Journal of Inequalities in Pure and Applied Mathematics ON THE SYMMETRY OF SQUARE-FREE SUPPORTED ARITHMETICAL FUNCTIONS IN SHORT INTERVALS volume 5, issue 2, article 33, 2004. GIOVANNI COPPOLA DIIMA-University of Salerno Via Ponte Don Melillo 84084 Fisciano(SA) - ITALY. EMail: gcoppola@diima.unisa.it Received 17 March, 2003; accepted 02 April, 2004. Communicated by: A. Fiorenza Abstract Contents JJ J II I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 059-04 Quit Abstract We study the links between additive and multiplicative arithmetical functions, say f, and their square-free supported counterparts, i.e. µ2 f (here µ2 is the square-free numbers characteristic function), regarding the (upper bound) estimate of their symmetry around x in almost all short intervals [x − h, x + h]. 2000 Mathematics Subject Classification: 11N37, 11N36 Key words: Symmetry, Square-free, Short intervals. The author wishes to thank Professor Saverio Salerno and Professor Alberto Perelli for friendly and helpful comments. Also, he wants to express his sincere thanks to Professor Henryk Iwaniec, for his warm and familiar welcome during his stay in Rutgers University as a Visiting Scholar (the present work was conceived and written during this period). On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Title Page Contents Contents 1 Introduction and Statement of the Results . . . . . . . . . . . . . . . . 3 2 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Proof of the Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 References JJ J II I Go Back Close Quit Page 2 of 22 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au 1. Introduction and Statement of the Results In this paper we study the symmetry, in almost all short intervals, of square-free supported arithmetical functions. In our previous paper [3] we applied elementary methods, i.e. the Large Sieve, in order to study the symmetry of distribution (around x) of the squarefree numbers in "almost all" the "short" intervals [x−h, x+h] (as usual, "almost all" means for all x ∈ [N, 2N ], except at most o(N ) of them; "short" means that h = h(N ) and h → ∞, h = o(N ), as N → ∞). As in [1], [2], [4], and [5] on (respectively) the prime-divisors function, von Mangoldt function, the divisor function and a wide class of arithmetical functions, we study the symmetry of our arithmetical function f . def def We define the "symmetry sum" of f as (here sgn(t) = t/|t|, sgn(0) = 0) X def Sf± (x) = f (n)sgn(n − x), |n−x|≤h and its mean-square as the "symmetry integral" of f : 2 X X def . If (N, h) = f (n)sgn(n − x) x∼N |n−x|≤h Here and hereafter x ∼ N stands for N < x ≤ 2N . We will connect (in Theorem 1.1 and Theorem 1.2) If (N, h) and Iµ2 f (N, h), for suitable f ; thus relating the symmetry of f to that of f on the square-free numbers (µ2 being their characteristic function). Thus, we can estimate just one On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Title Page Contents JJ J II I Go Back Close Quit Page 3 of 22 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au symmetry integral for two arithmetical functions, whenever they agree on the square-free numbers. As an example, for d(n) the divisor function, [4] estimates Id (N, h); then (using Theorem 1.4 to check the symmetry of d(n) in arithmetic progressions) in Theorem 1.3 we bound Iµ2 d (N, h) = Iµ2 2Ω (N, h), and then obtain information on I2Ω (N, h) by Theorem 1.1 (here the function 2Ω(n) is completely multiplicative, with 2Ω(p) = 2). We denote with F the set of arithmetical functions f : N → C and with B the set of f ∈ F, with |f | bounded (by an absolute constant); M denotes the multiplicative f ∈ F and A the additive ones. Also, we can define (∀α ∈]1, 2]) the set of "symmetric" arithmetical functions f as (where we assume: ∀E > 0 supN |f | N E ): N hα def cε Sα = f ∈ F : sup If (N, h, k, q) 2 ε ∀k ≤ N , for some c, ε > 0 k N q≤N cε (the -constant is absolute, as well as c > 0), where we have set 2 X X x def If (N, h, k, q) = f (n)sgn n − ; k x∼N |n−x/k|≤h/k n≡0(q) in the following, as here, we willl abbreviate n ≡ a(q) to mean n ≡ a(mod q). def On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Title Page Contents JJ J II I Go Back Close We start giving a first link between f and µ2 f (in the sequel L = log N ): Quit Theorem 1.1. Let N, h √∈ N, where h = h(N ), h/L2 → ∞ and h = o(N ) as N → ∞. Assume J Lh , J → ∞ as N → ∞. Let kf k∞ := supN |f |. Page 4 of 22 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au If f is completely multiplicative then X 2 If (N, h) L max (i) DJ 2 d Iµ2 f d∼D N h , d2 d2 N h , d2 d2 + N h2 kf k2∞ J2 + N h2 kf k2∞ . J2 and 2 Iµ2 f (N, h) L max (ii) DJ X 2 d If d∼D On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals If f is completely additive then 2 (i) If (N, h) L max DJ X 2 d Iµ2 f d∼D N h , d2 d2 + √ 2 N h2 + N J hL kf k2∞ 2 J Title Page and (ii) Giovanni Coppola 2 Iµ2 f (N, h) L max DJ X d∼D 2 d If N h , d2 d2 + N h2 2 + N JL kf k2∞ . J2 We generalize Theorem 1.1 to additive and to multiplicative functions: Theorem 1.2. Let f ∈ A ∪ M. Let N, h be natural numbers, with h = N θ (for 0 < θ < 1). Assume that f is supported over the cube-free numbers and that ∀E > 0, kf k∞ N E , as N → ∞. Choose ∀α ∈]1, 2] ε = θ(α−1) > 0. 3 Then f ∈ Sα ⇔ µ 2 f ∈ Sα . Contents JJ J II I Go Back Close Quit Page 5 of 22 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au We give a concrete example: the function f (n) = 2Ω(n) (where Ω(n) is the total number of prime divisors of n); in this case f ∈ Sα and µ2 f ∈ Sα ∀α > 23 , as we will prove directly, also to detail the (more delicate) estimates √ Theorem 1.3. Let N, h ∈ N, h = h(N ) ≥ L and h = o LN as N → ∞. Then 2 X X Ω(n) N h3/2 N ε 2 sgn(n − x) x∼N |n−x|≤h and 2 X X 2 Ω(n) µ (n)2 sgn(n − x) N h3/2 N ε . x∼N |n−x|≤h On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Remark 1.1. We explicitly remark that these bounds are non-optimal. Title Page This result is obtained directly upon estimating the mean-square of the symmetry sum for the divisor function over the arithmetic progressions: √ Theorem 1.4. Let N, h ∈ N, with h = h(N ) → ∞ and h = o LN as N → ∞. Then, uniformly ∀q ∈ N, 2 X X d(n)sgn(n − x) N hL3 + N L2 log2 q, |n−x|≤h x∼N n≡0(q) Contents JJ J II I Go Back Close Quit Page 6 of 22 where the O-constant does not depend on q. J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au The paper is organized as follows • In Section 2 we give the necessary lemmas; • In Section 3 we prove our theorems. On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Title Page Contents JJ J II I Go Back Close Quit Page 7 of 22 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au 2. Lemmas def Lemma 2.1. Let f ∈ F be an arithmetical function, kf k∞ = supN |f (n)|. Then, for N, h = h(N ) ∈ N and h → ∞, h = o(N ) as N → ∞: 2 X X X x b(m)f (md2 )sgn m − 2 N hL2 kf k2∞ , a(d) √ d √ x∼N 2h<d≤ x+h |m−dx2 |≤ dh2 uniformly ∀a, b ∈ B. (Actually, for our purposes, kf k∞ = max N −h≤n≤2N +h |f (n)|). Proof. Let Σ be the LHS. By a dyadic dissection and Cauchy inequality 2 X X X x 2 2 Σ L √ max√ a(d) b(m)f (md )sgn m − 2 d hD N x h x∼N d∼D |m− d2 |≤ d2 2 X X X x 2 2 L √ max√ D b(m)f (md )sgn m − 2 d hD N x∼N d∼D |m− x |≤ h d2 d2 X X X kf k2∞ L2 √ max√ D 1. hD N d∼D N −h N <x≤2N ≤m1 ,m2 ≤ 2N 2+h d2 d m1 d2 −h≤x≤m1 d2 +h m2 d2 −h≤x≤m2 d2 +h On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Title Page Contents JJ J II I Go Back Close Quit Page 8 of 22 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au Clearly, the limitations on x imply m1 − 2h ≤ m2 ≤ m1 + 2h (here we "reflect" d2 d2 √ 2 the "sporadicity") and this in turn, due to D h ⇒ d h, gives (∀m1 FIXED) O(1) possible values to m2 . Hence Σ is bounded by X X X kf k2∞ hL2 √ max√ D 1 N hL2 kf k2∞ . hD N d∼D N −h ≤m1 ≤ 2N 2+h d2 d |m2 −m1 |1 def Lemma 2.2. Assume f ∈ F is completely additive and kf k∞ = sup√ N |f |. Let N, h ∈ N with h = h(N ) → ∞, h = o(N ), as N → ∞. Then ∀J ≤ 2h On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals 2 X X X x 2 b(m)f (md )sgn m − 2 a(d) √ d h x x∼N d≤ 2h |m− d2 |≤ d2 2 X X X x 2 2 L max D kf k∞ b(m)sgn m − 2 DJ d d∼D x∼N |m− x |≤ h d2 d2 2 XX X x N h2 + b(m)f (m)sgn m − 2 + 2 kf k2∞ , d J d∼D x∼N |m− x |≤ h d2 d2 Giovanni Coppola uniformly ∀a, b ∈B (bounded arithmetical functions). Title Page Contents JJ J II I Go Back Close Quit Page 9 of 22 Proof. Let us call the left mean-square Σ. Then Σ is at most J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au 2 X X X N h2 x b(m)f (md2 )sgn m − 2 + 2 kf k2∞ . L2 max a(d) DJ d J x∼N d∼D |m− dx2 |≤ dh2 Since f is completely additive 2 X X X x Σ L2 max D b(m)f (m)sgn m − 2 DJ d x∼N d∼D |m− x |≤ h d2 d2 2 X X X x N h2 b(m)sgn m − 2 + 2 kf k2∞ , + kf k2∞ d J x∼N d∼D |m− x |≤ h 2 2 d d by the Cauchy inequality. The lemma is thus proved. Lemma 2.3. Let f be completely multiplicative. Then, if√N, h ∈ N, with h = h(N ) → ∞ and h = o(N ) (as N → ∞ ), we have ∀J ≤ 2h 2 X X X x a(d) b(m)f (md2 )sgn m − 2 √ d x∼N d≤ 2h |m− dx2 |≤ dh2 2 2 XX X x Nh kf k2∞ L2 maxD b(m)f (m)sgn m − 2 + 2 DJ d J d∼Dx∼N |m− x |≤ h 2 2 d d On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Title Page Contents JJ J II I Go Back Close Quit Page 10 of 22 uniformly ∀a, b ∈B. J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au Proof. Let us call the left mean-square Σ. Then 2 X X X x Σ L2 max a(d) b(m)f (md2 )sgn m − 2 DJ d x∼N d∼D |m− dx2 |≤ dh2 + N h2 kf k2∞ , J2 and being f completely multiplicative we get 2 X X X x b(m)f (m)sgn m − 2 Σ kf k2∞ L2 max D DJ d d∼D x∼N |m− x |≤ h d2 d2 + by the Cauchy inequality. The lemma is thus proved. N h2 kf k2∞ , J2 √ Lemma 2.4. Let N, h, J and D be as in Lemma 2.2, with D = o( h). Then 2 XX X x f (m)sgn m − 2 d d∼D x∼N |m− x |≤ h d2 d2 2 2 X X X h 2 + N D kf k2∞ . d f (m)sgn(m − y) + D d∼D y∼ N |m−y|≤h/d2 On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Title Page Contents JJ J II I Go Back Close Quit Page 11 of 22 d2 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au 2 Proof. Write d2 ) and let Σ be the left mean-square; since P x = yd P+ r (0 ≤ r < 2 we have x∼N = y∼ N2 +O (d ), then d 2 X X X X r h2 f (m)sgn m − y − 2 + kf k2∞ Σ d D d∼D 0≤r<d2 y∼ N |m−y− r |≤ h d2 d2 d2 2 (thus hD is due to x-range remainders); then correcting O(1) values of the msum gives as a remainder (due to h-range) ! X N 2 2 O d 2 kf k∞ = O N D kf k2∞ . d d∼D Gathering the estimates we then obtain the lemma. On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Title Page Contents JJ J II I Go Back Close Quit Page 12 of 22 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au 3. Proof of the Theorems We start by proving Theorem 1.1. Proof. In both cases (f completely additive or completely multiplicative) we use the hypothesis on f to "separate variables" after having expressed the symmetry of f by that of µ2 f (for i), say) and the symmetry of µ2 f by that of f (for ii), say). Thus, to prove i) it will suffice to remember that each natural number n = md2 , where m and d are natural and µ2 (m) = 1, i.e. m is square-free: X X X x f (n)sgn(n − x) = µ2 (m)f (md2 )sgn m − 2 . d √ |n−x|≤h d≤ x+h |m− x2 |≤ h2 d d On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Instead, to prove ii) we simply use the following formula (see [7]): X µ2 (n) = µ(d) ∀n ∈ N d2 |n to get X |n−x|≤h µ2 (n)f (n)sgn(n − x) = X √ d≤ x+h Title Page Contents X µ(d) |m− dx2 |≤ dh2 x f (md2 )sgn m − 2 . d As for the additional terms in the completely additive case, they come from the estimate of the square-free symmetry sum as in [3]. Putting together Lemmas 2.1, 2.2, 2.3 and 2.4, the theorem is proved. JJ J II I Go Back Close Quit Page 13 of 22 We now come to the proof of Theorem 1.2. J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au Proof. We first prove that f ∈ S ⇒ µ2 f ∈ S. As before, we split at D (to be chosen); say (here [a, b] is the l.c.m. of a, b) def Σ= X X µ(d) | d≤D f (n)sgn(n − x) | n− x ≤ h k k n≡0([q,d2 ]) = X µ(d) X X t|[q,d2 ] m− x ≤ h kt2 g kt2 g g=[q,d2 ]/t (m,g)=1 d≤D x f (mt g)sgn m − 2 kt g 2 and observe that, since f is supported over the cube-free numbers, Σ is X X X X x 2 µ(d) f (t g) µ(j) f (m)sgn m − 2 kt g 2 d≤D t|[q,d ] j|g g=[q,d2 ]/t m− x ≤ h kt2 g kt2 g m≡0(j) X1 X x δ kf k∞ N d max2 f (m)sgn m − , 2 d j,t≤qd kt[q, d ] d≤D m− x 2 ≤ h 2 kt[q,d ] kt[q,d ] On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Title Page Contents JJ J II I m≡0(j) Go Back by (see [7]) the estimate ∀δ > 0 d(n) nδ ; using the hypothesis f ∈ Sα we get, by Cauchy inequality Close X x∼N |Σ|2 kf k2∞ N 2δ α X 1 X N hα 2 Nh d d2 d≤D k 2 d4 N ε k2N ε d≤D Quit Page 14 of 22 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au Hence, it remains to prove that the mean-square of, say def X Σ0 = √ | f (n)sgn(n − x) | n− x ≤ h k k n≡0([q,d2 ]) D<d≤ x+h is X µ(d) N hα |Σ | 2 ε . k N x∼N X 0 2 By the Cauchy inequality and a "sporadicity" argument as in the proof of Lemma 2.1, X |Σ0 |2 kf k2∞ x∼N X x∼N X D<d≤ √h 2 h +1 2 kd Giovanni Coppola k 2 + kf k2∞ L2 √ max √ J h J k δ N N Hence h2 h + 2 2 k D k N hα |Σ | 2 ε k N x∼N X 0 2 N XX d∼J x∼N + N δ √ max √ J h J k N On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals X m− k[dx2 ,q] ≤ k[dh2 ,q] Contents 1 X X d∼J N −h <m≤ 2N2+h k[d2 ,q] k[d ,q] N δ+ε h2−α 1−α δ+ε +h N k . D2 Title Page h. JJ J II I Go Back Close Quit Page 15 of 22 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au In order to obtain the above required estimate we need ε ≤ θ(α−1) (for the II 3 0 term in brackets) and, comparing the mean-squares of Σ and of Σ , we come to 4−α the choice D = N 2(α−1) ε (I term). This proves the first implication. As for the reverse implication µ2 f ∈ S ⇒ f ∈ S we do not need the hypothesis on the support of f and we use the same method (but using n = md2 instead of the identity for µ2 ). This finally proves Theorem 1.2. We now prove Theorem 1.4. Proof. First of all, let us call Iq (N, h) the mean-square to evaluate. We will closely follow the proof of Theorem 1 in [4]. In fact, we start from the "flipping" property to write: X d(n)sgn(n − x) |n−x|≤h n≡0(q) Giovanni Coppola Title Page 1X X h X = eq (rn) 2 1 sgn(n − x) + O √ + 1 , q r≤q N d|n |n−x|≤h √ d≤ n having used the orthogonality of the additive characters (see [7]). By our hypothesis on h (see [4] for the details) X 2X X X d(n)sgn(n − x) = eq (rn)sgn(n − x) + O(1) q r≤q √ |n−x|≤h |n−x|≤h n≡0(q) On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals d≤ x n≡0(d) Contents JJ J II I Go Back Close Quit Page 16 of 22 (here the constant is independent of q, like all the others following). J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au Next, write n − x = s to get (again by orthogonality) X X eq (rn)sgn(n − x) = eq (rx) eq (rs)sgn(s) |n−x|≤h n≡0(d) |s|≤h s≡−x(d) X eq (rx) X ed (jx) eq (rs)ed (js)sgn(s) d j≤d |s|≤h X = eq (rx) cj,d (q, r)ed (jx), = On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals j≤d say, where def 2i cj,d (q, r) = d X s≤h sin 2πs r j + q d . Giovanni Coppola Here (w.r.t. the quoted [4, Theorem 1]) we have the dependence of the Fourier coefficients on q and r; also, while cd,d = 0 there, here (by the estimate in of [6, Chap. 25]) 2πsr q 2i X sin . cd,d (q, r) = d s≤h q rd Hence, this term’s contribute to the mean-square Iq (N, h) is: 2 X 1 X X X eq (rx) cd,d (q, r)ed (jx) q √ x∼N r≤q x∼N d≤ x X1 r≤q r Contents JJ J II I Go Back Close !2 L Title Page 2 2 N L log q Quit Page 17 of 22 (that is why we have this additional remainder, here!). J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au Henceforth, we can rely upon the proof of [4, Theorem 1], the only difference being the r, s dependence: (*) 2 X 1 X XX c (q, r)e (jx) e (rx) j,d d q q √ x∼N r≤q d≤ x j<d 2 X X X X 1 cj,d (q, r)ed (jx) q r≤q x∼N √ j<d d≤ x (we have used the Cauchy inequality). We apply, then, exactly the same estimates; while there we get (we are quoting inequalities to ease comparison) X j<d |cj,d |2 ≤ X |cj,d |2 ≤ j≤d 2h , d here we have (the constant c > 0 is ininfluent) X r j 1 X |cj,d (q, r)| = c 2 sgn(s1 )sgn(s2 ) e (s1 − s2 ) + d q d j≤d j≤d |s1 |,|s2 |≤h X c X = sgn(s1 ) sgn(s2 )eq (r(s1 − s2 )), d |s |≤h X 2 |s1 |≤h 2 s2 ≡s1 (d) On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Title Page Contents JJ J II I Go Back Close Quit Page 18 of 22 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au whence, by (*), we get (see [4, Theorem 1]), ignoring the remainder O(N L2 log2 q): X 1 X X 1X N L2 sgn(s1 ) sgn(s2 )eq (r(s1 − s2 )) Iq (N, h) q r≤q d √ |s |≤h d≤ 2N = N L2 |s1 |≤h 2 s2 ≡s1 (d) X X 1 X sgn(s1 ) sgn(s2 ) d √ |s |≤h d≤ 2N |s1 |≤h 2 s2 ≡s1 (d) s2 ≡s1 (q) X 1 N L2 h+ h d d≤ L [d,q]≤ h L X h <d≤ L √ 2N 1 d 2 h + h . d On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Thus Iq (N, h) N hL3 + N L2 log2 q. Title Page Contents We now prove Theorem 1.3. Proof. We first show the second estimate. First of all, we observe that µ2 (n)2Ω(n) = µ2 (n)d(n), ∀n ∈ N; here we will apply the flipping property of the divisor function as in [4]. Then, we will try to link our symmetry integral (for µ2 2Ω ) with that of d(n). Writing µ2 (n) as before X X X µ2 (n)d(n)sgn(n − x) = µ(d) d(n)sgn(n − x). |n−x|≤h √ d≤ x+h |n−x|≤h n≡0(d2 ) JJ J II I Go Back Close Quit Page 19 of 22 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au Splitting the range at D = D(x) ≤ def Σ1 (x) = X √ µ(d) x + h (to be chosen later), we treat, say X d(n)sgn(n − x) |n−x|≤h n≡0(d2 ) d≤D by the Cauchy inequality and Theorem 1.4 to get 2 X X X X |Σ1 (x)|2 D d(n)sgn(n − x) |n−x|≤h x∼N d≤D x∼N 2 On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals n≡0(d ) 2 3 N D L (h + L) N D2 hL3 , Giovanni Coppola by our hypothesis on h. It remains to bound the mean-square of, say Title Page def X Σ2 (x) = √ D<d≤ x+h µ(d) X d(n)sgn(n − x). Contents |n−x|≤h n≡0(d2 ) √ We split again at 2h (to distinguish non-sporadic and sporadic terms). Since by the classical estimate d(n) nε (see [7]; here ε > 0 will not be the same at each occurrence) we estimate trivially (the non-sporadic terms) JJ J II I Go Back Close X √ D<d≤ 2h µ(d) X |n−x|≤h n≡0(d2 ) d(n)sgn(n − x) X √ D<d≤ 2h ε 2 hN Nh ε N 2 d D2 Quit Page 20 of 22 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au we get, together with (the sporadic terms, treated by Lemma 2.1) 2 X X X µ(d) d(n)sgn(n − x) N hN ε , √ √ |n−x|≤h x∼N 2h<d≤ x+h 2 n≡0(d ) that X 2 |Σ2 (x)| x∼N N h2 + N h N ε. D2 Thus, comparing the mean-squares of Σ1 (x) and Σ2 (x) we make the best choice D = h1/4 , finally proving the second estimate. Writing I2Ω for the symmetry integral of 2Ω , we apply Theorem 1.1 to this function; then, i) gives us N h3/2 N h2 I2Ω (N, h) L2 max d2 2 3 N ε + 2 N ε N h3/2 N ε , DJ d d J d∼D X by the choice J = Theorem 1.3. √ h. This gives the first estimate, hence finally proving On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Title Page Contents JJ J II I Go Back Close Quit Page 21 of 22 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au References [1] G. COPPOLA, On the symmetry of distribution of the prime-divisors function in almost all short intervals, to appear. [2] G. COPPOLA, On the symmetry of primes in almost all short intervals, Ricerche Mat., 52(1) (2003), 21–29. [3] G. COPPOLA, On the symmetry of the square-free numbers in almost all short intervals, submitted. [4] G. COPPOLA AND S. SALERNO, On the symmetry of the divisor function in almost all short intervals, to appear in Acta Arithmetica. [5] G. COPPOLA AND S. SALERNO, On the symmetry of arithmetical functions in almost all short intervals, submitted. [6] H. DAVENPORT, Multiplicative Number Theory, Springer Verlag, New York 1980. [7] G. TENENBAUM, Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics, 46, Cambridge University Press, 1995. On the Symmetry of Square-Free Supported Arithmetical Functions in Short Intervals Giovanni Coppola Title Page Contents JJ J II I Go Back Close Quit Page 22 of 22 J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004 http://jipam.vu.edu.au