arXiv subjects

Lorenz Milla

Publications and source records attributed to Lorenz Milla.

6 recordsLinked to original sources

arXiv2025

Easy Proof of Three Recursive $π$-Algorithms -- Einfacher Beweis dreier rekursiver $π$-Algorithmen

This paper consists of three independent parts: First we use only elementary algebra to prove that the quartic algorithm of the Borwein brothers has exactly the same output as the Brent-Salamin algorithm, but that the latter needs twice as many iterations. Second we use integral calculus to prove that the Brent-Salamin algorithm approximates $π$. Combining these results proves that the Borwein brothers' quartic algorithm also approximates $π$. Third, we prove the quadratic convergence of the Brent-Salamin algorithm, which also proves the quartic convergence of Borwein's algorithm. -- -- Dieses Paper besteht aus drei unabhängigen Teilen: Erstens beweisen wir mit elementarer Algebra, dass der Borwein-Algorithmus vierter Ordnung die gleichen Ergebnisse liefert wie der Brent-Salamin-Algorithmus, wobei letzterer doppelt so viele Iterationen benötigt. Zweitens beweisen wir mit Integralrechnung, dass der Brent-Salamin-Algorithmus gegen $π$ konvergiert. Hieraus folgt, dass der Borwein-Algorithmus vierter Ordnung ebenfalls gegen $π$ konvergiert. Drittens beweisen wir die quadratische Konvergenz des Brent-Salamin-Algorithmus und somit auch die quartische Konvergenz des Borwein-Algorithmus.

math.NT↗

arXiv2024

Asymptotic expansions of truncated hypergeometric series for $1/π$

In this paper, we consider rational hypergeometric series of the form \[\frac{p}π= \sum_{k=0}^\infty u_k\quad\text{with}\quad u_k=\frac{\left(\frac{1}{2}\right)_k \left(q\right)_k \left(1-q\right)_k}{(k!)^3}(r+s\,k)\,t^k,\] where $(a)_k$ denotes the Pochhammer symbol and $p,q,r,s,t$ are algebraic coefficients. Using only the first $n+1$ terms of this series, we define the remainder \[\mathcal{R}_n = \frac{p}π - \sum_{k=0}^n u_k=\sum_{k=n+1}^\infty u_k.\] We consider an asymptotic expansion of $\mathcal{R}_n$. More precisely, we provide a recursive relation for determining the coefficients $c_j$ such that \[ \mathcal{R}_n = \frac{\left(\frac{1}{2}\right)_n \left(q\right)_n \left(1-q\right)_n}{n!^3}nt^n\left(\sum_{j=0}^{J-1}\frac{c_j}{n^j}+\mathcal{O}\left(n^{-J}\right)\right),\qquad n \rightarrow \infty.\] Here we need $J<\infty$ to approximate $\mathcal{R}_n$, because (like the Stirling series) this series diverges if $J\rightarrow\infty$. By applying our recursive relation to the Chudnovsky formula, we solve an open problem posed by Han and Chen.

math.NT↗

arXiv2022

Asymptotic expansions for the truncation error in Ramanujan-type series

Many of the fastest known algorithms to compute $π$ involve generalized hypergeometric series, such as the Ramanujan-Sato series. In this paper, we investigate the rates of convergence for several such series and we give asymptotic expansions for the error of finite approximation. For example, when using the first $n$ terms of the Chudnovskys' series, we obtain the finite approximation $π_n\approx π$. It is known that the truncation error satisfies $|π_n-π|\approx 53360^{-3n}.$ In this paper, we prove that the asymptotic expansion for the truncation error in the Chudnovskys' series is $$\left|π_n-π\right|=53360^{-3n}\cdot\frac{{106720}\sqrt{{10005}π}}{{1672209}\sqrt{n}}\cdot\exp\left(\frac{A_1}{n}+\frac{A_2}{n^2}+\frac{δ_n}{n^3}\right),$$ with ${0.006907}<δ_n<{0.008429}$ and the exact rational values of $A_1$ and $A_2$: $$A_1= -\frac{1781843197433}{7456754505816},$$ $$A_2= -\frac{1080096011925710088395}{3475199235000451148614116}.$$ Thus we demonstrate how to establish precise error bounds for the approximations for $π$ obtained through Ramanujan-like series for $1/π$. We also give asymptotic expansions for all known rational hypergeometric series for $1/π$ in the appendix.

math.NT↗

arXiv2021

A detailed proof of the Chudnovsky formula with means of basic complex analysis -- Ein ausführlicher Beweis der Chudnovsky-Formel mit elementarer Funktionentheorie

In this paper we give another proof of the Chudnovsky formula for calculating $π$ - a proof in detail with means of basic complex analysis. With the exception of the tenth chapter, the proof is self-contained, with proofs provided for all the advanced theorems we use (e.g. for the Clausen formula and for the Picard-Fuchs differential equation). -- In diesem Aufsatz wird die Chudnovsky-Formel zur Berechnung von $π$ erneut bewiesen - wesentlich ausführlicher, mit elementaren Methoden der Funktionentheorie und der Analysis. Die benötigten fortgeschrittenen Sätze (z.B. die Clausen-Formel und die Picard-Fuchs-Differentialgleichung) werden ihrerseits ausführlich bewiesen. Nur im zehnten Kapitel verweisen wir auf externe Quellen.

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arXiv2020

The Transcendence of $π$ and the Squaring of the Circle -- Die Transzendenz von $π$ und die Quadratur des Kreises

In this paper we prove the transcendence of $π$ using Hilbert's method. We also prove that all points constructible with compass and straightedge have algebraic coordinates. Thus we give a self-contained proof that squaring the circle is impossible, requiring only basic linear algebra, analysis and Cauchy's Integral Theorem. -- In diesem Aufsatz beweisen wir mit Hilberts Methode, dass $π$ transzendent ist. Weiter beweisen wir, dass alle mit Zirkel und Lineal konstruierbaren Punkte algebraische Koordinaten haben. Somit beweisen wir, dass die Quadratur des Kreises unmöglich ist. Der vorliegende Beweis ist in sich abgeschlossen und setzt nur grundlegende lineare Algebra und Analysis und den Cauchy'schen Integralsatz voraus.

math.HO↗

arXiv2013

Smallest Squared Squares

In this paper we have a look at squared squares with small integer sidelengths, where the only restriction is that any two subsquares of the same size are not allowed to share a full border. We prove that there are exactly two such squared squares (and their mirrored versions) up to and including size 17x17. They are shown in Figure 1 (page 2).

math.CO↗