• Corpus ID: 118256023

Modular Functions In Analytic Number Theory

  title={Modular Functions In Analytic Number Theory},
  author={Marvin Isadore Knopp},
Knopp's engaging book presents an introduction to modular functions in number theory by concentrating on two modular functions, $\eta(\tau)$ and $\vartheta(\tau)$, and their applications to two number-theoretic functions, $p(n)$ and $r_s(n)$. They are well chosen, as at the heart of these particular applications to the treatment of these specific number-theoretic functions lies the general theory of automorphic functions, a theory of far-reaching significance with important connections to a… 

Quantum Modular Forms and Singular Combinatorial Series with Distinct Roots of Unity

Understanding the relationship between mock modular forms and quantum modular forms is a problem of current interest. Both mock and quantum modular forms exhibit modular-like transformation

Some New Old-Fashioned Modular Identities

AbstractThis paper uses modular functions on the theta group to derive an exact formula for the sum $$\sum\limits_{\left| j \right| \leqslant n^{{1 \mathord{\left/ {\vphantom {1 2}} \right.

Elementary Methods in Number Theory

Elementary Methods in Number Theory begins with "a first course in number theory" for students with no previous knowledge of the subject. The main topics are divisibility, prime numbers, and

Survey of Modular Functions of One Variable

In this paper a survey is given of the theory of modular functions of one variable, including Dirichlet series, functional equations, compactifications, Hecke operators, Eisenstein series and the

A Hecke correspondence theorem for automorphic integrals with rational period functions

1. Following Riemann, Hecke in his celebrated work [4], [5] uncovered-using the Mellin transform and its inversemthe systematic theory relating automorphic (in particular, modular) forms to Dirichlet

A Single-Variable Proof of the Omega SPT Congruence Family Over Powers of 5

In 2018 Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third order mock theta function $\omega(q)$. Their

Finding Modular Functions for Ramanujan-Type Identities

This paper is concerned with a class of partition functions $a(n)$ introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins,


It is widely recognized that the work of Ramanujan deeply influenced the direction of modern number theory. This influence resonates clearly in the “Ramanujan conjectures”. Here I will explore

Continued fractions and modular functions

It is widely recognized that the work of Ramanujan deeply influenced the direction of modern number theory. This influence resonates clearly in the “Ramanujan conjectures.” Here I will explore

H-convex standard fundamental domain of a subgroup of a modular group

AbstractThe two major ways of obtaining fundamental domains for discrete subgroups of SL(2,ℝ) are the Dirichlet Polygon construction (see Lehner in Discontinuous Groups and Automorphic Functions,