PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 41, Number 1, November 1973
ON 2o-*(rt) AND Tç>*(n) n^x n R. SITA RAMA CHANDRA RAO AND D. SURYANARAYANA Abstract. Let o*(n) and q>*{n)be the unitary analogues of o(n) 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 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 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 2 2 (2.2) 2° (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 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 = (1 + P + • • • + pa) - p(l + p + • • • + p*-2) = 1+P* = o*ÍPa). Hence the lemma follows. Lemma 2.4. = (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) -«»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 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 =|^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