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Wiener–Ikehara Theorem
The Wiener–Ikehara theorem is a Tauberian theorem, originally published by Shikao Ikehara, a student of Norbert Wiener's, in 1931. It is a special case of Wiener's Tauberian theorems, which were published by Wiener one year later. It can be used to prove the prime number theorem (Chandrasekharan, 1969), under the assumption that the Riemann zeta function has no zeros on the line of real part one. Statement Let ''A''(''x'') be a non-negative, monotonic function, monotonic nondecreasing function of ''x'', defined for 0 ≤ ''x'' 1 to the function ''ƒ''(''s'') and that, for some non-negative number ''c'', :f(s) - \frac has an extension as a continuous function for ℜ(''s'') ≥ 1. Then the Limit of a function, limit as ''x'' goes to infinity of ''e''−''x'' ''A''(''x'') is equal to c. One Particular Application An important number-theoretic application of the theorem is to Dirichlet series of the form :\sum_^\infty a(n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tauberian Theorem
In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series convergent series, converges to some limit then its Abel sum is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/''n'')) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods. There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shikao Ikehara
was a Japanese mathematician. He was a student of Norbert Wiener at MIT (PhD 1930). Career Following Wiener in 1928, in 1931 Ikehara used Wiener's Tauberian theory to derive another proof of the prime number theorem, demonstrated solely via the non-vanishing of the zeta function on the line Re s = 1. An improved version of Ikehara's 1931 result by Wiener in 1932 is now known as the Wiener–Ikehara theorem. Proofs of the prime number theorem before 1928 and only using the behaviour of the zeta function on the line Re s = 1 (as the 1908 proof of Edmund Landau), also appealed to some bound on the order of growth of the zeta function on this line. Returning to Japan after studying with Dr Wiener, he taught at Osaka University The , abbreviated as UOsaka or , is a List of national universities in Japan, national research university in Osaka, Japan. The university traces its roots back to Edo period, Edo-era institutions Tekijuku (1838) and Kaitokudō, Kaitokudo (1724), ... an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norbert Wiener
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American computer scientist, mathematician, and philosopher. He became a professor of mathematics at the Massachusetts Institute of Technology ( MIT). A child prodigy, Wiener later became an early researcher in stochastic and mathematical noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems. Wiener is considered the originator of cybernetics, the science of communication as it relates to living things and machines, with implications for engineering, systems control, computer science, biology, neuroscience, philosophy, and the organization of society. His work heavily influenced computer pioneer John von Neumann, information theorist Claude Shannon, anthropologists Margaret Mead and Gregory Bateson, and others. Wiener is credited as being one of the first to theorize that all intelligent behavior was the result of feedback mechanisms, tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener's Tauberian Theorem
In mathematical analysis, Wiener's Tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in L^1 or L^2 can be approximated by linear combinations of translations of a given function. Informally, if the Fourier transform of a function f vanishes on a certain set Z, the Fourier transform of any linear combination of translations of f also vanishes on Z. Therefore, the linear combinations of translations of f cannot approximate a function whose Fourier transform does not vanish on Z. Wiener's theorems make this precise, stating that linear combinations of translations of f are dense if and only if the zero set of the Fourier transform of f is empty (in the case of L^1) or of Lebesgue measure zero (in the case of L^2). Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the L^1 group ring L^1(\mathbb) of the grou ... [...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] |
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] |
Monotonic Function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if it is either entirely non-decreasing, or entirely non-increasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is termed ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\right), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Of A Function
Although the function is not defined at zero, as becomes closer and closer to zero, becomes arbitrarily close to 1. In other words, the limit of as approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which may or may not be in the domain of the function. Formal definitions, first devised in the early 19th century, are given below. Informally, a function assigns an output to every input . We say that the function has a limit at an input , if gets closer and closer to as moves closer and closer to . More specifically, the output value can be made ''arbitrarily'' close to if the input to is taken ''sufficiently'' close to . On the other hand, if some inputs very close to are taken to outputs that stay a fixed distance apart, then we say the limit ''does not exist''. The notion of a limit has many applications in modern calc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Series
In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. Specifically, the Riemann zeta function ''ζ(s)'' is the Dirichlet series of the constant unit function ''u(n)'', namely: \zeta(s) = \sum_^\infty \frac = \sum_^\infty \frac = D(u, s), where ''D(u, s)'' denotes the Dirichlet series of ''u(n)''. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet. Combinatorial importance Dirichlet series can be used as generating series for counting weighted sets of objects with respect to a weight which is combined multiplicatively when taking Cartesian product ... [...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] |
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] |
