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 whose domain is not bounded above to be the infimum of all real numbers such that is . The Riemann hypothesis, for example, is equivalent to the statement that the degree of the function is , where is the prime counting function and 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…
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