ON A GOLDEN PAIR OF IDENTITIES IN THE THEORY OF NUMBERS ROBERT PETER SCHNEIDER
We exploit two useful, general lemmas involving a change in the order of certain double summations, to prove an interesting relationship between the golden ratio, the MĂśbius function, the Euler totient function and the natural logarithm—central players in the theory of numbers. Original problems are included, as our purpose in writing this paper is to encourage the reader to gain facility with the lemmas to discover new identities, as much as it is to highlight the principal theorem. Let ℤ+ denote the positive integers. Let đ?œ™ denote the golden ratio đ?œ™=
1+ 5 , 2
a constant that has historically attracted much attention; which obeys the well-known identity [4] đ?œ™ =1+
1 . đ?œ™
Let đ?œ‘ denote the Euler totient function, such that đ?œ‘(đ?‘›) denotes the number of positive integers less than đ?‘› ∈ ℤ+ , that are co-prime to đ?‘› [2, p. 233]. It is a pleasing coincidence of notation that the present theorem involves two different number theoretic quantities, đ?œ™ and đ?œ‘, denoted by the Greek letter phi; hence our use of both stylistic variants of the letter. Let đ?œ‡ denote the MĂśbius function, defined [2, p. 234] as đ?œ‡ đ?‘› =
0 if đ?‘› is non − squarefree, 1 if đ?‘› is squarefree, having an even number of prime factors, −1 if đ?‘› is squarefree, having an odd number of prime factors. vv
Theorem We have the pair of reciprocal identities ∞
đ?œ™=− đ?‘˜=1
1 =− đ?œ™
∞
đ?‘˜=1
đ?œ‘(đ?‘˜) 1 log 1 − đ?‘˜ , đ?‘˜ đ?œ™ đ?œ‡(đ?‘˜) 1 log 1 − đ?‘˜ đ?‘˜ đ?œ™
highlighting a connection between the golden ratio đ?œ™ and the factorization of integers that is not obvious; and displaying a sort of inverse relationship between the MĂśbius function đ?œ‡ and Euler totient function đ?œ‘, with respect to đ?œ™ and the natural logarithm.