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Hans Carl Friedrich Von Mangoldt
Hans Carl Friedrich von Mangoldt (1854 in Weimar– 1925 in Danzig) was a German mathematician who contributed to the solution of the prime number theorem. Biography Mangoldt completed his Doctorate of Philosophy (Ph.D) in 1878 at the University of Berlin, where his supervisors were Ernst Kummer and Karl Weierstrass. He contributed to the solution of the prime number theorem by providing rigorous proofs of two statements in Bernhard Riemann's seminal paper " On the Number of Primes Less Than a Given Magnitude". Riemann himself had only given partial proofs of these statements. Mangoldt worked as professor at the RWTH Aachen and was succeeded by Otto Blumenthal. See also * Prime-counting function In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ). History Of great interest in number theory is t ... * Cartan–Hadamard theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weimar
Weimar is a city in the state (Germany), state of Thuringia, Germany. It is located in Central Germany (cultural area), Central Germany between Erfurt in the west and Jena in the east, approximately southwest of Leipzig, north of Nuremberg and west of Dresden. Together with the neighbouring cities of Erfurt and Jena, it forms the central metropolitan area of Thuringia, with approximately 500,000 inhabitants. The city itself has a population of 65,000. Weimar is well known because of its large cultural heritage and its importance in German history. The city was a focal point of the German Enlightenment and home of the leading figures of the literary genre of Weimar Classicism, writers Johann Wolfgang von Goethe and Friedrich Schiller. In the 19th century, noted composers such as Franz Liszt made Weimar a music centre. Later, artists and architects such as Henry van de Velde, Wassily Kandinsky, Paul Klee, Lyonel Feininger, and Walter Gropius came to the city and founded the Ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RWTH Aachen
RWTH Aachen University (), also known as North Rhine-Westphalia Technical University of Aachen, Rhine-Westphalia Technical University of Aachen, Technical University of Aachen, University of Aachen, or ''Rheinisch-Westfälische Technische Hochschule Aachen'', is a German public research university located in Aachen, North Rhine-Westphalia, Germany. With more than 47,000 students enrolled in 144 study programs, it is the largest technical university in Germany. In 2018, the university was ranked 31st in the world university rankings in the field of engineering and technology, and 36th world-wide in the category of natural sciences.Daten & Fakten – RWTH AACHEN UNIVERSITY – Deutsch Rwth-aachen.de (12 December 2011). Retrieved on 2013-09-18. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theorists
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations ( Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
19th-century German Mathematicians
The 19th (nineteenth) century began on 1 January 1801 ( MDCCCI), and ended on 31 December 1900 ( MCM). The 19th century was the ninth century of the 2nd millennium. The 19th century was characterized by vast social upheaval. Slavery was abolished in much of Europe and the Americas. The First Industrial Revolution, though it began in the late 18th century, expanding beyond its British homeland for the first time during this century, particularly remaking the economies and societies of the Low Countries, the Rhineland, Northern Italy, and the Northeastern United States. A few decades later, the Second Industrial Revolution led to ever more massive urbanization and much higher levels of productivity, profit, and prosperity, a pattern that continued into the 20th century. The Islamic gunpowder empires fell into decline and European imperialism brought much of South Asia, Southeast Asia, and almost all of Africa under colonial rule. It was also marked by the collapse of the la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1925 Deaths
Nineteen or 19 may refer to: * 19 (number), the natural number following 18 and preceding 20 * one of the years 19 BC, AD 19, 1919, 2019 Films * ''19'' (film), a 2001 Japanese film * ''Nineteen'' (film), a 1987 science fiction film Music * 19 (band), a Japanese pop music duo Albums * ''19'' (Adele album), 2008 * ''19'', a 2003 album by Alsou * ''19'', a 2006 album by Evan Yo * ''19'', a 2018 album by MHD * ''19'', one half of the double album '' 63/19'' by Kool A.D. * '' Number Nineteen'', a 1971 album by American jazz pianist Mal Waldron * ''XIX'' (EP), a 2019 EP by 1the9 Songs * "19" (song), a 1985 song by British musician Paul Hardcastle. * "Nineteen", a song by Bad4Good from the 1992 album ''Refugee'' * "Nineteen", a song by Karma to Burn from the 2001 album ''Almost Heathen''. * "Nineteen" (song), a 2007 song by American singer Billy Ray Cyrus. * "Nineteen", a song by Tegan and Sara from the 2007 album '' The Con''. * "XIX" (song), a 2014 song by Slip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1854 Births
Events January–March * January 4 – The McDonald Islands are discovered by Captain William McDonald aboard the ''Samarang''. * January 6 – The fictional detective Sherlock Holmes is perhaps born. * January 9 – The Teutonia Männerchor in Pittsburgh, U.S.A. is founded to promote German culture. * January 20 – The North Carolina General Assembly in the United States charters the Atlantic and North Carolina Railroad, to run from Goldsboro through New Bern, to the newly created seaport of Morehead City, near Beaufort. * January 21 – The iron clipper runs aground off the east coast of Ireland, on her maiden voyage out of Liverpool, bound for Australia, with the loss of at least 300 out of 650 on board. * February 11 – Major streets are lit by coal gas for the first time by the San Francisco Gas Company; 86 such lamps are turned on this evening in San Francisco, California. * February 13 – Mexican troops force William Wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Mangoldt Function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive. Definition The von Mangoldt function, denoted by , is defined as :\Lambda(n) = \begin \log p & \textn=p^k \text p \text k \ge 1, \\ 0 & \text \end The values of for the first nine positive integers (i.e. natural numbers) are :0 , \log 2 , \log 3 , \log 2 , \log 5 , 0 , \log 7 , \log 2 , \log 3, which is related to . Properties The von Mangoldt function satisfies the identityApostol (1976) p.32Tenenbaum (1995) p.30 :\log(n) = \sum_ \Lambda(d). The sum is taken over all integers that divide . This is proved by the fundamental theorem of arithmetic, since the terms that are not powers of primes are equal to . For example, consider the case . Then :\begin \sum_ \Lambda(d) &= \Lambda(1) + \Lambda(2) + \Lambda(3) + \Lambda(4) + \Lambda(6) + \Lambda(12) \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–von Mangoldt Formula
In mathematics, the Riemann–von Mangoldt formula, named for Bernhard Riemann and Hans Carl Friedrich von Mangoldt, describes the distribution of the zeros of the Riemann zeta function. The formula states that the number ''N''(''T'') of zeros of the zeta function with imaginary part greater than 0 and less than or equal to ''T'' satisfies :N(T)=\frac\log-\frac+O(\log). The formula was stated by Riemann in his notable paper "On the Number of Primes Less Than a Given Magnitude" (1859) and was finally proved by Mangoldt in 1905. Backlund gives an explicit form of the error for all ''T'' > 2: :\left\vert\right\vert < 0.137 \log T + 0.443 \log\log T + 4.350 \ . Under the Lindelöf and Riemann
Georg Friedri ...
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Cartan–Hadamard Theorem
In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point. It was first proved by Hans Carl Friedrich von Mangoldt for surfaces in 1881, and independently by Jacques Hadamard in 1898. Élie Cartan generalized the theorem to Riemannian manifolds in 1928 (; ; ). The theorem was further generalized to a wide class of metric spaces by Mikhail Gromov in 1987; detailed proofs were published by for metric spaces of non-positive curvature and by for general locally convex metric spaces. Riemannian geometry The Cartan–Hadamard theorem in conventional Riemannian geometry asserts that the universal covering space of a connected complete Riemannian manifold of non-positive sectional curvature is diffeomorphic to R''n''. In fact, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime-counting Function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ). History Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately : \frac x where log is the natural logarithm, in the sense that :\lim_ \frac=1. This statement is the prime number theorem. An equivalent statement is :\lim_\pi(x) / \operatorname(x)=1 where li is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859. Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Otto Blumenthal
Ludwig Otto Blumenthal (20 July 1876 – 12 November 1944) was a German mathematician and professor at RWTH Aachen University. Biography He was born in Frankfurt, Hesse-Nassau. A student of David Hilbert, Blumenthal was an editor of ''Mathematische Annalen''. When the Civil Service Act of 1933 became law in 1933, after Hitler became Chancellor, Blumenthal was dismissed from his position at RWTH Aachen University. He was married to Amalie Ebstein, also known as 'Mali' and daughter of Wilhelm Ebstein. Blumenthal, who was of Jewish background, emigrated from Nazi Germany to the Netherlands, lived in Utrecht and was deported via Westerbork to the concentration camp, Theresienstadt in Bohemia (now Czech Republic), where he died. In 1913, Blumenthal made a fundamental, though often overlooked, contribution to applied mathematics and aerodynamics by building on Joukowsky's work to extract the complex Complex commonly refers to: * Complexity, the behaviour of a system whose co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
