A Lambert series generating function is a special series summed over an arithmetic function $f$ defined by \[ L_f(q) := \sum_{n \geq 1} \frac{f(n) q^n}{1-q^n} = \sum_{m \geq 1} (f \ast 1)(m) q^m. \] Because of the way the left-hand-side terms of this type of generating function generate divisor sums of $f$ convolved by Dirichlet convolution with one, these expansions are natural ways to enumerate the ordinary generating functions of many multiplicative special functions in number theory. We present an overview of key properties of Lambert series generating function expansions, their more combinatorial generalizations, and include a compendia of tables illustrating known formulas for special cases of these series. In this sense, we focus more on the formal properties of the sequences that are enumerated by the Lambert series, and do not spend significant time treating these series as analytic objects subject to rigorous convergence constraints.
The first question one might ask before reading this document is: Why has is catalog of interesting Lambert series identities compiled? As with the indispensible reference by H. W. Gould and T. Shonhiwa, A catalog of interesting Dirichlet series, for Dirichlet series (DGF) identities, there are many situations in which one needs a summary reference on Lambert series and their properties. New work has been done recently tying Lambert series expansions to partition functions by expansions of their generating functions. In addition to these new expansions and providing an introduction to Lambert series, we have listings of classically relevant and "odds and ends'' examples for Lambert series summations that are occasionally useful in applications. If you see any topics or identities the author has missed, please contact us over email to append to this reference.
Factorization Theorems for Relatively Prime Divisor Sums, GCD Sums and Generalized Ramanujan Sums
We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$ . Our new factorization theorems provide analogs to these established expansions generating sums of the form $\sum_{d: (d,n)=1} f(d)$ (type I) and the Anderson-Apostol sums $\sum_{d|(m,n)} f(d) g(n/d)$ (type II) for any arithmetic functions $f$ and $g$ . Our treatment of the type II sums includes a matrix-based factorization method relating the partition function $p(n)$ to arbitrary arithmetic functions $f$ . We also concl…
