By Knopp M.I. (ed.)

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2 The M¨ obius function The sum μ d| n λ 19 n =1 λd if and only if n = λ and in every other case it is equal to zero. Hence, for n = λ we obtain f (d) = g(n). ✷ d|n Historical Remark. August Ferdinand M¨ obius, born on the 17th of November 1790 in Schulpforta, was a German mathematician and theoretical astronomer. He was first introduced to mathematical notions by his father and later on by his uncle. During his school years (1803–1809), August showed a special skill in mathematics. In 1809, however, he started law studies at the University of Leipzig.

3 The Euler function 23 and therefore, for any pair of positive integers n1 , n2 it holds φ(n1 n2 ) = φ(n1 )φ(n2 ) d , φ(d) where d = gcd(n1 , n2 ). Proof. We can write 1− 1 p1 1− 1 p2 ··· 1 − 1 pk (−1)λ , p m1 p m2 · · · p mλ =1+ where mi are λ distinct integers in the set {1, 2, . . , k} and hence the sum extends over all possible products of the prime divisors of n. However, by the definition of the M¨ obius function we know that μ(pm1 pm2 . . pmλ ) = (−1)λ , where μ(1) = 1 and μ(r) = 0 if the positive integer r is divisible by the square of any of the prime numbers p1 , p2 , .

Let n ∈ N. If g(n) = f (d), d|n then f (n) = n g(d). d μ d|n The inverse also holds. Proof. • Generally, for every arithmetic function m(n), it holds m(d) = d|n m d|n n , d since n/d = d and d is also a divisor of n. Therefore, it is evident that μ d|n n g(d) = d μ(d)g d|n n . d ⎞ ⎛ But μ(d)g d|n n = d (1) f (λ)⎠ . ⎝μ(d) · (2) λ| n d d|n At this point, we are going to express (2) in an equivalent form, where there will be just one sum at the left-hand side. In order to do so, we must find a common condition for the sums d|n and λ| n .

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Analytic Number Theory: Proceedings by Knopp M.I. (ed.)


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