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  • '''Rational approximation''' is the application of [[Rational approximation|Dirichlet's theorem]] which shows that, for each irrational number <math>x\in\mathbb R< ...p}\right\rfloor</math> where <math>\lfloor\cdot\rfloor</math> is the floor function, we have
    8 KB (1,431 words) - 12:48, 26 January 2008
  • A (weak) '''multiplicative function''' <math>f: \mathbb{N} \to \mathbb{C}</math> is one such that <math>f(a\cdo '''Euler's totient function''': <math>\varphi (n) = \phi (n) : = \left| \{ k \in \mathbb{N} \ : \ k \le
    8 KB (1,401 words) - 16:49, 10 January 2025
  • ...hus we begin with an informal explanation: a limit is the value to which a function grows close when its argument is near (but not at!) a particular value. For ...m_{x\to 2}x^2=4</cmath> because whenever <math>x</math> is close to 2, the function <math>f(x)=x^2</math> grows close to 4.
    7 KB (1,327 words) - 17:39, 28 September 2024
  • Let <math>\mathcal{F}</math> be the set of weak [[multiplicative function]]s mapping the positive [[integer]]s into themselves. Then the elements of
    6 KB (994 words) - 05:16, 8 April 2015
  • ...th> to the [[order]] of the [[root|zero]] of the associated <math>L</math>-function <math>L(E, s)</math> at <math>s = 1</math>. An L-function
    7 KB (1,102 words) - 16:23, 6 September 2008
  • ...convolution of functions <math> \displaystyle f, g </math> usually means a function of the form <math> \int f(\tau) g(t-\tau) d\tau </math>.
    524 bytes (81 words) - 18:12, 7 June 2007
  • The function <math> \displaystyle \psi </math> from the set <math> \mathbf{N} </math> of ...ath>m </math> and relatively prime to <math>m </math> (the [[Euler Totient Function]]). It follows that
    6 KB (1,007 words) - 08:10, 29 August 2011
  • ...>n=1 </math>, and 0 otherwise. Not all functions have inverses (e.g., the function <math>f(n) : n \mapsto 0 </math> has no inverse, as <math>f*g = f </math>, ...math>a </math>. For relatively prime <math>m,n </math>, we claim that the function <math> p : (d_m,d_n) \mapsto d_md_n </math> is a [[bijection]] from <math>
    3 KB (613 words) - 20:40, 21 June 2009
  • ...th>, then <math>q</math> is a divisor of some term of the given sequence. Dirichlet's Theorem guarantees that within the arithmetic sequence <cmath>q,\quad 2q-1,
    4 KB (571 words) - 20:21, 22 November 2018
  • ...f the natural numbers makes it immediately obvious that the [[Riemann zeta function]] will be of great importance in number theory, especially dealing with pri
    708 bytes (116 words) - 20:15, 13 August 2015
  • ...ntable cardinality, <math>f</math> is still integrable. Essentially, every function which had been considered up to the 20th century is Lebesgue integrable.
    892 bytes (130 words) - 20:10, 20 February 2009
  • '''Chebyshev's theta function''', denoted <math>\vartheta</math> or sometimes <math>\theta</math>, is a function of use in [[analytic number theory]].
    2 KB (264 words) - 12:28, 1 April 2014
  • ...5})=a^2+5b^2</math>. It can be shown that the [[norm]] is [[multiplicative function|multiplicative]], and so if <math>1+\sqrt{-5}=ab</math> for some <math>a,b ==Dirichlet's unit theorem==
    10 KB (1,646 words) - 14:04, 28 May 2020
  • ...h>\mathbb{Q^+}</math> be the set of positive rational numbers. Construct a function <math>f :\mathbb{Q^+}\rightarrow\mathbb{Q^+}</math> such that <math>f(xf(y) ...xtrm{an element from } S_2)</math> can hold, there will exist a piece-wise function (possibly with a greater than countable infinity number of pieces) that sat
    5 KB (1,045 words) - 10:01, 30 September 2022
  • ...s]]. In mathematical notation we would say that a Dirichlet character is a function <math>\chi: \mathbb{Z} \to \mathbb{R}</math> such that
    1 KB (170 words) - 19:55, 2 September 2019
  • ...se where <math>a</math> is prime and <math>a \neq 0,1 \pmod{43}</math>. By Dirichlet's Theorem (Refer to the <b>Remark</b> section.), such <math>a</math> always e ...zation of <math>a</math>. Since <math>\sigma(n)</math> is [[multiplicative function|multiplicative]], we have
    13 KB (2,185 words) - 01:28, 13 November 2023
  • ...eta function''' is a result due to analytic continuation of [[Riemann zeta function]]: There are multiple proofs for the functional equation for Riemann zeta function, and this page presents a light-weighted approach which merely relies on th
    4 KB (682 words) - 02:56, 13 January 2021