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- 34 bytes (3 words) - 11:49, 26 June 2006
- 33 bytes (3 words) - 15:17, 15 August 2006
- ...functions <math> f, g : \mathbb{N} \rightarrow \mathbb{C} </math>, the '''Dirichlet convolution''' (or simply '''convolution''', when the context is clear) <ma3 KB (613 words) - 20:40, 21 June 2009
- 805 bytes (138 words) - 11:19, 30 May 2019
- ...r]]s to the [[real numbers]]. In mathematical notation we would say that a Dirichlet character is a function <math>\chi: \mathbb{Z} \to \mathbb{R}</math> such t The Dirichlet characters with period <math>q</math> have been completely classified. They1 KB (170 words) - 19:55, 2 September 2019
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- In older texts, the principle may be referred to as the '''Dirichlet box principle'''. A common phrasing of the principle uses balls and boxes a11 KB (1,986 words) - 18:13, 19 June 2024
- ==Dirichlet's theorem== ===Proof of Dirichlet's theorem===7 KB (1,290 words) - 11:18, 30 May 2019
- ...prove the infinitude of twin primes is an idea adopted from the proof of [[Dirichlet's Theorem]]. If one can show that the sum2 KB (307 words) - 20:11, 28 December 2024
- ==Proof via Dirichlet's simultaneous [[rational approximation]] theorem==2 KB (333 words) - 12:50, 26 June 2006
- '''Rational approximation''' is the application of [[Rational approximation|Dirichlet's theorem]] which shows that, for each irrational number <math>x\in\mathbb8 KB (1,431 words) - 12:48, 26 January 2008
- For two functions <math>f,g: \mathbb{N} \to \mathbb{C}</math> the '''Dirichlet convolution''' <math>f*g</math> is defined as <math>f*g(n) : = \sum_{d|n} f '''Dirichlet's density'''(does not always exist): <math>\delta(A) : = \lim_{s \to 1+0} \8 KB (1,401 words) - 16:49, 10 January 2025
- ...hen <math>f</math> is defined at <math>c</math>. For example, consider the Dirichlet function, <math>D(x)</math>, defined to be 0 when <math>x</math> is irratio7 KB (1,327 words) - 17:39, 28 September 2024
- On the other hand, by Dirichlet's theorem, <math>a_i \le 2</math>, since9 KB (1,699 words) - 12:48, 11 April 2020
- ...mathcal{F}</math> form a [[pseudo-ring]], with multiplication defined as [[Dirichlet convolution]], i.e.,6 KB (994 words) - 05:16, 8 April 2015
- ...th>L</math>-function is analogous to the [[Riemann zeta function]] and the Dirichlet <math>L</math>-series that is defined for a binary quadratic form. It is a7 KB (1,102 words) - 16:23, 6 September 2008
- ...ons <math> f,g : \mathbb{N} \rightarrow \mathbb{C} </math> usually means [[Dirichlet convolution]], defined as <math> \displaystyle f * g = \sum_{d\mid n} f(d)g524 bytes (81 words) - 18:12, 7 June 2007
- In other words, <math>\psi </math> is the [[Dirichlet convolution | convolution]] of the identity function <math> f: n \mapsto n6 KB (1,007 words) - 08:10, 29 August 2011
- ...functions <math> f, g : \mathbb{N} \rightarrow \mathbb{C} </math>, the '''Dirichlet convolution''' (or simply '''convolution''', when the context is clear) <ma3 KB (613 words) - 20:40, 21 June 2009
- By [[Dirichlet's Theorem]], there are infinitely many primes congruent to 1 (mod 3). Let11 KB (1,964 words) - 02:38, 17 August 2019
- ...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-4 KB (571 words) - 20:21, 22 November 2018
- ...<math>p</math>. The Euler Product is a specific case of the more general [[Dirichlet series]], which is a generalization of the Euler Product for any general [[708 bytes (116 words) - 20:15, 13 August 2015
- ...emann's integral. These functions include such pathological functions as [[Dirichlet's function]]. In fact, even if <math>D_f</math> is an uncountable cardinali892 bytes (130 words) - 20:10, 20 February 2009
- * [[Dirichlet's Theorem]]520 bytes (69 words) - 14:01, 29 March 2009
- * [[Dirichlet's Theorem]]2 KB (264 words) - 12:28, 1 April 2014
- Hence, by the Dirichlet Box Principle (pigeonhole principle), some two of them S_i and S_j, where i629 bytes (125 words) - 05:42, 4 March 2010