PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 41, Number 1, November 1973 ON 2o-*(rt) AND Tç>*(n) n^x n<Lx R. SITA RAMA CHANDRA RAO AND D. SURYANARAYANA Abstract. Let o*(n) and q>*{n)be the unitary analogues of o(n) and <p(ri) respectively. It is known that E(x)=^n¿x <J*(n)— (7t2a2/12í(3)) = 0(x log2 x) and F(x) = 2 9>*(») - i«*2 = °(x '°g2 *). n^.x where a is a positive constant. In this paper we improve the order estimates of E(x) and F(x) to E(x) = 0(x log573x) and F(x) = 0(x log5/3 x(log log x)il3). 1. Introduction. It is well known that a divisor d>0 of the positive integer « is called unitary, if dô=n and id, ô)=l. Let ct*in) denote the sum of the unitary divisors of n. For integers a, b, Z>>0, let (a, b)* denote the greatest divisor of a which is a unitary divisor of b. If (a, b)* = 1, then a is said to be semiprime to b. The symbol (a, b)* was introduced by E. Cohen (cf. [1, §1]). Let <p*(«) denote the number of positive integers a—^n such that a is semiprime to n. E. Cohen (cf. [1, Corollaries 4.1.1 and 4.1.2]) established the following asymptotic formulae : (1.1) 2 o*in) = ^r + 0(x log2 x), nSx 1¿4(J) where l(s) denotes the Riemann Zeta function defined by £(■?)= 2n=i ^ln¡> for j> 1 ; (1.2) 2 V*(n) = lax2 + 0(x log2 x), where the product being extended over all primes p. Received by the editors December 18, 1972 and, in revised form, March 19, 1973. AMS (MOS) subject classifications (1970). Primary 10H25. Key words and phrases. Unitary divisor, a is semiprime to b, abscissa of absolute convergence, partial summation. (c^ American Mathematical Society 1973 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use61 62 R. SITA RAMA CHANDRA RAO AND D. SURYANARAYANA [November The object of the present paper is to improve the order estimates of the error terms in the asymptotic formulae (1.1) and(1.2). In fact, we prove that -rr2V2 (1.4) 2ff*(«) = ^7T X + 0(xl°g5/3^ „s* 12£(3) (1.5) 2 <P*W- ^ + °(x loë5/3x(lo8 lo8 x)4/3)> nt^x where a is the constant given by (1.3). 2. Prerequisites. In this section, we prove some lemmas which are needed in the present discussion. Let p(n) denote the Möbius function, cp(n) denote the Euler totient function, a(n) denote the sum of the divisors of n and y(n) denote the core (the maximal square-free divisor) of n. Let a(n) be the arithmetical function defined by a(l)=l and a(n) = nLi.K-i)if«=nLi/^. It is known (cf. [1, Corollary 2.2.1]) that cp*(ri) is multiplicative, and so we have (2.1) q>*(n)= nf\(\- A ¿=i V Pi I if/!=nLi/>¿'- We use the following best-known estimates due to A. Walfisz [4] concerning the averages of the functions a(ri) and <p(n): Lemma 2.1 (cf. [4, Satz 4, p. 99]). For x^2, 2 2 (2.2) 2°<n) = ^-77 X + 0(xlog2/3x). nSx 12 Lemma 2.2 (cf. [4, Satz 1, p. 144]). For x^3, (2.3) 2 fOO - 3 + °(x lo82/3x(lo8 l0S *)4/3)- nSx W Lemma 2.3. c7*(/j)=2d2j=n(it(ûr)i/o-(o). Proof. Since /¿(/i)« and o-(n) are multiplicative, it follows (cf. [3, Lemma 2.4]) that 2d2<s=npid) rfo"(<5)is a multiplicative function of«. Also, o*(ri) is multiplicative (cf. [1, Lemma 6.1]). Hence, it is enough if we verify the identity for n=p", a prime power. We have 2 p{d) da(ô) = p(l)o(p) = 1 + p = o*(p), 2 (i ô=p License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1973] ON 2„sx a*(n) AND y„fix (p*(n) 63 and for <x=2, 2 Kd) daiô) = p(l)o(p«) + pip)paip*-2) = (1 + P + • • • + pa) - p(l + p + • • • + p*-2) = 1+P* = o*ÍPa). Hence the lemma follows. Lemma 2.4. <p*in)= £«_. (<ft(d)a(S)<pid)y(&yß)- Proof. Since airi), yin), tpiri) and cp*in) are multiplicative, it is enough, if we verify the identity for n=p", a prime power. We have y cpid)aiô)<piô)yiô) = <pil)aip")ip - 1) + <pip)aip*-l)ip - 1) + • • • + cpip°-2)aip2)ip- 1) + KpXU = (p - l){(a - 1) + p(l - l/p)(a - 2) + • • • + p*"2(l - 1/p)} + cpip*) = ip - l)(a - 1) + (p - l)2 X {(oc- 2) + p(oc - 3) + • • • + p^} + p\\ - 1/p) = (p - l)(a - 1) + (p - l)2 a(pa-2 - 1) 1 1 - oip"-1 + (a - l)p*) + -- , L-— + (P*- P* ) p - 1 p pip- l)2 = (p - l)(a - 1) + (p«-1 + p + a - ap - 2) + (p< - p*~l) = p" - 1 = <p*ip"). Hence the lemma follows. Lemma 2.5. Â(x)=~2,naxa(n)y(n)=0(x). Proof. We have lli(s)=Yl„ (1 -l/ps) for .v> 1 (cf. [2, Theorem 280]). Hence 1 A a(»)y(n) C(2s-1)¿, »s (2.4) » i p3s p4s p5s = > —-, say. n=l n License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 64 R. SITA RAMA CHANDRA RAO AND D. SURYANAKAYANA [November Hence •£ a(n)y(n) % bin) ^ n ^ b(n) n=l " n=l " n=l " n=l " so that (2.5) ain)yin) = 2 db(à). d o=n Also, by (2.4) we see that the abscissa of absolute convergence ß of 2"=i (b(n)¡ns) is ß=%. Hence, we have (2.6) 2 fo(") = 0(x3/4+£), for every £ > 0. nikx Hence by (2.5) and (2.6), we have 2 a(n)y(n)= 2 db(ô)= £ d 2 W) n^x d ô^x d^x ô^x/d - 0(x3/4+£(x1/2)1/2-2e) = 0(x). Lemma 2.6. 2n=i (a(n)<p(n)y(n)ln3)= £(2)<x, where a is given by (1.3). Proof. Applying Lemma 2.5 and partial summation, it is easy to show that 2«äs (a(n)y(n)lni) = 0(l). Now since cp(n)^n, it follows that the series 2n=i ia(n)cp(n)y(n)ln3) 's absolutely convergent. Further, the general term of the series is multiplicative. Hence the series can be expanded into an infinite product of Euler type (cf. [2, Theorem 286]), so that we have y a(n)<p(n)y(n)a(n)cp[n)y(n) _= j-rVT /,L , Ä(iy - !)(/> - 1) P2i TT 11 + P-1 j = n,(l -2//+1/P3) "VI /a-ir-W' n.o-i//)2 -«»n(ip V -|+i.)-Ä2) p p7 níi-T-zrlp \ p(p + 1)/ Hence the lemma follows. Lemma 2.7. For x^.2, (2.7) 2^ = 0(1) n>x n \xl License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1973] ON 2n<x °*in) AND y„<x 9»*(«) 65 and (2-8) 2a-^^ = 0(logx). nSx '' Proof. Both (2.7) and (2.8) follow by partial summation and Lemma 2.5. 3. Proofs of (1.4) and (1.5). By Lemmas 2.3 and 2.1, we have 2 o*(n)= 2 P(d)do(ñ) = 2 Kd)d2 °Í») n<x d â^x d^x S^x/d 12 „Sa;i/2 n" \ nS(Bx/>«/ tt2x2 ^ pin) 2^? + 0(x22 ^+G(xlog-x) 12 + 0(x) + 0(x log573 x), 12£(3) since 2Li 0"(«)/"3) = l/£(3) (cf. [2, Theorem 287]). Hence (1.4) follows. By Lemmas 2.4 and 2.2, we have 2 ?*(«)- 2 c<d)a(ó)9<ÓMá) »il dSix ° =2a(ô)(piô)7(ô)Z <pw =|^i|+o(,logii)(loglog(,)nj _ 3x2 y a(n)cp(n)y(n) **T2 »Si ^ «M3 + 0(xlog2^(loglogx)^2fl(")y(f(")) V nâx n I (31) *2 V a(w)y(/iMw) Q /^ y a(w)y(")\ 2^(2)6 n3 + I £ n2 j + o(xlog273x(logIogxr2^MîL)), since (pin)^n. Now, the first O-term in (3.1) is 0(x2 • l/x) = 0(x), by License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 66 R. SITA RAMA CHANDRA RAO AND D. SURYANARAYANA (2.7) and the second 0-term in (3.1) is 0(x Iog5/3 x(log log a-)1/3), by (2.8). Hence (1.5) follows by Lemma 2.6. References 1. E. Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Z. 74 (1960), 66-80. MR 22 #3707. 2. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Oxford Univ. Press, London, 1960. 3. D. Suryanarayana, The number of k-ary divisors of an integer, Monatsh. Math. 72 (1968), 445-450. MR 38 #4428. 4. A. Val'fis, Weylsche Exponentialsummen in der neuren Zahlentheorie, Mathema- tische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963. MR 36 #3737. Department of Mathematics, Andhra University, Waltair, India License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use.