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Lambert Series
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form :S(q)=\sum_^\infty a_n \frac . It can be resumed formally by expanding the denominator: :S(q)=\sum_^\infty a_n \sum_^\infty q^ = \sum_^\infty b_m q^m where the coefficients of the new series are given by the Dirichlet convolution of ''a''''n'' with the constant function 1(''n'') = 1: :b_m = (a*1)(m) = \sum_ a_n. \, This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform. Examples Since this last sum is a typical number-theoretic sum, almost any natural multiplicative function will be exactly summable when used in a Lambert series. Thus, for example, one has :\sum_^\infty q^n \sigma_0(n) = \sum_^\infty \frac where \sigma_0(n)=d(n) is the number of positive divisors of the number ''n''. For the higher order sum-of-divisor functions, one has :\sum_^\infty q^n \sigma_\alpha(n) = \sum_^\infty \frac where \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan Theta Function
In mathematics, particularly -analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after mathematician Srinivasa Ramanujan. Definition The Ramanujan theta function is defined as :f(a,b) = \sum_^\infty a^\frac \; b^\frac for . The Jacobi triple product identity then takes the form :f(a,b) = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty. Here, the expression (a;q)_n denotes the -Pochhammer symbol. Identities that follow from this include :\varphi(q) = f(q,q) = \sum_^\infty q^ = and :\psi(q) = f\left(q,q^3\right) = \sum_^\infty q^\frac = and :f(-q) = f\left(-q,-q^2\right) = \sum_^\infty (-1)^n q^\frac = (q;q)_\infty This last being the Euler function, which is closely related to the Dedekind eta function. The Jacobi theta functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruce Berndt
Bruce Carl Berndt (born March 13, 1939, in St. Joseph, Michigan) is an American mathematician. Berndt attended college at Albion College, graduating in 1961, where he also ran track. He received his master's and doctoral degrees from the University of Wisconsin–Madison. He lectured for a year at the University of Glasgow and then, in 1967, was appointed an assistant professor at the University of Illinois at Urbana-Champaign, where he has remained since. In 1973–74 he was a visiting scholar at the Institute for Advanced Study in Princeton. He is currently () Michio Suzuki Distinguished Research Professor of Mathematics at the University of Illinois. Berndt is an analytic number theorist who is probably best known for his work explicating the discoveries of Srinivasa Ramanujan. He is a coordinating editor of The Ramanujan Journal and, in 1996, received an expository Steele Prize from the American Mathematical Society for his work editing ''Ramanujan's Notebooks''. A Lester ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polylogarithm
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams. The polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special cases of the Lerch transcendent. Polylogarithms should not be confused with polylogarithmic functions nor with the offset logarithmic integral which h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeta Constants
In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted and is named after the mathematician Bernhard Riemann. When the argument is a real number greater than one, the zeta function satisfies the equation \zeta(s) = \sum_^\infty\frac \, . It can therefore provide the sum of various convergent infinite series, such as \zeta(2) = \frac + \frac + \frac + \ldots \, . Explicit or numerically efficient formulae exist for at integer arguments, all of which have real values, including this example. This article lists these formulae, together with tables of values. It also includes derivatives and some series composed of the zeta function at integer arguments. The same equation in above also holds when is a complex number whose real part is greater than one, ensuring that the infinite sum still converges. The zeta function can then be extended to the whole of the complex plane by analytic continuation, e ... [...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 \operatorname(s) > 1 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 is consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nome (mathematics)
In mathematics, specifically the theory of elliptic functions, the nome is a special function that belongs to the non-elementary functions. This function is of great importance in the description of the elliptic functions, especially in the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents and the Weber modular functions, that are used for solving equations of higher degrees. Definition The nome function is given by :q =\mathrm^ =\mathrm^ =\mathrm^ \, where ''K'' and iK' are the quarter periods, and \omega_1 and \omega_2 are the fundamental pair of periods, and \tau=\frac=\frac is the half-period ratio. The nome can be taken to be a function of any one of these quantities; conversely, any one of these quantities can be taken as functions of the nome. Each of them uniquely determines the others when 0 |
Jordan's Totient Function
Let k be a positive integer. In number theory, the Jordan's totient function J_k(n) of a positive integer n equals the number of k-tuples of positive integers that are less than or equal to n and that together with n form a coprime set of k+1 integers. Jordan's totient function is a generalization of Euler's totient function, which is given by J_1(n). The function is named after Camille Jordan. Definition For each k, Jordan's totient function J_k is multiplicative and may be evaluated as :J_k(n)=n^k \prod_\left(1-\frac\right) \,, where p ranges through the prime divisors of n. Properties * \sum_ J_k(d) = n^k. \, :which may be written in the language of Dirichlet convolutions as :: J_k(n) \star 1 = n^k\, :and via Möbius inversion as ::J_k(n) = \mu(n) \star n^k. :Since the Dirichlet generating function of \mu is 1/\zeta(s) and the Dirichlet generating function of n^k is \zeta(s-k), the series for J_k becomes ::\sum_\frac = \frac. * An average order of J_k(n) is ::\frac. * ... [...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. Thereby \omega(n) (little omega) counts each ''distinct'' prime factor, whereas the related function \Omega(n) (big omega) counts the ''total'' number of prime factors of n, honoring their 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 respective 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. \omega(n)=\sum_ 1 If p divides n at least once we count it only once, e.g. \omega(12)=\omega(2^2 3)=2. \Omega(n) =\sum_ 1 =\sum_\alpha If p divides n \alpha \geq 1 times then we count the exponent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when is the identity function, the equality is true for all values of to which can be applied. Definition Formally, if is a set, the identity function on is defined to be a function with as its domain and codomain, satisfying In other words, the function value in the codomain is always the same as the input element in the domain . The identity function on is clearly an injective function as well as a surjective function, so it is bijective. The identity function on is often denoted by . In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or ''diagonal'' of . Algebraic properties If is any function, then we have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum Of Squares Function
In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer as the sum of squares, where representations that differ only in the order of the summands or in the signs of the numbers being squared are counted as different, and is denoted by . Definition The function is defined as :r_k(n) = , \, where , \,\ , denotes the cardinality of a set. In other words, is the number of ways can be written as a sum of squares. For example, r_2(1) = 4 since 1 = 0^2 + (\pm 1)^2 = (\pm 1)^2 + 0^2 where each sum has two sign combinations, and also r_2(2) = 4 since 2 = (\pm 1)^2 + (\pm 1)^2 with four sign combinations. On the other hand, r_2(3) = 0 because there is no way to represent 3 as a sum of two squares. Formulae ''k'' = 2 The number of ways to write a natural number as sum of two squares is given by . It is given explicitly by :r_2(n) = 4(d_1(n)-d_3(n)) where is the number of divisors of wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theta Function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory. The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called ), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent. One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions". Throughout this article, (e^)^ should b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
