# Questions tagged [asymptotics]

Asymptotic behavior of functions, asymptotic series and related topics

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### Heuristics for constrained maximal volumes in hypercubes as $n \to \infty$

It can be shown that there is a unique maximal surface of revolution with constant positive Gaussian curvature embedded in $[0,1]^3$ with a pair of antipodal points as cone points which attain the ...
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### Asymptotics of a strange oscillatory function

Consider the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=\sum_{n\geq 1}\sin(x/n^2)$. It is easy to see that $f(x) = O(\sqrt{x})$ for large real $x$. Is it true that $f(x)>0$ for $x>0$...
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### Multivariate Jackson inequality for Chebyshev approximation

There is an approximation of a multivariate function by a Chebyshev polynomial of degree n. One needs to understand how the approximation error behaves depending on the degree of the polynomial or ...
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### Error function of the second moment of the divisor function

It is easy to show that the second moment of the divisor function has asymptotics: $$\sum_{n\leq x} d_0(n)^2 = xP(\log(x))+E_2(x)$$ Where $P$ is some polynomial and that: $$E_2 = o(x)$$ Previously, ...
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I'm a rather inexperienced researcher, I've been stuck on a question for a while. I would like to find the largest $N = f(n)$ that satisfies the following inequality: $$\sum_{j = 0} ^ n p^{n - j} (-1)... 6 votes 2 answers 648 views ### On the asymptotic behaviour of the series \sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right ) near s=0 I am interested in determining the behaviour of the the series/function$$f(s)=\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$$near s=0. It is clear that f(0) is undefined.... • 397 1 vote 0 answers 197 views ### Asymptotic behaviour of a sum involving Möbius function (This is a cross-posted simplification of this question posted in MSE which did not have a complete answer.) I am trying to get the asymptotic behaviour when n grows to infinity of a partial sum of ... • 229 -1 votes 1 answer 164 views ### Is it possible to consistently and naturally define this subset of Hardy field? Consider Hardy field H, the field of germs of functions at positive infinity. Can we define H_I\subset H, such that it would have the following properties: H_I is an integral domain. For each ... • 9,496 5 votes 2 answers 286 views ### Small parameter expansion of an integral I am trying analyze an integral of the form$$I(\varepsilon)=\int_0^\infty f(t,\varepsilon) \,dt$$where \varepsilon is a small real parameter. The function f(t,\varepsilon) is very complicated, ... 5 votes 1 answer 383 views ### Asymptotic solution of a system of ODEs I have asked this question on math.stackexchange, however, have not got any answer. Therefore, I suspect that this system of ordinary differential equations cannot be solved analytically. But I still ... • 482 7 votes 1 answer 464 views ### On a paper by Dimitrie Pompéiu and on one (in two parts) by Edmund Landau To celebrate the new year and the future of mathematics (or the mathematics of future), I see no better way to ask a question stemming from my researches on power series. The two papers the title ... • 5,672 0 votes 0 answers 20 views ### Asymptotic behavior of a truncated MGF of a quadratic form of a standard gaussian Assume X follows an n dimensional standard Gaussian distribution and A is an n\times n semi-definite matrix. I am trying to analyze the asymptotic behavior of \mathbb{E} e^{-X^TAX} \mathbf{1}... • 101 2 votes 1 answer 92 views ### Show v(x,t) \in L^2([0,T];H^2(\mathbb{R})) when v(x,t) is a transformation of a L^2([0,T];H^2(\mathbb{R})) function Context: I am reading a paper on Long-Time Asymptotics of the thin film equations, in which the authors consider the strong solutions of the thin film equation in 1-D and transform them using a time-... • 23 0 votes 1 answer 153 views ### Asymptotic behavior of the polylogarithm function and generalisation So, right now I am writing my master thesis and I need to find a reference for a formula I found in a paper:$$ \sum_{k=1}^{\infty}k^{-\alpha}(1-\varepsilon)^k\sim b+c\Gamma(1-\alpha)\varepsilon^{\...
Let $p$ be prime and $r$ be a positive integer. I am interested in asymptotics for the number of imaginary quadratic discriminants $d$ such that $p$ does not divide the conductor of $d$, $p$ splits ...