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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] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from 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 can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Inversion
Moebius, Mœbius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Friedrich Möbius (art historian) (1928–2024), German art historian and architectural historian * Theodor Möbius (1821–1890), German philologist, son of August Ferdinand * Karl Möbius (1825–1908), German zoologist and ecologist * Paul Julius Möbius (1853–1907), German neurologist, grandson of August Ferdinand * Dieter Moebius (1944–2015), Swiss-born German musician * Mark Mobius (born 1936), emerging markets investments pioneer * Jean Giraud (1938–2012), French comics artist who used the pseudonym Mœbius Fictional characters * Mobius M. Mobius, a character in Marvel Comics * Mobius, also known as the Anti-Monitor, a supervillain in DC Comics * Johann Wilhelm Möbius, a character in the play '' The Physicists'' * Moebius, the main antagonistic faction in the video game ''Xenoblade Chronicles 3'' * Mobius, or Dr. Ignatio Mobiu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Order Of An Arithmetic Function
In number theory, a normal order of an arithmetic function is some simpler or better-understood function which "usually" takes the same or closely approximate values. Let ''f'' be a function on the natural numbers. We say that ''g'' is a normal order of ''f'' if for every ''ε'' > 0, the inequalities : (1-\varepsilon) g(n) \le f(n) \le (1+\varepsilon) g(n) hold for ''almost all In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...'' ''n'': that is, if the proportion of ''n'' ≤ ''x'' for which this does not hold tends to 0 as ''x'' tends to infinity. It is conventional to assume that the approximating function ''g'' is continuous and monotone. Examples * The Hardy–Ramanujan theorem: the normal order of ω(''n''), the number of distinct prime factors o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor Summatory Function
In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function. The various studies of the behaviour of the divisor function are sometimes called divisor problems. Definition The divisor summatory function is defined as :D(x)=\sum_ d(n) = \sum_ 1 where :d(n)=\sigma_0(n) = \sum_ 1 is the divisor function. The divisor function counts the number of ways that the integer ''n'' can be written as a product of two integers. More generally, one defines :D_k(x)=\sum_ d_k(n)= \sum_\sum_ d_(n) where ''d''''k''(''n'') counts the number of ways that ''n'' can be written as a product of ''k'' numbers. This quantity can be visualized as the count of the number of lattice points fenced off by a hyperbolic surface in ''k'' dimensions. Thus, for ''k'' = 2, ''D''(''x'') = ''D''2(''x'') counts the number of points on a square lattice bounded on the left by the v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the RS ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler 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] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a constant called the ''center'' of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center ''c'' is equal to zero, for instance for Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Product
In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. Definitions The Cauchy product may apply to infinite series or power series. When people apply it to finite sequences or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution). Convergence issues are discussed in the next section. Cauchy product of two infinite series Let \sum_^\infty a_i and \sum_^\infty b_j be two infinite series with complex terms. The Cauchy product of these two infinite series is defined by a discrete convolution as follows: :\left(\sum_^\infty a_i\right) \cdot \left(\sum_^\infty b_j\right) = \sum_^\infty c_k where c_k=\sum_^k a_l b_. Cauchy product of two power series Consider the following two power series :\sum_^\infty a_i x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer n is divisible by a nonzero integer m if there exists an integer k such that n=km. This is written as : m\mid n. This may be read as that m divides n, m is a divisor of n, m is a factor of n, or n is a multiple of m. If m does not divide n, then the notation is m\not\mid n. There are two conventions, distinguished by whether m is permitted to be zero: * With the convention without an additional constraint on m, m \mid 0 for every integer m. * With the convention that m be nonzero, m \mid 0 for every nonzero integer m. General Divisors can be negative as well as positive, although often the term is restricted to posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Function
In number theory, a multiplicative function is an arithmetic function f of a positive integer n with the property that f(1)=1 and f(ab) = f(a)f(b) whenever a and b are coprime. An arithmetic function is said to be completely multiplicative (or totally multiplicative) if f(1)=1 and f(ab) = f(a)f(b) holds ''for all'' positive integers a and b, even when they are not coprime. Examples Some multiplicative functions are defined to make formulas easier to write: * 1(n): the constant function defined by 1(n)=1 * \operatorname(n): the identity function, defined by \operatorname(n)=n * \operatorname_k(n): the power functions, defined by \operatorname_k(n)=n^k for any complex number k. As special cases we have ** \operatorname_0(n)=1(n), and ** \operatorname_1(n)=\operatorname(n). * \varepsilon(n): the function defined by \varepsilon(n)=1 if n=1 and 0 otherwise; this is the unit function, so called because it is the multiplicative identity for Dirichlet convolution. Sometimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Polynomials
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings. A closely related notion is that of the ring of polynomial functions on a vector ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the integers mod n, integers mod p when p is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p and every positive integer k there are fields of order p^k. All finite fields of a given order are isomorphism, isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set that is a fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
