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Dirichlet Multiplication : Number Theory #Dirichlet #Multiplication #: #Number #Theory

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The Dirichlet product of arithmetical functions :   Definition :                    If f and g are two arithmetical functions then we define their Dirichlet product ( or Dirichlet convolution )  to be the arithmetical function f *g  defined by the equation     (f * g)(n) = 𝜮 f(d) g(n/d) for d/n Notation :               The symbol N will be used for the arithmetical function for which N(n) = n ∀ n.     From this notation , we can relate the Mobius function ( 𝝁 ) and Euler totient function ( Ჶ ) by        Ჶ = 𝝁 * N. Note :              The Concept Dirichlet  multiplication helps clarify interrelations between various arithmetical                    functions. Result :              Dirichlet multiplication is commutative a...
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The Mangoldt Function ⴷ ( n ) : Number Theory #The #Mangoldt #Function #ⴷ ( n ) : #Number #Theory

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 Mangoldt Function :  Definition             For every n ≥ 1 we define                       ⴷ ( n )  =  { log p    if n = p m for some  prime p and some m   ≥ 1              =  0    Otherwise    Since 1, 6, 10 can not be expressed as prime power , ⴷ( 0) = ⴷ(6) = ⴷ(10 ) = 0.     For n = 2, we have 2 = 2 1 = P m , here m = 1                 ∴    ⴷ ( 2 ) = log 2.     For n = 3 , we have 3 = 3 1 = p m , here m = 1                   ∴   ⴷ ( 3 ) = log 3     For n = 4, we have 4 = 2 2 = p m , here m = 2        ...
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The Euler Totient Function Ჶ(n) : Number Theory #The #Euler #Totient #Function #Ჶ(n) #Number #Theory

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The Euler Totient Function Ჶ(n) :                                                      If n ≥ 1 , the Euler Totient Function Ჶ(n) is defined to              be the number of positive integers not exceeding n which are relatively prime to             n ;  thus ,                         Ჶ ( n ) = 𝚺 1 for 1 ≤ k ≤ n, here 𝚺 indicates that the sum is extended over                                                                       those k relatively prime to n.  For Example :      Suppose n = 1.       ...
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The Mobius Function μ (n) : Number Theory #The #Mobius #Function #μ (n) : #Number #Theory

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Number theory , like many other branches of mathematics , is often concerned  with sequences of real or complex numbers. In number theory such sequences are called arithmetic functions. Definition :                    A real or complex - valued functions defined on  the positive integers is called an arithmetical function or a number- theoretic function. The Mobius Function μ (n) :                    The Mobius function μ is defined as follows :                         μ(1) = 1 ;                     If n > 1 , write n = p 1 a1 + p 2 a2 + … + p k ak . Then                                           μ( n) = ( -1 ) k if a 1 = a 2 = … = a k =1     ...
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G.C.D of more than 2 numbers: Number Theory #GCD #of #numbers #theory #2

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                                  Greatest Common Divisor for more than two numbers :                                  The greatest common divisor of three numbers ab,c is denoted by    ( a, b, c ) and is defined by gcd { a,b,c } = (a,b,c)                                                                     = ( a, ( b,c) )                                                                     = ( (a,b) , c).                i.e g.c.d depe...
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Euclidean Algorithm - Number Theory #Euclidean #Algorithm #- #Number #Theory

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  Euclid Algorithm or Division Algorithm :  Statement :               Given integers a and b with b > 0 , there exists a unique pair of integers q and r such that               a = bq + r , with 0 ≤ r < b. Moreover , r = 0 if, and  only, if b/a. Proof :              Suppose a and b are two integers with b > 0.               Let S = { y / y= a - bx , x is an integer, y ≥ 0 } .              Clearly S is a non - empty set of non - negative integers.              ∴ By well- ordering Principle , S must have a smallest member , say a - bq for                   some integer q .              Let r = a - bq.              Hence r is the sm...
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