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| In [[mathematics]], the '''von Mangoldt function''' is an [[arithmetic function]] named after [[Germany|German]] mathematician [[Hans Carl Friedrich von Mangoldt|Hans von Mangoldt]].
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| ==Definition==
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| The von Mangoldt function, conventionally written as Λ(''n''), is defined as
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| :<math>\Lambda(n) = \begin{cases} \log p & \mbox{if }n=p^k \mbox{ for some prime } p \mbox{ and integer } k \ge 1, \\ 0 & \mbox{otherwise.} \end{cases}</math>
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| which is a sequence starting:
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| :<math>\log 1 , \log 2 , \log 3 , \log 2 , \log 5 , \log 1 , \log 7 , \log 2 , \log 3,...</math>
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| [http://oeis.org/A014963 oeis sequence A014963]
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| It is an example of an important arithmetic function that is neither [[multiplicative function|multiplicative]] nor [[additive function|additive]].
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| The von Mangoldt function satisfies the identity<ref name=Apo76>{{Apostol IANT}}</ref>
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| :<math>\log n = \sum_{d\,\mid\,n} \Lambda(d),\,</math>
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| that is, the sum is taken over all [[integer]]s ''d'' that [[division (mathematics)|divide]] ''n''. This is proved by the [[fundamental theorem of arithmetic]], since the terms that are not powers of primes are equal to 0. | |
| :For instance, let '''n=12'''. Recall the prime factorization of 12, 12=2<sup>2</sup>·3, which will turn up in the example.
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| :Take the summation over all distinct positive divisors d of n:
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| : <math>\sum_{d\,\mid\,12} \Lambda(d) = \Lambda(1) + \Lambda(2) + \Lambda(3) + \Lambda(4) + \Lambda(6) + \Lambda(12) </math>
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| ::: <math>= \Lambda(1) + \Lambda(2) + \Lambda(3) + \Lambda(2^2) + \Lambda(2 \times 3) + \Lambda(2^2 \times 3) </math>
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| ::: <math>= 0 + \log 2 + \log 3 + \log 2 + 0 + 0 \,\! </math>
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| ::: <math>=\log (2 \times 3 \times 2) = \log 12. \,\! </math>
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| :This provides an example of how the summation of the von Mangoldt function equals log (n).
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| The '''summatory von Mangoldt function''', ψ(''x''), also known as the [[Chebyshev function]], is defined as | |
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| :<math>\psi(x) = \sum_{n\le x} \Lambda(n).</math>
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| von Mangoldt provided a rigorous proof of an explicit formula for ψ(''x'') involving a sum over the non-trivial zeros of the [[Riemann zeta function]]. This was an important part of the first proof of the [[prime number theorem]].
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| ==Dirichlet series==
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| The von Mangoldt function plays an important role in the theory of [[Dirichlet series]], and in particular, the [[Riemann zeta function]]. In particular, one has
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| :<math>\log \zeta(s)=\sum_{n=2}^\infty \frac{\Lambda(n)}{\log(n)}\,\frac{1}{n^s}</math>
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| for <math>\Re(s) > 1</math>. The [[logarithmic derivative]] is then
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| :<math>\frac {\zeta^\prime(s)}{\zeta(s)} = -\sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}.</math>
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| These are special cases of a more general relation on Dirichlet series.<ref name=Apo76 /> If one has
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| :<math>F(s) =\sum_{n=1}^\infty \frac{f(n)}{n^s}</math>
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| for a [[completely multiplicative function]] <math>f(n)</math>, and the series converges for <math>\Re(s) > \sigma_0</math>, then
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| :<math>\frac {F^\prime(s)}{F(s)} = - \sum_{n=1}^\infty \frac{f(n)\Lambda(n)}{n^s}</math>
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| converges for <math>\Re(s) > \sigma_0</math>.
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| ==Mellin transform==
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| The [[Mellin transform]] of the Chebyshev function can be found by applying [[Perron's formula]]:
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| :<math>\frac{\zeta^\prime(s)}{\zeta(s)} = - s\int_1^\infty \frac{\psi(x)}{x^{s+1}}\,dx</math>
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| which holds for <math>\Re(s)>1</math>.
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| ==Exponential series==
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| [[G. H. Hardy|Hardy]] and [[J. E. Littlewood|Littlewood]] examine the series<ref>{{Cite journal |first=G. H. |last=Hardy |lastauthoramp=yes |first2=J. E. |last2=Littlewood |url=http://www.ift.uni.wroc.pl/%7Emwolf/Hardy_Littlewood%20zeta.pdf |title=Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes |journal=Acta Mathematica |volume=41 |issue= |year=1916 |pages=119–196 |doi=10.1007/BF02422942 }}</ref>
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| :<math>F(y)=\sum_{n=2}^\infty \left(\Lambda(n)-1\right) e^{-ny}</math>
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| in the limit <math>y\to 0^+</math>. Assuming the [[Riemann hypothesis]], they demonstrate that | |
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| :<math>F(y)=\mathcal{O}\left(\sqrt{\frac{1}{y}}\right).</math>
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| Curiously, they also show that this function is oscillatory as well, with diverging oscillations. In particular, there exists a value <math>K>0</math> such that
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| :<math>F(y)< -\frac{K}{\sqrt{y}}</math> and <math>F(y)> \frac{K}{\sqrt{y}}</math>
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| infinitely often. The graphic to the right indicates that this behaviour is not at first numerically obvious: the oscillations are not clearly seen until the series is summed in excess of 100 million terms, and are only readily visible when <math>y<10^{-5}</math>.
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| ==Riesz mean==
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| The [[Riesz mean]] of the von Mangoldt function is given by
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| :<math>
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| \sum_{n\le \lambda} \left(1-\frac{n}{\lambda}\right)^\delta \Lambda(n)
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| = - \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty}
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| \frac{\Gamma(1+\delta)\Gamma(s)}{\Gamma(1+\delta+s)}
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| \frac{\zeta^\prime(s)}{\zeta(s)} \lambda^s ds</math>
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| :::::::<math>= \frac{\lambda}{1+\delta} +
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| \sum_\rho \frac {\Gamma(1+\delta)\Gamma(\rho)}{\Gamma(1+\delta+\rho)}
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| +\sum_n c_n \lambda^{-n}.
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| </math>
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| Here, <math>\lambda</math> and <math>\delta</math> are numbers characterizing the Riesz mean. One must take <math>c>1</math>. The sum over <math>\rho</math> is the sum over the zeroes of the Riemann zeta function, and
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| :<math>\sum_n c_n \lambda^{-n}\,</math>
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| can be shown to be a convergent series for <math>\lambda > 1</math>. | |
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| ==Expansion of terms==
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| {{confusing|section|reason=(1) No acceptable sources are provided, i.e., from textbooks (excluding blogs) or peer-reviewed journals; and (2) this appears to be an [[Wikipedia:no original research|original theory]]|date=February 2014}}
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| The terms of the von Mangoldt function can be expanded into series which have numerators that form an infinite [[Greatest common divisor]] pattern symmetric matrix starting:
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| :<math> T = \begin{bmatrix} 1&1&1&1&1&1 \\ 1&-1&1&-1&1&-1 \\ 1&1&-2&1&1&-2 \\ 1&-1&1&-1&1&-1 \\ 1&1&1&1&-4&1 \\ 1&-1&-2&-1&1&2 \end{bmatrix} </math>
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| This matrix <ref>Mats Granvik, ''[http://oeis.org/A191898 On-line Encyclopedia of Integer Sequences Sequence A191898]'' (2011)</ref> is defined by the recurrence:
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| :<math> T(n,1)=1,\;T(1,k)=1,\;n \geq k: T(n,k) = -\sum\limits_{i=1}^{k-1} T(n-i,k),\;n<k: T(n,k) = -\sum\limits_{i=1}^{n-1} T(k-i,n) </math>
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| or:
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| :<math> T(n,k) = a(GCD(n,k))</math>
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| where "a" is the Dirichlet inverse of the [[Euler's totient function]], and
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| for <math> n>1 </math>:
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| :<math> a(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \mu(d)\exp(d)^{(s-1)}</math>
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| The von Mangoldt function can then for <math> n>1 </math> be calculated as:<ref>joriki, ''[http://math.stackexchange.com/questions/48946/do-these-series-converge-to-the-mangoldt-function Do these series converge to the Mangoldt function]'' (2011)</ref>
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| :<math> \Lambda(n) = \sum\limits_{k=1}^{\infty}\frac{T(n,k)}{k} </math>
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| Or:
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| :<math> \Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(s-1)}}</math>
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| where
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| <math> \zeta(s)</math>
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| is the [[Riemann zeta function]], | |
| <math> \mu</math> is the [[Möbius function]] and
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| <math> d</math> is a divisor.
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| ==Approximation by Riemann zeta zeros==
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| The real part of the sum over the zeta zeros:{{citation needed|reason=this formula is not included in the reference given below|date=February 2014}}
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| :<math>\Lambda(n) = -\sum\limits_{i=1}^{\infty} n^{\rho(i)} </math>
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| where | |
| :<math>\rho(i)</math> is the i-th zeta zero,
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| approximates the von Mangoldt function by summing several waves onto each other [http://www.ams.org/notices/200303/fea-conrey-web.pdf](page 346).
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| [[File:Real part of n raised to first zeta zero.svg|thumb|right|The first Riemann zeta zero wave in the sum that approximates the von Mangoldt function]]
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| The Fourier transform of the von Mangoldt function gives a spectrum with spikes at ordinata equal to imaginary part of the Riemann zeta function zeros. This is sometimes called a duality.
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| [[File:Von Mangoldt function Fourier transform zeta zero duality.PNG|frame|The Fourier transform of the von Mangoldt function gives a spectrum with imaginary parts of Riemann zeta zeros as spikes at the x-axis ordinata (right), while the von Mangoldt function can be approximated by Zeta zero waves (left)]]
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| ==Recurrence==
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| The exponentiated von Mangoldt function can be described by a recurrence in a two dimensional matrix.<ref>Mats Granvik, ''[http://math.stackexchange.com/questions/164767/prime-number-generator-how-to-make/164829#164829 Mathematics Stack Exchange, Prime number generator how to make]'' (2011)</ref>{{citation needed|reason=math.stackexchange is not an acceptable source|date=February 2014}}
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| ==See also==
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| * [[Prime-counting function]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * Allan Gut, ''[http://www.math.uu.se/research/pub/Gut10.pdf Some remarks on the Riemann zeta distribution]'' (2005)
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| * {{springer|id=m/m062200|author=S.A. Stepanov|title=Mangoldt function}}
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| * Chris King, ''[http://the-messiahs-blog.blogspot.fi/2010/03/primes-out-of-thin-air.html Primes out of thin air]'' (2010)
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| * Heike, ''[http://stackoverflow.com/questions/8934125/how-plot-the-riemann-zeta-zero-spectrum-with-the-fourier-transform-in-mathematic How plot Riemann zeta zero spectrum in Mathematica?]'' (2012)
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| [[Category:Arithmetic functions]]
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' Brushing - A simple technique stimulates blood and lymph flow, removes dead skin cells and encourages new growth. Charles is a certified wellness practitioner and a certified nutritionist. Some celebrities that have to deal with cellulite are considered perfect to the point that some people just can't help but get surprised when discovering their flaws. How delightfully it would be if you could lose weight by eating. It predominantly occurs in women, and is visible mainly within the pelvic buttocks, and thigh regions.
Each of these anti-cellulite creams brings something different to the reduction cellulite. The excess will be stored by our bodies for later use. But at least she was trying and that was the beginning of the uphill climb. Compare product reviews to find something that works for you. This article will examine the basics of the diet and which foods are good or bad for use in this diet.
The damage to the circulatory system may prevent the effective transport of oral medication. The workout moves were simple enough that just about anyone could do them, even a complete beginner. For more information about our comprehensive range of cellulite removal treatments and products, visit the cellulite treatment page on our website. It enhances the total toxin removal from your body. Cellulite gels and creams have long been used to reduce the appearance of cellulite.
Muscles in our body are covered with a padding of fatty connective tissue. In this case too much fluid reaches the cell area, stopping the body's normal exchange of substances, namely the elimination of toxins. Therefore, supplementing with "body repair nutrients" makes sense. Stale air can give your house a stuffy, musty odor. Massage and even invasive treatment methods will only provide temporary relief.
As an instructor, I can tell you that a very targeted weight lifting program is the most effective way to get get rid of cellulite on the hip and legs. Get the head approximately 7 ins away from the floor. As a result, which is below the skin, or the structure of bones, and circulation system of fat cells will become more evident after menopause. Cellulite is actually a muscle issue, not a skin problem. Lotions that have all-natural herbal ingredients work wonders for getting rid of fat from thighs and buttocks.
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