parameter involving ideal counting functions.

Derived upper bounds for the sums as the upper index tends to infinity.

Linked classical divisor sum identities to algebraic number field problems.

Abstract

In a one-page fragment published with his lost notebook, Ramanujan stated two double series identities associated, respectively, with the famous Gauss Circle and Dirichlet Divisor problems. The identities contain an "extra" parameter, and it is possible that Ramanujan derived these identities with the intent of attacking these famous problems. Similar famous unsolved problems are connected with $f_K(n)$, the number of integral ideals of norm $n$ in an algebraic number field $K$. In this paper we establish Riesz sum identities containing an "extra" parameter and involving $f_K(n)$, or divisor functions associated with $K$. Upper bounds for the sums as the upper index tends to infinity are also established.