Analytic Number Theory and Algebraic Asymptotic Analysis

TL;DR

This monograph introduces the concepts of degree and logexponential degree to analyze the growth of functions in analytic number theory, providing new perspectives on longstanding conjectures like the Riemann hypothesis and abc conjecture.

Contribution

It defines and applies the notion of logexponential degree to extend results in algebraic asymptotic analysis and relate key number theory conjectures to function growth degrees.

Findings

Degree of $ ext{pi}(x)- ext{li}(x)$ is 1/2 if Riemann hypothesis holds

The abc conjecture corresponds to a function having degree 1

Logexponential degree offers a unified framework for growth analysis in number theory

Abstract

This monograph elucidates and extends many theorems and conjectures in analytic number theory and algebraic asymptotic analysis via the natural notion of "degree" and a more general notion that we call "logexponential degree." Specifically, we define the \emph{degree} of a real function $f$ whose domain is not bounded above to be the infimum of all real numbers $t$ such that $f(x)$ is $O(x^t)$. The Riemann hypothesis, for example, is equivalent to the statement that the degree of the function $\pi(x)- \operatorname{li}(x)$ is $1/2$, where $\pi(x)$ is the prime counting function and $\operatorname{li}(x)$ is the logarithmic integral function; likewise, the abc conjecture is equivalent to the statement that a particular function has degree 1. Part 1 of the text is a survey of analytic number theory, Part 2 introduces the notion of logexponential degree and uses it to extend results in algebraic asymptotic analysis, and Part 3 applies the results of Part 2 to the various functions that figure most prominently in analytic number theory and Diophantine analysis. Central to the notion of logexponential degree are Hardy's \emph{logarithmico-exponential functions}, which are real functions defined in a neighborhood of $\infty$ that can be built from $\operatorname{id}$, $\exp$, and $\log$ using the operations $+$, $\cdot$, $/$, and $\circ$. Such functions are natural benchmarks for the orders of growth of functions in analytic number theory. The main goal of Part 3 is to express the logexponential degree of various functions in analytic number theory in terms of as few "logexponential primitives" as possible.