Related papers: Lambert series in analytic number theory
We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions…
The present paper presents some reflections of the author on divergent series and their role and place in mathematics over the centuries. The point of view presented here is limited to differential equations and dynamical systems.
We use a generalized Lambert series identity due to the first author to present q-series proofs of recent results of Imamoglu, Raum and Richter concerning recursive formulas for the coefficients of two 3rd order mock theta functions.…
We study the Lambert series $\mathscr{L}_q(s,x) = \sum_{k=1}^\infty k^s q^{k x}/(1-q^k)$, for all $s \in \mathbb{C}$. We obtain the complete asymptotic expansion of $\mathscr{L}_q(s,x)$ near $q=1$. Our analysis of the Lambert series yields…
A wardian calculus of sequences started almost seventy years ago constitutes the general scheme for extensions of the classical umbral operator calculus considered by many afterwards . At the same time this calculus is an example of the…
We prove several new variants of the Lambert series factorization theorem established in the first article "Generating special arithmetic functions by Lambert series factorizations" by Merca and Schmidt (2017). Several characteristic…
The main purpose of this survey is to introduce an inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic…
A series of lecture notes on the elementary theory of algebraic numbers, using only knowledge of a first-semester graduate course in algebra (primarily groups and rings). No prerequisite knowledge of fields is required. Based primarily on…
After defining in detail the Lambert $W$-function branches, we give a large number of exact identities involving (infinite) symmetric functions of these branches, as well as geometrically convergent series for all the branches. In doing so,…
The paper reviews various arithmetic analogues of Hamiltonian systems and presents some new facts suggesting ways to relate/unify these examples.
This paper analyzes several interest rates time series from the United Kingdom during the period 1999 to 2014. The analysis is carried out using a pioneering statistical tool in the financial literature: the complexity-entropy causality…
A categorized bibliography of combinators is given, providing what is believed to be a largely complete coverage of publications from the origination of combinators in 1920 to the present day.
In Early Transcendentals (The American Mathematical Monthly, Vol. 104, No 7) Steven Weintraub presents a rigorous justifcation of the "early transcendental" calculus textbook approach to the exponential and logarithmic functions. However,…
This is a report on the work of Robert Langlands, following his award of the Abel Prize in 2018. It includes his contributions to the general areas of Representation Theory, Automorphic Forms, Number Theory and Arithmetic Geometry. We have…
We retrace the recent history of the Umbral Calculus. After studying the classic results concerning polynomial sequences of binomial type, we generalize to a certain type of logarithmic series. Finally, we demonstrate numerous typical…
In this article, we prove two identities of generalized Lambert series. By introducing what we call $\mathcal{S}$-series, we establish relationships between multiple generalized Lambert series and multiple infinite products. Compared with…
In this text, we wish to provide the reader with a short guide to recent works on the theory of dilatations in Commutative Algebra and Algebraic Geometry. These works fall naturally into two categories: one emphasises foundational and…
The article gives a survey of mathematical proofs that rely on computer calculations and formal proofs.
This article was prepared in connection with the 2009 Barnett lecture at the University of Cincinnati, and deals with various classes of fractal sets and analysis on them.
This work presents an introduction to feature-based time-series analysis. The time series as a data type is first described, along with an overview of the interdisciplinary time-series analysis literature. I then summarize the range of…