On the distribution of consecutive square-free

Transcription

1 On the distribution of consecutive square-free numbers of the form [n],[n]+ S. I. Dimitrov Faculty of Applied Mathematics and Informatics, Technical University of Sofia 8, St.Kliment Ohridsi Blvd. 756 Sofia, BULGARIA arxiv: v2 [math.nt] 22 Mar 209 Abstract: In the present paper we show that there exist infinitely many consecutive square-free numbers of the form [n], [n] +, where > is irrational number with bounded partial quotient or irrational algebraic number. Keywords: Consecutive square-free numbers, Asymptotic formula. AMS Classification: L05 N25 N37. Notations Let N be a sufficiently large positive integer. By ε we denote an arbitrary small positive number, not necessarily the same in different occurrences. We denote by µ(n) the Möbius function and by τ(n) the number of positive divisors of n. As usual [t] and {t} denote the integer part, respectively, the fractional part of t. Let t be the distance from t to the nearest integer. Instead of m n (mod ) we write for simplicity m n(). Moreover e(t)=exp(2πit) and ψ(t) = {t} /2. Let > be irrational number with bounded partial quotient or irrational algebraic number. Denote σ = ( 2p ). () 2 p We define the characteristic functionω (x) in the interval(0,] as follows, if < x < ; ω (x) = 2, ifx = or x = ; 0, otherwise; (2) and we extend it periodically to all real line.

2 2 Introduction and statement of the result The problem for the consecutive square-free numbers arises in 932 when Carlitz [3] proved that µ 2 (n)µ 2 (n+) = σn +O ( N 2/3+ε), (3) whereσ is denoted by (). Subsequently in 949 Mirsy [9] improved the error term of (3) to µ 2 (n)µ 2 (n+) = σn +O ( N 2/3 (logn) 4/3). (4) Further in 984 Heath-Brown [7] improved the error term of (4) to µ 2 (n)µ 2 (n+) = σn +O ( N 7/ (logn) 7). (5) Finally in 204 Reuss [0] improved the error term of (5) to µ 2 (n)µ 2 (n+) = σn +O ( N ) ( )/8+ε (6) and this is the best result up to now. In 2008 Güloğlu and Nevans [6] showed that there exist infinitely many square-free numbers of the form[n], where > is irrational number of finite type. More precisely they proved that the asymptotic formula µ 2 ([n]) = 6 ( ) N loglogn π 2N +O logn holds. On the other hand in 2009 Abercrombie and Bans[] showed that for almost all > the asymptotic formula µ 2 ([n]) = 6 ( ) π 2N +O N 2 3 +ε holds, however this result provides no specific value of. Subsequently in 203 Victorovich [3] proved that when > is irrational number with bounded partial quotient or irrational algebraic number, then the asymptotic formula µ 2 ([n]) = 6 ) (AN π 2N +O 5 6 log 5 N holds. HereA = A(N) = max m N 2τ(m). In 208 the author [4] showed that for any fixed < c < 22/3 there exist infinitely many consecutive square-free numbers of the form[n c ],[n c ]+. Recently the author [5] proved that there exist infinitely many consecutive square-free numbers of the form x 2 +y 2 +,x 2 +y

3 Define S(N,) = µ 2 ([n])µ 2 ([n]+). (7) Motivated by these results and following the method of Victorovich [3] we shall prove the following theorem. Theorem. Let > be irrational number with bounded partial quotient or irrational algebraic number. Then for the sums(n,) defined by (7) the asymptotic formula ) S(N,) = σn +O (N 5 6 +ε (8) holds. Hereσ is defined by (). 3 Lemmas Lemma. For the functionω (x) defined by (2) the formula ω (x) = ( +ψ(x) ψ x+ ) holds. Proof. See ([2], p. 480 ). Lemma 2. For every J 2, we have ψ(t) = ( ) a()e(t) + O b()e(t), a() /, b() /J. Proof. See []. J J Lemma 3. IfX, then n X e(n) ( X, ). 2 Proof. See ([8], Ch. 6, 2). Lemma 4. Suppose thatx,y, λ = a + θ, q, (a,q) =, θ. Then q q 2 ( ) Y, XY +(X +q)log2q. λn q n X Proof. See ([2], Lemma ). 3

4 4 Proof of the Theorem { m m The equality m = [n] is tantamount to n < m < n, } < n < m +, i.e. >. Then from (7) we get S(N,) = µ 2 ([n])µ 2 ([n]+) = m N µ 2 (m)µ 2 (m+) = m N Now (9) and Lemma give us S(N,) = m N { m }> ( m ) µ 2 (m)µ 2 (m+)ω. (9) µ 2 (m)µ 2 (m+)+ m N [ ( m µ 2 (m)µ 2 (m+) ψ ψ ) ( )] m+ = S (N,)+S 2 (N,), (0) where S (N,) = m N S 2 (N,) = m N µ 2 (m)µ 2 (m+), () [ ( m µ 2 (m)µ 2 (m+) ψ ψ ) ( )] m+. (2) Estimation ofs (N,) Bearing in d (6) and () we obtain whereσ is denoted by (). Estimation ofs 2 (N,) Let S (N,) = σn +O ( N ( )/8+ε ), (3) J = N. (4) From (2) and Lemma 2 it follows S 2 (N,) = ( )[ ( )] m µ 2 (m)µ 2 (m+) a()e e m N J ( +O µ 2 (m)µ 2 (m+) ( )[ ( )] ) m b()e +e m N J = [ ( )] ( ) m a() e µ 2 (m)µ 2 (m+)e J m N ( [ ( )] ( ) ) m +O b() +e µ 2 (m)µ 2 (m+)e J m N ( ) = S 3 (N,)+O S 4 (N,), (5) 4

5 where S 3 (N,) = J S 4 (N,) = b() J [ a() e [ +e ( )] ( ) m µ 2 (m)µ 2 (m+)e, (6) m N )] ( ) m µ 2 (m)µ 2 (m+)e. (7) ( m N Using (6) and Lemma 2 we find S 3 (N,) J J From (4), (7) and Lemma 2 we get S 4 (N,) J m N m N µ 2 (m)µ 2 (m+)e µ 2 (m)µ 2 (m+)e ( ) m. (8) ( ) m + N. (9) In order to estimate the sumss 3 (N,) and S 4 (N,) we shall prove the following lemma. Lemma 5. Let > be irrational number with bounded partial quotient or irrational algebraic number. Then for the sum Σ = µ 2 (m)µ 2 (m+)e(λm). (20) J m N where λ =, the estimation Σ N 5 6 +ε, holds. Proof. Using (20) and the well-nown identityµ 2 (m) = d 2 m µ(d) we write Σ = J 2 mµ(d) µ(t) e(λm) m N d t 2 m+ = µ(d) µ(t) e(λrd 2 ) J d N t N+ r N d 2 rd 2 + 0(t 2 ) e(λrd 2 ). J d N t N+ r N d 2 rd 2 + 0(t 2 ) Splitting the range of, d and t into dyadic subintervals we obtain Σ (logn) 3 max K J/2 D N/2 T N+/2 Σ 0 (K,D,T), (2) 5

6 where Σ 0 (K,D,T) = K 2K D d 2D T t 2T r N d 2 rd 2 + 0(t 2 ) e(λrd 2 ). (22) If (d, t) > then the sum Σ 0 (K,D,T) is empty. Suppose now that (d, t) =. Then the congruencerd 2 + 0(t 2 ) is equivalent tor r 0 (t 2 ), wherer 0 is some integer with r 0 t 2. From the last consideration and (22) it follows Σ 0 (K,D,T) K 2K Consider two cases. Case. DT (N) /6. The inequality (23) and Lemma 3 give us Σ 0 (K,D,T) K 2K D d 2D T t 2T D d 2D T t 2T K 2K D d 2D T t 2T Replacing m = d 2 t 2 from (24) we get Σ 0 (K,D,T) K 2K D d 2D T t 2T KD 2 T 2 m 32KD 2 T 2 s N r 0 d2 d 2 t 2 e(λsd 2 t 2 ). (23) ( ) N d 2 t 2, λd 2 t 2 ( ) N d 2 t 2, λd 2 t 2 ( ) N KD 2 T 2,. (24) λd 2 t 2 ( )( ) d 2 m d 2D t 2 m t 2T ( ) N KD 2 T 2, λm ( ) N τ 2 (m) KD 2 T 2, λm KD 2 T 2 m 32KD 2 T 2 ( ) N N ε KD 2 T 2,. (25) λm KD 2 T 2 m 32KD 2 T 2 Since ( therefore λ = ) is irrational number with bounded partial quotient or irrational algebraic number then λ can be represented in the form λ = a + θ, (N) q q 2 2 ε q (N) 2, (a,q) =, θ. This follows for example from ([3], Ch.2, Lemma.5, Lemma.6). Bearing in d these considerations, (4), (25), Lemma 4, the inequalities DT (N) /6 and K J we find ( ) N Σ 0 (K,D,T) N ε +KD 2 T 2 +q logn N 5 6 +ε. (26) q 6

7 Case 2. DT > (N) /6. Using (4), (23), the trivial estimate, the inequalitiesdt > (N) /6 and K J we obtain Σ 0 (K,D,T) K 2K D d 2D T t 2T N d 2 t 2 N DT logk N 5 6 +ε. (27) From (2), (26) and (27) it follows The lemma is proved. Σ N 5 6 +ε. On the one hand (8) and Lemma 5 give us On the other hand (9) and Lemma 5 imply S 4 (N,) µ 2 (m)µ 2 (m+)e J By (5), (28) and (29) we find m N S 3 (N,) N 5 6 +ε. (28) ( ) m + N N 5 6 +ε. (29) S 2 (N,) N 5 6 +ε. (30) The end of the proof Bearing in d (0), (3) and (30) we obtain the asymptotic formula (8). The theorem is proved. Acnowledgments. The author thans Professor Stephen Choi for his helpful comments and suggestions, that led to improvement of the reder term in the asymptotic formula (8). References [] A. G. Abercrombie, W. D Bans, I. E. Shparlinsi, Arithmetic functions on Beatty sequences, Acta Arith., 36, (2009), [2] G. I. Arhipov, V. A. Sadovnichy, V. N. Chubariov, Lectures on mathematical analysis, Vysshaya Shola, Moscow, (999), (in Russian). [3] L. Carlitz, On a problem in additive arithmetic II, Quart. J. Math., 3, (932), [4] S. I. Dimitrov, Consecutive square-free numbers of the form [n c ],[n c ] +, JP Journal of Algebra, Number Theory and Applications, 40, 6, (208), [5] S. I. Dimitrov, On the number of pairs of positive integers x,y H such that x 2 + y 2 +,x 2 +y 2 +2 are square-free, arxiv: v [math.nt] 5 Jan 209. [6] A. M. Güloğlu, C. W. Nevans, Sums of multiplicative functions over a Beatty sequence, Bull. Austral. Math. Soc., 78, (2008),

8 [7] D. R. Heath-Brown, The Square-Sieve and Consecutive Square-Free Numbers, Math. Ann., 266, (984), [8] A. Karatsuba, Principles of the Analytic Number Theory, Naua, Moscow, (983), (in Russian). [9] L. Mirsy, On the frequency of pairs of square-free numbers with a given difference, Bull. Amer. Math. Soc., 55, (949), [0] T. Reuss, Pairs of -free Numbers, consecutive square-full Numbers, arxiv:22.350v2 [math.nt] 9 Mar 204. [] J. D. Vaaler, Some extremal problems in Fourier analysis, Bull. Amer. Math. Soc. 2, (985), [2] R. C. Vaughan, On the distribution ofp modulo, Mathematia, 24, (977), [3] G. D. Victorovich, On additive property of arithmetic functions, Thesis, Moscow State University, (203), (in Russian). 8