Scourfield, Eira J.

Numérisation de Documents Anciens Mathématiques

AstérisqueEIRA J. SCOURFIELDOn the property ( f (n),g(n)) = 1 for certainmultiplicative functionsAstérisque, tome 41-42 (1977), p. 273-276<http://www.numdam.org/item?id=AST_1977__41-42__273_0>© Société mathématique de France, 1977, tous droits réservés.L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/Publications/Asterisque/) implique l’accord avec les conditions générales d’uti-lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ouimpression systématique est constitutive d’une infraction pénale. Toute copieou impression de ce fichier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/Société Mathématique de France Astérisque 41-42 (1977) p.273-276 ON THE PROPERTY (f(xi),g(n)) = 1 FOR CERTAIN MULTIPLICATIVE FUNCTIONS Eira J. SCOURFIELD The problem of investigating the sum \ ( x ) = n 1 (1) n £ x (n,h(n))=l for certain integer-valued arithmetic functions h has been considered by several authors in cases when the arithmetic properties of n and h(n) are not too closely related, and the expected resuit ẐU) - 6X / T I 2 established ; for références, see this author's paper [2] . Multiplicative func-tions, however, présent a rather différent problem, and in 1948 [1], Erdos obtained the resuit S (x) - e TX/log log log x as x 00 for Euler's function <p . In [2], we considered the sum (1 ) for a class of integer-valued multiplicative functions, called polynomial-like, that includes <p and the divisor functions o*̂  (v > 0) ; f is polynomial-like if there exists a polynomial W C Z[x] such that 273 E.J. SCOURFIELD f(p) = W(p) for ail primes p . (2) For thèse functions, we proved in [ 2 ] : THEOREM 1. If the polynomial ¥ of. (2) satisfies deg W > 0 , w(o) ̂  0 , then there exist constants C > 0 , X ( 0 < X < 1 , X rational), depending on f , such that E f (x) ~ C x (log log log x) ^ as_ x -> 00 . If ¥ is a non-zero constant, then there exists a constant C (0 < C < 1) such that Z (x) ~ C x as x -» 00 . If ¥(0) = 0 , Z f (x) = O(x^) . Example, f = a (v > 0) . For v odd, X = 1 , C = e ^ , whilst for v even, X = 2~ P , where 2P(|v • ¥e obtain a generalization of the sum in theorem 1 by noting that n itself is a polynomial-like multiplicative function. Let f , g be multiplicative polynomial-like functions, and let ¥^ , ¥^ £ Z[x] be the polynomials such that f(p) = W (p) , g(p) = W 2(p) for ail primes p . Suppose that the following conditions hold : (i) deg ¥ ± > 0 (i = 1,2) ; (ii) w (x) = x°V*(x) where a > 0 , ¥*(o) ^ 0 , deg ¥* > 0 , and W 2 ( o ) ^ 0 ; (iii) w , ¥^ are coprime polynomials, It follows from (iii) that the set 274 MULTIPLICATIVE FUNCTIONS S Q = {p : p|(f(q),g(q)) for ail primes q ^ p} of primes is finite (possibly empty). If P G S q , p | (f(n),g(n)) whenever there exist s a prime q / p with q||n , and hence for "most" n . This suggests that for our generalization of the sum E^(x) , we consider E f , g ( x ) = C p|(f(n),g(n)) Vp/3 o Using results from sieve theory, we can prove THEOREM 2. If conditions (i), (ii), (iii) above hold, there exist constants C > 0 , X (0 < X < 1 , X rational) such that — — log log x « £ (x) « — — exp ( -) . l 0 g X f ' g l 0 g X (log log log x ) X Conditions (i), (ii), (iii) ensure that the sum (x) is not too small and does not reduce to the sum considered in theorem 1 or in other published papers. Examples, (i) f = cp , g = CJ (v > o) , when S Q = {2} , X = 2^ where 2̂ ||v . (ii) f = a , g = o^ (V , H > 0 , p > y > where 2̂ ||v , 2 Y||K) , when S Q = {2} , X = 2~? . The method used to prove the upper bound in theorem 2 also establishes THEOREM 3. The number of positive integers n < x with the property that n does  not have a prime divisor in every residue class (mod p) coprime to p for any  odd prime p is. x / B log log x \ « exp U - = ) , log x log log log x where B > 0 is constant. 275 E.J. SCOURFIELD BIBLIOGRAFHY [l] P. ERDÖS.- Some asymptotic formulas in number theory. J. Indian Math. Soc. (N.S.), 12 (1948), PP. 75-78. [2] E.J. SCOURFIELD.- An asymptotic formula for the property (n,f(n)) = 1 for a class of multiplicative functions. Acta Arithmetica, 29 (1976), pp. 401-423. Eira J. SCOURFIELD Westfield Collège Department of Mathematics LONDON NW3 7ST, England U.K. 276 

On the property $(f(n), g(n)) = 1$ for certain multiplicative functions

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On the property $(f(n), g(n)) = 1$ for certain multiplicative functions

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On the property (f(n),g(n))=1(f(n), g(n)) = 1(f(n),g(n))=1 for certain multiplicative functions

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On the property $(f(n), g(n)) = 1$ for certain multiplicative functions