|
Mertens's Theorems
In analytic number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens.F. Mertens. J. reine angew. Math. 78 (1874), 46–6Ein Beitrag zur analytischen Zahlentheorie/ref> In the following, let p\le n mean all primes not exceeding ''n''. First theorem Mertens' first theorem is that : \sum_ \frac - \log n does not exceed 2 in absolute value for any n\ge 2. () Second theorem Mertens' second theorem is :\lim_\left(\sum_\frac1p -\log\log n-M\right) =0, where ''M'' is the Meissel–Mertens constant (). More precisely, Mertens proves that the expression under the limit does not in absolute value exceed : \frac 4 +\frac 2 for any n\ge 2. Proof The main step in the proof of Mertens' second theorem is :O(n)+n\log n=\log n! =\sum_ \lfloor n/p^k\rfloor\log p = \sum_ \left(\frac+O(1)\right)\log p= n \sum_\frac\ + O(n) where the last equality needs \sum_\log p =O(n) which follows from \sum_\log p\le \log=O(n). Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Number Theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet ''L''-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Branches of analytic number theory Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. * Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. *Additive numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number Theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is , where is the prime-counting function (the number of primes less than or equal to ''N'') and is the natural logarithm of . This means that for large enough , the probability that a random integer not greater than is prime is very close to . Consequently, a random integer with at most digits (for large enough ) is about half as likely to be prime as a random integer with at most digits. For example, among the positive integers of at most 1000 digits, about on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series (mathematics)
In mathematics, a series is, roughly speaking, an addition of Infinity, infinitely many Addition#Terms, terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance. Among the Ancient Greece, Ancient Greeks, the idea that a potential infinity, potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the Quadrature of the Parabola, quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isaak Moiseevich Yaglom
Isaak Moiseevich Yaglom (; 6 March 1921 – 17 April 1988) was a Soviet mathematician and author of popular mathematics books, some with his twin Akiva Yaglom. Yaglom received a Ph.D. from Moscow State University in 1945 as student of Veniamin Kagan. As the author of several books, translated into English, that have become academic standards of reference, he has an international stature. His attention to the necessities of learning (pedagogy) make his books pleasing experiences for students. The seven authors of his Russian obituary recount "…the breadth of his interests was truly extraordinary: he was seriously interested in history and philosophy, passionately loved and had a good knowledge of literature and art, often came forward with reports and lectures on the most diverse topics (for example, on Alexander Blok, Anna Akhmatova, and the Dutch painter M. C. Escher), actively took part in the work of the cinema club in Yaroslavl and the music club at the House of Composer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Akiva Moiseevich Yaglom
Akiva Moiseevich Yaglom (; 6 March 1921 – 13 December 2007) was a Soviet and Russian physicist, mathematician, statistician, and meteorologist. He was known for his contributions to the statistical theory of turbulence and theory of random processes. Yaglom spent most of his career in Russia working in various institutions, including the Institute of Theoretical Geophysics. From 1992 until his death, Yaglom worked at the Massachusetts Institute of Technology as a research fellow in the Department of Aeronautics and Astronautics. He authored several popular books in mathematics and probability, some of them with his twin brother and mathematician Isaak Yaglom. Education and career Akiva Yaglom was born on 6 March 1921 in Kharkiv, Ukraine to the family of an engineer. He had a twin brother Isaak. The family moved to Moscow when the Yaglom brothers were five years old. During their school years they were keen on mathematics. In 1938 they shared the first prize at the Moscow mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Mascheroni Constant
Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by : \begin \gamma &= \lim_\left(-\log n + \sum_^n \frac1\right)\\ px&=\int_1^\infty\left(-\frac1x+\frac1\right)\,\mathrm dx. \end Here, represents the floor function. The numerical value of Euler's constant, to 50 decimal places, is: History The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled ''De Progressionibus harmonicis observationes'' (Eneström Index 43), where he described it as "worthy of serious consideration". Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Euler used the notations and for the constant. The Italian mathematician Lorenzo Mascheroni attempted to calculate the constant to 32 dec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number Theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is , where is the prime-counting function (the number of primes less than or equal to ''N'') and is the natural logarithm of . This means that for large enough , the probability that a random integer not greater than is prime is very close to . Consequently, a random integer with at most digits (for large enough ) is about half as likely to be prime as a random integer with at most digits. For example, among the positive integers of at most 1000 digits, about on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Summation By Parts
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826. Statement Suppose \ and \ are two sequences. Then, :\sum_^n f_k(g_-g_k) = \left(f_g_ - f_m g_m\right) - \sum_^n g_(f_- f_). Using the forward difference operator \Delta, it can be stated more succinctly as :\sum_^n f_k\Delta g_k = \left(f_ g_ - f_m g_m\right) - \sum_^ g_\Delta f_k, Summation by parts is an analogue to integration by parts: :\int f\,dg = f g - \int g\,df, or to Abel's summation formula: :\sum_^n f(k)(g_-g_)= \left(f(n)g_ - f(m) g_m\right) - \int_^n g_ f'(t) dt. An alternative statement is :f_n g_n - f_m g_m = \sum_^ f_k\Delta g_k + \sum_^ g_k\Delta f_k + \sum_^ \Delta f_k \Delta g_k which is analogous to the integration by parts formula for semimartingales ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edmund Landau
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopold Landau, a gynecologist, and his mother was Johanna Jacoby. Landau studied mathematics at the University of Berlin, receiving his doctorate in 1899 and his habilitation (the post-doctoral qualification required to teach in German universities) in 1901. His doctoral thesis was 14 pages long. In 1895, his paper on scoring chess tournaments is the earliest use of eigenvector centrality. Landau taught at the University of Berlin from 1899 to 1909, after which he held a chair at the University of Göttingen. He married Marianne Ehrlich, the daughter of the Nobel Prize-winning biologist Paul Ehrlich, in 1905. At the 1912 International Congress of Mathematicians Landau listed four problems in number theory about primes that he said were pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a member of a #Related asymptotic notations, family of notations invented by German mathematicians Paul Gustav Heinrich Bachmann, Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '':wikt:Ordnung#German, Ordnung'', meaning the order of approximation. In computer science, big O notation is used to Computational complexity theory, classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetic function, arithmetical function and a better understood approximation; one well-known exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions. Scope Construction of the real numbers The theorems of real analysis rely on the properties of the (established) real number system. The real number system consists of an uncountable set (\mathbb), together with two binary operations denoted and \cdot, and a total order denoted . The operations make the real numbers a field, and, along with the order, an ordered field. The real number system is the unique '' complete ordered field'', in the sense that any other complete ordered field is isomorphic to it. Intuitively, completenes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
