Von Mangoldt Function (Essay Sample)

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The Von Mangoldt function
Introduction
There are various formulas used to calculate and explain various topics. Von Mangoldt is a function mainly employed in the mathematics field. The Von function is an arithmetic function that was named after a German mathematician called Hans Von Mangoldt. The Von function satisfies the identity:
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The sum is in use over all integers d that divide n. That is shown by the significant theorem of arithmetic as the terms, which are not prime powers are equal to 0. For instance, take the case n = 12 = 22 × 3. Then
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The function is denoted by:
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The function is also called the lambda function  INCLUDEPICTURE "http://mathworld.wolfram.com/images/equations/MangoldtFunction/Inline1.gif" \* MERGEFORMATINET 
 and the precise representation is:
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Where  INCLUDEPICTURE "http://mathworld.wolfram.com/images/equations/MangoldtFunction/Inline2.gif" \* MERGEFORMATINET 
 refers to the least common multiple. The beginning few values of  INCLUDEPICTURE "http://mathworld.wolfram.com/images/equations/MangoldtFunction/Inline3.gif" \* MERGEFORMATINET 
 for  INCLUDEPICTURE "http://mathworld.wolfram.com/images/equations/MangoldtFunction/Inline4.gif" \* MERGEFORMATINET 
, 2,…, plotted over are 1,2,3,2,5,1,7,2.
The Mangoldt function is executed in mathematics as mangoldt lambda[n]. There are various functions that relates to the Von function. Some of the functions that relates to the Von function is the mobius function, the divisor function and the phi function amongst others.
The Von function satisfies the divisor sums
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Where  INCLUDEPICTURE "http://mathworld.wolfram.com/images/equations/MangoldtFunction/Inline17.gif" \* MERGEFORMATINET 
denotes the  HYPERLINK "http://mathworld.wolfram.com/MoebiusFunction.html" Möbius function (Hardy and Wright 1979, p. 254). By mobius inversion we get:
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There is a relation between the Mangoldt function and the  HYPERLINK "http://mathworld.wolfram.com/RiemannZetaFunction.html" Riemann zeta function  INCLUDEPICTURE "http://mathworld.wolfram.com/images/equations/MangoldtFunction/Inline18.gif" \* MERGEFORMATINET 
by
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Where  INCLUDEPICTURE "http://mathworld.wolfram.com/images/equations/MangoldtFunction/Inline19.gif" \* MERGEFORMATINET 
(Hardy 1999, p. 28; Krantz 1999, p. 161; Edwards 2001, p. 50).
In number theory, the phi function φ(n), is a part of the arithmetic function that counts the n totatives, that is, the positive integers that are less than or equal to n that are mainly prime to n. If n is a positive integer, then φ(n) is the integer number k in the range 1 ≤ k ≤ n for which the biggest common divisor gcd (n,k)=1. The totient function refers to a multiplicative function that means that where two numbers n and m are relatively prim, then φ(mn) = Ï†(m)φ(n). The Euler’s product formula states
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Where the product is exceeding the individual prime number dividing n (Havil, p 109).
A divisor function is also a part of the arithmetic function linked to the integer divisors. When denoted as the divisor function, it states the integer divisors number. It comes out as a remarkable identities involving relationships with the Eisenstein series and Riemann zeta function of modular forms. A linked formula is the divisor summatory function that like the name is the sum exceeding the divisor function. The Von function is related to the divisor function through the summatory function. The subsequent chebyshey function π(x) is the HYPER13 HYPERLINK "http://en.wikipedia.org/w/index.php?title=Summatory_function&action=edit&redlink=1" \o "Summatory function (page does not exist)" summatory function associated with the von Mangoldt function:
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There are various aspects that surround the Von function. Many questions have been asked as to whether the Von function is multiplicative. The von function is one of the most significant arithmetic function that cannot be classified as additive or multiplicative. An arithmetic function a is...