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Divisor Sum Identities
The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number n, or equivalently the Dirichlet convolution of an arithmetic function f(n) with one: :g(n) := \sum_ f(d). These identities include applications to sums of an arithmetic function over just the proper prime divisors of n. We also define periodic variants of these divisor sums with respect to the greatest common divisor function in the form of :g_m(n) := \sum_ f(d),\ 1 \leq m \leq n Well-known inversion relations that allow the function f(n) to be expressed in terms of g(n) are provided by the Möbius inversion formula. Naturally, some of the most interesting examples of such identities result when considering the average order summatory functions over an arithmetic function f(n) defined as a divisor sum of another arithmetic function g(n). Particular examples of divisor sums involvi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Average Order Of An Arithmetic Function
In number theory, an average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average". Let f be an arithmetic function. We say that an ''average order'' of f is g if \sum_ f(n) \sim \sum_ g(n) as x tends to infinity. It is conventional to choose an approximating function g that is Continuous function, continuous and Monotonic function, monotone. But even so an average order is of course not unique. In cases where the limit \lim_ \frac\sum_ f(n)=c exists, it is said that f has a mean value (average value) c. If in addition the constant c is not zero, then the constant function g(x)=c is an average order of f. Examples * An average order of , the Divisor function, number of divisors of , is ; * An average order of , the Divisor function, sum of divisors of , is ; * An average order of , Euler's totient function of , is ; * An average order of , the number of ways of expressing as a sum of two squares, is ; * The avera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LEAD
Lead () is a chemical element; it has Chemical symbol, symbol Pb (from Latin ) and atomic number 82. It is a Heavy metal (elements), heavy metal that is density, denser than most common materials. Lead is Mohs scale, soft and Ductility, malleable, and also has a relatively low melting point. When freshly cut, lead is a shiny gray with a hint of blue. It tarnishes to a dull gray color when exposed to air. Lead has the highest atomic number of any stable nuclide, stable element and three of its isotopes are endpoints of major nuclear decay chains of heavier elements. Lead is a relatively unreactive post-transition metal. Its weak metallic character is illustrated by its Amphoterism, amphoteric nature; lead and lead oxides react with acids and base (chemistry), bases, and it tends to form covalent bonds. Lead compounds, Compounds of lead are usually found in the +2 oxidation state rather than the +4 state common with lighter members of the carbon group. Exceptions are mostly limited ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Inverse
In mathematics, Dirichlet convolution (or divisor convolution) is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet. Definition If f , g : \mathbb\to\mathbb are two Arithmetic function, arithmetic functions, their Dirichlet convolution f*g is a new arithmetic function defined by: : (f*g)(n) \ =\ \sum_ f(d)\,g\!\left(\frac\right) \ =\ \sum_\!f(a)\,g(b), where the sum extends over all positive divisors d of n, or equivalently over all distinct pairs (a,b) of positive integers whose product is n. This product occurs naturally in the study of Dirichlet series such as the Riemann zeta function. It describes the multiplication of two Dirichlet series in terms of their coefficients: :\left(\sum_\frac\right) \left(\sum_\frac\right) \ = \ \left(\sum_\frac\right). Properties The set of arithmetic functions forms a commutative ring, the , with addition given by pointwise addition and multiplicat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Omega Function
In number theory, the prime omega functions \omega(n) and \Omega(n) count the number of prime factors of a natural number n. The number of ''distinct'' prime factors is assigned to \omega(n) (little omega), while \Omega(n) (big omega) counts the ''total'' number of prime factors with multiplicity (see arithmetic function). That is, if we have a prime factorization of n of the form n = p_1^ p_2^ \cdots p_k^ for distinct primes p_i (1 \leq i \leq k), then the prime omega functions are given by \omega(n) = k and \Omega(n) = \alpha_1 + \alpha_2 + \cdots + \alpha_k. These prime-factor-counting functions have many important number theoretic relations. Properties and relations The function \omega(n) is additive and \Omega(n) is completely additive. Little omega has the formula \omega(n)=\sum_ 1, where notation indicates that the sum is taken over all primes that divide , without multiplicity. For example, \omega(12)=\omega(2^2 3)=2. Big omega has the formulas \Omega(n) =\sum_ 1 ... [...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] |
Möbius Function
The Möbius function \mu(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted \mu(x). Definition The Möbius function is defined by :\mu(n) = \begin 1 & \text n = 1 \\ (-1)^k & \text n \text k \text \\ 0 & \text n \text > 1 \end The Möbius function can alternatively be represented as : \mu(n) = \delta_ \lambda(n), where \delta_ is the Kronecker delta, \lambda(n) is the Liouville function, Prime omega function, \omega(n) is the number of distinct prime divisors of n, and Prime omega function, \Omega(n) is the number of prime factors of n, counted with multiplicity. Another characterization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completely Multiplicative
In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime numbers, and such functions are called multiplicative functions. Outside of number theory, the term "multiplicative function" is often taken to be synonymous with "completely multiplicative function" as defined in this article. Definition A completely multiplicative function (or totally multiplicative function) is an arithmetic function (that is, a function whose domain is the natural numbers), such that ''f''(1) = 1 and ''f''(''ab'') = ''f''(''a'')''f''(''b'') holds ''for all'' positive integers ''a'' and ''b''. In logic notation: f(1) = 1 and \forall a, b \in \text(f), f(ab) = f(a)f(b). Without the requirement that ''f''(1) = 1, one could still have ''f''(1) = 0, but then ''f''(''a'') = 0 for all positive integers ''a'', so this is n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function f. Generally speaking, is said to be of class if its first k derivatives f^(x), f^(x), \ldots, f^(x) exist and are continuous over the domain of t ... [...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] |
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 . It is denoted by (unrelated to the number ). A symmetric variant seen sometimes is , which is equal to if is exactly a prime number, and equal to otherwise. That is, the number of prime numbers less than , plus half if equals a prime. Growth rate 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 where is the natural logarithm, in the sense that \lim_ \frac=1. This statement is the prime number theorem. An equivalent statement is \lim_\frac=1 where 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. Proof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel's Summation Formula
In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in analytic number theory and the study of special functions to compute series. Formula Let (a_n)_^\infty be a sequence of real or complex numbers. Define the partial sum function A by :A(t) = \sum_ a_n for any real number t. Fix real numbers x . Then: :\sum_ a_n\phi(n) = A(y)\phi(y) - A(x)\phi(x) - \int_x^y A(u)\phi'(u)\,du. The formula is derived by applying integration by parts for a Riemann–Stieltjes integral to the functions A and \phi. Variations Taking the left endpoint to be -1 gives the formula :\sum_ a_n\phi(n) = A(x)\phi(x) - \int_0^x A(u)\phi'(u)\,du. If the sequence (a_n) is indexed starting at n = 1, then we may formally define a_0 = 0. The previous formula becomes :\sum_ a_n\phi(n) = A(x)\phi(x) - \int_1^x A(u)\phi'(u)\,du. A common way to apply Abel's summation formula is to take the limit of one of these formulas as x \to \infty. The resulting formulas are :\b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
