Note on a Result of the Lemmas of Our Previous Paper

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NOTE ON A RESULT OF THE LEMMAS OF OUR PREVIOUS PAPER ROBERT PETER SCHNEIDER Following up on our previous report [2], we note a pair of identities that convert summations involving the Euler totient function into summations involving the MĂśbius function, and vice versa in special cases. Let đ??šđ?‘› đ?‘? denote the finite sum đ??šđ?‘› đ?‘? =

đ?‘“(đ?‘?đ?‘—) 1≤đ?‘— ≤đ?‘›

where đ?‘“ is an arbitrary function defined on integer arguments, and đ?‘? is constant with respect to the index of summation đ?‘—. If this sum converges as đ?‘› approaches infinity, let đ??š(đ?‘?) denote the limit đ??š đ?‘? = lim đ??šđ?‘› đ?‘? . đ?‘›â†’∞

Theorem We have the identity between finite summations đ?‘›

đ?‘˜=1

đ?‘›

đ?œ‘ đ?‘˜ đ?‘“ đ?‘˜ = đ?‘˜

đ?‘˜=1

đ?œ‡ đ?‘˜ đ??šđ?‘› đ?‘˜ , đ?‘˜ đ?‘˜

which holds true as đ?‘› approaches infinity, yielding the identity between infinite sums ∞

đ?‘˜=1

đ?œ‘ đ?‘˜ đ?‘“ đ?‘˜ = đ?‘˜

∞

đ?‘˜=1

đ?œ‡ đ?‘˜ đ??š đ?‘˜ , đ?‘˜

when both sides converge; displaying a sort of inverse relationship between đ?œ‘ and đ?œ‡, with respect to đ?‘“ and đ??š, of a more general nature than that seen in the principal theorem of [2]. PROOF. We begin with the well-known relation [1, p. 235] đ?œ‘ đ?‘› =đ?‘› đ?‘‘|đ?‘›

đ?œ‡ đ?‘‘ , đ?‘‘

which results from applying MĂśbius inversion to the identity đ?‘› =

đ?‘‘|đ?‘›

đ?œ‘ đ?‘‘ .

The first identity of the theorem follows from Lemma 1 of [2], as đ?‘›

đ?‘˜=1

đ?œ‘ đ?‘˜ đ?‘“ đ?‘˜ = đ?‘˜

đ?‘›

đ?‘˜=1

đ?‘“ đ?‘˜ đ?‘˜

đ?‘˜ đ?‘‘|đ?‘˜

đ?œ‡ đ?‘‘ đ?‘‘

đ?‘›

=

đ?‘“ đ?‘˜ đ?‘˜=1

đ?‘‘|đ?‘˜

đ?œ‡ đ?‘‘ đ?‘‘


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Note on a Result of the Lemmas of Our Previous Paper by Robert Schneider - Issuu