Analytic Number Theory and Algebraic Asymptotic Analysis - Jesse Elliott

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Résumé :
This monograph elucidates and extends many theorems and conjectures in analytic number theory and algebraic asymptotic analysis via the natural notions of degree and logexponential degree. 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. 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.Central to the notion of logexponential degree are G H Hardy's 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. The logexponential degree of the function eγπp

Biographie:
-- it provides a formal algebraic language for comparing growth rates, interpreting numerical evidence, and connecting longstanding problems, including the Riemann hypothesis. The new degree formalism incorporates Hardy's logarithmico-exponential functions, built from id, exp, and log using the operations of addition, multiplication, division, and composition, as benchmarks for comparison. The monograph develops foundational results about the structure and algebra of the degree formalism, including its relation to Karamata theory, Hardy fields, transseries, and asymptotic differential algebra. While not offering proofs of major conjectures, it proposes a new way of establishing interdependencies among error terms. Applications to summatory functions, prime gaps, the Riemann zeta function, and Diophantine approximation demonstrate the framework's reach and utility. These applications reduce error terms in analytic number theory to core set of primitives, including the function π...

Sommaire:
This monograph introduces a unified framework for analyzing and comparing the asymptotic growth of number-theoretic functions through the novel notions of degree and logexponential degree. Extending the asymptotic calculus shaped by du Bois-Reymond, Landau, and Hardy -- rooted in notations such as O, o, and ∼...