Journal of Inequalities in Pure and Applied Mathematics CORRIGENDUM TO “ON SHORT SUMS OF CERTAIN MULTIPLICATIVE FUNCTIONS” volume 5, issue 3, article 81, 2004. OLIVIER BORDELLÈS 2 Allée de la Combe 43000 AIGUILHE FRANCE. EMail: borde43@wanadoo.fr Received 27 August, 2003; accepted 28 April, 2004. Communicated by: S.S. Dragomir Abstract Contents JJ J II I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 118-03 Quit Abstract This note is a corrigendum of the main result of the paper ”On short sums of certain multiplicative functions” (J. Ineq. Pure & Appl. Math., 3(5), Art. 70 (2002)). 2000 Mathematics Subject Classification: 11N37, 11P21 Key words: Corrigendum, Short sums, Integer points Corrigendum to “On Short Sums of Certain Multiplicative Functions” The author is indebted to the referee for insightful comments. Olivier Bordellès The purpose of this note is both to give a corrected statement of the main result in [1] and to provide the necessary changes to the arguments in that paper to justify this corrected statement. The result we now assert is the following : Theorem 1. Let ε, c0 > 0 and 2 6 y 6 c0 x1/2 be real numbers. Let f be a multiplicative function satisfying 0 6 f (n) 6 1 for any positive integer n and f (p) = 1 for any prime number p. We have as x → +∞ : X f (n) = yP (f ) + Oε x1/15+ε y 2/3 , x<n6x+y Contents JJ J II I Go Back Close Quit where P (f ) := Title Page Y p 1 1− p ∞ X f pl 1+ pl l=1 Page 2 of 7 ! . J. Ineq. Pure and Appl. Math. 5(3) Art. 81, 2004 http://jipam.vu.edu.au Before giving the proof, we note that if y < x1/5 , then x1/15 > y 1/3 so that the expression x1/15+ε y 2/3 in the error term exceeds the main term (as well as the trivial bound of y + 1 on the sum). Proof. On page 5 of [1], the sum S1 := X |g (d)| y<d6x+y d squarefull h i x+y x − d d Corrigendum to “On Short Sums of Certain Multiplicative Functions” has been bounded by the sum S2 := X b6(x+y)1/3 X 1/2 ( by3 ) 1/2 −3 (x + y) b−3 xb − 2 a a2 <a6( x+y 3 ) Olivier Bordellès Title Page b by using d = a2 b3 with µ2 (b) = 1, and S2 has been bounded by x y ε ε x max R 3 , B, 3 b B 16B6(x+y)1/3 Contents JJ J II I for any (small) positive real number ε, where we defined R (f, N, δ) to be the number of integer points P(n, m) verifying n ∈ ]N ; 2N ] and |f (n) − m| 6 δ. Unfortunately, the part b6y1/3 of S2 cannot be estimated by Go Back y max R 3 , B, 3 , b B 16B6(x+y)1/3 Page 3 of 7 x Close Quit J. Ineq. Pure and Appl. Math. 5(3) Art. 81, 2004 http://jipam.vu.edu.au hence we have to proceed differently: for any positive integer r, we set X τ(r) (n) := 1 dr |n and recall that τ(r) (n) ε nε/r for any positive integers n, r. In S1 , d = a2 b3 > y implies a > y 1/5 or b > y 1/5 . Since −2 X X (x + y) a−2 xa − 3 b b3 y 1/5 <a6(x+y)1/2 ( y )1/3 <b6( x+y )1/3 a2 a2 X X 6 Corrigendum to “On Short Sums of Certain Multiplicative Functions” Olivier Bordellès τ(3) (n) x+y y 1/5 <a6(x+y)1/2 ax2 <n6 a2 Contents 1/5 and we have the same if b > y , then X X S1 6 τ(2) (n) + x+y y 1/5 <b6(x+y)1/3 bx3 <n6 b3 X = X X X y 1/5 <a6(2y)1/2 x <n6 x+y a2 a2 := Σ1 + Σ2 + Σ3 + Σ4 . X τ(3) (n) x+y y 1/5 <a6(x+y)1/2 ax2 <n6 a2 X τ(2) (n) + x+y y 1/5 <b6(2y)1/3 bx3 <n6 b3 + X (2y) 1/3 τ(3) (n) + X Title Page JJ J II I Go Back τ(2) (n) x+y <b6(x+y)1/3 bx3 <n6 b3 X X (2y)1/2 <a6(x+y)1/2 x <n6 x+y a2 a2 Close Quit τ(3) (n) Page 4 of 7 J. Ineq. Pure and Appl. Math. 5(3) Art. 81, 2004 http://jipam.vu.edu.au • For Σ1 and Σ3 we use the trivial bound: h i X x+y x ε/2 − 3 Σ1 + Σ3 ε x 3 b b y 1/5 <b6(2y)1/3 h i X x+y x ε/3 +x − 2 2 a a y 1/5 <a6(2y)1/2 X 1 X 1 ε y 4/5 xε/2 . ε xε/2 y + 3 2 b a 1/5 1/5 b>y a>y • For Σ2 , we use the method of [1] to get Σ2 ε xε x1/6 + y Corrigendum to “On Short Sums of Certain Multiplicative Functions” Olivier Bordellès 1/3 . Title Page 1/2 • For Σ4 , if we suppose y 6 c0 x (where c0 > 0 is sufficiently small), we have using Lemmas 2.1 and 2.2 of [1]: ( ) x x y y Σ 4 ε x ε max R 2 , A, 2 + −1 max R 2 , A, 2 a A a A c0 y<A6x1/2 (2y)1/2 <A6c−1 0 y ε xε (xy)1/6 + x1/5 + x1/15 y 2/3 . Hence we finally have S1 ε xε x1/15 y 2/3 + y 4/5 + (xy)1/6 + x1/5 + x1/6 + y 1/3 ε xε x1/15 y 2/3 + y 4/5 Contents JJ J II I Go Back Close Quit Page 5 of 7 J. Ineq. Pure and Appl. Math. 5(3) Art. 81, 2004 http://jipam.vu.edu.au if y > x1/5 . Note that y 4/5 x1/15 y 2/3 if y 6 c0 x1/2 and that X f (n) − yP (f ) y x1/15 y 2/3 x<n6x+y if y < x1/5 . This concludes the proof of the theorem. Corrigendum to “On Short Sums of Certain Multiplicative Functions” Olivier Bordellès Title Page Contents JJ J II I Go Back Close Quit Page 6 of 7 J. Ineq. Pure and Appl. Math. 5(3) Art. 81, 2004 http://jipam.vu.edu.au References [1] O. BORDELLÈS, On short sums of certain multiplicative functions, J. Inequal. Pure and Appl. Math., 3(5) (2002), Art. 70. [ONLINE: http: //jipam.vu.edu.au/article.php?sid=222] Corrigendum to “On Short Sums of Certain Multiplicative Functions” Olivier Bordellès Title Page Contents JJ J II I Go Back Close Quit Page 7 of 7 J. Ineq. Pure and Appl. Math. 5(3) Art. 81, 2004 http://jipam.vu.edu.au