1 ) / '

Suryanarayana and Prasad^ The number of k-ary, (k + lyfree divisors 19

Proof , Each of the above series is absolutely convergent for 5 ^ e > 0, so that we get Lemma 3. 2 by termwise differentiation of the series in Lemma 3. 1 with respect to s.

As particular cases of (3. 1) and (3. 4) for s = 2k — 1, we have by Remark 2. 1, (2. 2) and (2. 3),

( 3 . 7) ^ -----тт-йТ = ^k + oi -J-yV

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and

^ ' n% %+i(«)«'*-' * p j9*(/>*+^ — 1) — (/) — 1) ^ \ a;2*-i / •

As a particular case of (3. 5) for s = a, we have by Lemma 2. 1 and (2. 3),

( 3 . 10 ) ^ JÇ^^

" г /'(;''* —/'*-' — i)/T/?*(/'*+^—!) — (/' —1) "^ I ж^* /•

Again , as a particular case of (3. 6) for s = 2k — 1, we have by Remark 2. 1, Lemma 2. 1 and (2. 3),

П in У /<"(") Jog » _ гr^ J___________P~'^__________

^'' - ' „f. гр,^,(п)Н,^,{п)п^''-^' %\'^ p>'i(p>'+^ _ 1)2 _p(p _ 1)]

У v___________(p — l)logp_____________^ / log a;

P ?"[{?'+'-ir-Pip-i)l + (p-i) "^

Lemma 3. 3. For /c ^ 1,

( 3 - 12) i , ^(74_, = C(Ä + 1) n(l ^"' + ^ ^ •^*+^('^)

( m , n ) ==l

( 3 . 13 ) 1 —r'#^-т-l^=c(k+i)п(i—^—£^^1„\л±пп)

( »n , n ) = l

Proof . If ^(m) = 1 or 0 according as л = 1 or j^ > 1, then the above series would become

J lii(m)Q((m,n)) ^^^ ^ ___f^(m)Q({m,n))

These two series are absolutely convergent by Remark 2. 1 and Lemma 2. 1 and ther, the general terms of the series are multiplicative functions of m. Hence by ing the series into infinite products of Euler type (cf. [2], Theorem 286) we get Lemma 3. 3.