Euler-type Recurrence Relation for Arbitrary Arithmetical Function

TL;DR

This work establishes a universal Euler-type recurrence framework for arbitrary arithmetic functions by fusing Lambert series with Euler's Pentagonal Number Theorem. The core result expresses $\sum_{k=1}^{n} g(k)\omega(n-k)$ in terms of a base function $f$ through a divisor-sum relation $g(n)=\sum_{d|n} f(d)$, enabling recurrences for a broad class of functions, including $\phi$, $\tau$, $\lambda$, and $\mu$, as well as partition-related and sum-of-divisors quantities. The paper also develops alternative derivations via Jacobi and Gauss identities to access recurrences for $q(n)$, $qq(n)$, $p(n)$, and representation counts $r_k(n)$, and extends these ideas with Euler's logarithmic derivative to obtain additional identities and congruences. Overall, the results unify multiple known recurrences and provide a versatile toolkit for deriving Euler-type recurrences across arithmetic and combinatorial contexts, linking divisor sums, partition theory, and sums of squares. These methods promise new insights into arithmetic function behavior and enable efficient computation of various recurrence relations without explicit factorization.

Abstract

An interplay between the Lambert series and Euler's Pentagonal Number Theorem gives an Euler-type recurrence relation for any given arithmetical function. As consequences of this, we present Euler-type recurrence relations for some well-known arithmetic functions. Furthermore, we derive Euler-type recurrence relations for certain partition functions and sum-of-divisors functions using infinite product identities of Jacobi and Gauss.

Theorems & Definitions (32)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Theorem 2.2
• proof