Related papers: Lambert series in analytic number theory
We establish two identities for Lambert series and double Lambert series, thereby resolving conjectures of Andrews, Dixit, Schultz and Yee (Acta Arith.~181:253--286, 2017), as well as Amdeberhan, Andrews and Ballantine (J Combin Theory…
The translated logarithmic Lambert function is defined and basic analytic properties of the function are obtained including the derivative, integral, Taylor series expansion, real branches and asymptotic approximation of the function.…
We describe an algorithm to evaluate all the complex branches of the Lambert W function with rigorous error bounds in interval arithmetic, which has been implemented in the Arb library. The classic 1996 paper on the Lambert W function by…
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…
We consider special Lambert series as generating functions of divisor sums and determine their complete transseries expansion near rational roots of unity. Our methods also yield new insights into the Laurent expansions and modularity…
Let $\alpha$ and $\beta$ be two nonnegative integers such that $\beta < \alpha$. For an arbitrary sequence $\{a_n\}_{n\geqslant 1}$ of complex numbers, we consider the generalized Lambert series in order to investigate linear combinations…
We prove new variants of the Lambert series factorization theorems studied by Merca and Schmidt (2017) which correspond to a more general class of Lambert series expansions of the form $L_a(\alpha, \beta, q) := \sum_{n \geq 1} a_n q^{\alpha…
Using elementary means, we prove an identity giving the infinite product form of a sum of Lambert series originally stated by Venkatachaliengar, then rediscovered by Andrews, Lewis, and Liu. Then we derive two identities expressing certain…
We survey some recent developments in the analytic theory of multiple Dirichlet series with arithmetical coefficients on the numerators.
We connect a primitive operation from arithmetic -- summing the digits of a base-$B$ integer -- to $q$-series and product generating functions analogous to those in partition theory. We find digit sum generating functions to be intertwined…
In this paper, we study a class of double Lambert series and establish several identities and transformation relations for them. These formulae provide useful tools for reducing certain double Lambert series to single Lambert series. As…
We summarize the known useful and interesting results and formulas we have discovered so far in this collaborative article summarizing results from two related articles by Merca and Schmidt arriving at related so-termed Lambert series…
We study some series expansions for the Lambert $W$ function. We show that known asymptotic series converge in both real and complex domains. We establish the precise domains of convergence and other properties of the series, including…
We first summarize joint work on several preliminary canonical Lambert series factorization theorems. Within this article we establish new analogs to these original factorization theorems which characterize two specific primary cases of the…
Andrews, Dixit, Schultz, and Yee conjecture the parity of a double Lambert series. In 2026, Amdeberhan, Andrews, and Ballantine offer some ideas that are pointing in the right direction for the proof. In this paper, we complete the rest of…
In my 2011 Annals of Applied Statistics article [Goerg (2011)] I wrote that "Whereas the Lambert $W$ function plays an important role in mathematics, physics, chemistry, biology and other fields, it has not yet been used in statistics."…
The history of the development of the concept of complex numbers from the 16th to 19th centuries. The origin and refinement of the geometric and physical meaning of complex numbers, the emergence of vectoral analysis.
A new algebraic treatment of dependent type theory is proposed using ideas derived from topos theory and algebraic set theory.
We show how the generalized Lambert series sum(n>=1, x*q^n/(1-x*q^n)) can be computed with Theta convergence. This allows the computation of the sum of the inverse Fibonacci numbers without splitting the sum into even and odd part. The…
We provide an annotated bibliography for the study of Hamilton cycles in random graphs and hypergraphs.