Papers › Analytic Number Theory and Algebraic Asymptotic Analysis
Analytic Number Theory and Algebraic Asymptotic Analysis
Jesse Elliott
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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(xᵗ). The Riemann hypothesis, for example, is equivalent to the statement that the degree of the function π(x)- li(x) is 1/2, where π(x) is the prime counting function and 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 ∞ that can be built from id, exp, and log using the operations +, ·, /, and ∘. 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.
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