Asymptotic analysis

Résumé

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function f (n) as n becomes very large. If f(n) = n2 + 3n, then as n becomes very large, the term 3n becomes insignificant compared to n2. The function f(n) is said to be "asymptotically equivalent to n2, as n → ∞". This is often written symbolically as f (n) ~ n2, which is read as "f(n) is asymptotic to n2". An example of an important asymptotic result is the prime number theorem. Let π(x) denote the prime-counting function (which is not directly related to the constant pi), i.e. π(x) is the number of prime numbers that are less than or equal to x. Then the theorem states that \pi(x)\sim\frac{x}{\ln x}. Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is o

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https://en.wikipedia.org/wiki/Asymptotic_analysis

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The Shifted Convolution of Generalized Divisor Functions

Berke Topacogullari

We prove an asymptotic formula for the shifted convolution of the divisor functions d(k)(n) and d(n) with k >= 4, which is uniform in the shift parameter and which has a power saving error term, improving results obtained previously by Fouvry and Tenenbaum and, more recently, by Drappeau.

OXFORD UNIV PRESS2018

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The fourth moment of individual Dirichlet L-functions on the critical line

Berke Topacogullari

We prove an asymptotic formula for the second moment of a product of two Dirichlet L-functions on the critical line, which has a power saving in the error term and which is uniform with respect to the involved Dirichlet characters. As special cases we give uniform asymptotic formulae for the fourth moment of individual Dirichlet L-functions and for the second moment of Dedekind zeta functions of quadratic number fields on the critical line.

2020

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Non-Exponential Variations for Classical Results in First Passage Percolation

Jacques Saliba

We study in this thesis the asymptotic behavior of optimal paths on a random graph model, the configuration model, for which we assign continuous random positive weights on its edges. We start by describing the asymptotic behavior of the diameter and the flooding time on the graph for a set of light-tailed edge-weights, and we prove that it is the largest class of densities for which these precise asymptotic expressions for the diameter/flooding time hold. We then show how the weighted optimal path can be constructed using a particular positive recurrent Markov chain. We finally study, in the last chapter, the noise sensitivity of the model by replacing every edge-weight independently by a new realization of the same distribution, with probability epsilon. We show that the optimal paths between two vertices before and after this modification are asymptotically ''independent'' conditioning on the graph, as the number of vertices tends to infinity.

EPFL2020

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