The Formulas for the Distribution of the 3-Smooth, 5-Smooth, 7-Smooth and all other Smooth Numbers Raphael Schumacher [email protected] Abstract In this paper we present and prove rapidly convergent formulas for the distribution of the 3-smooth, 5-smooth, 7-smooth and all other smooth numbers. One of these formulas is another version of a formula due to Hardy and Littlewood for the arithmetic p q function Na,b(x), which counts the number of positive integers of the form a b less than or equal to x. 1 Introduction Let a N≥2 be a fixed natural number. ∈ p Let Na(x) denote the number of natural numbers of the form a which are smaller or equal to x, where p N0. By definition [∈1, 2], we have that the 2-smooth numbers are just the powers of 2, namely S := 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,... 2 { } and that the formula for their distribution is log(x) 1 log(x) N (x)= + B . 2 log(2) 2 − 1 log(2) This follows directly from the more general formula arXiv:1608.06928v4 [math.NT] 23 Oct 2016 log(x) 1 log(x) N (x)= + B . a log(a) 2 − 1 log(a) p q Numbers of the form 2 3 , where p N0 and q N0 are called 3-smooth numbers [1, 2, 3], because these numbers are exactly the∈ numbers∈ which have no prime factors larger than 3. We will denote the sequence of 3-smooth numbers by S2,3. Thus, we have that S := 2p3q : p N , q N 2,3 { ∈ 0 ∈ 0} = 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144,... { } 1 More generally, let a, b N be fixed natural numbers such that a < b and gcd(a, b) = 1. ∈ p q Let Na,b(x) denote the number of natural numbers of the form a b which are smaller or equal to x, where p, q N . Furthermore, we denote by χ (x) the characteristic function ∈ 0 Sa,b of the natural numbers of the form apbq. In his first letter to Hardy [4, 5, 6, 7], Ramanujan gave the formula 1 log(2x) log(3x) 1 N (x) + χ (x), 2,3 ≈ 2 log(2) log(3) 2 S2,3 which provides a very close approximation to the number N2,3(x) of 3-smooth numbers less than or equal to x. In his notebooks [4, 7], Ramanujan later generalized this expression to all a, b N with ∈ gcd(a, b) = 1, namely 1 log(ax) log(bx) 1 N (x) + χ (x), a,b ≈ 2 log(a) log(b) 2 Sa,b which is again a very close approximation to Na,b(x). The analog formula for Na,b,c(x)[8, 9, 10] is 3 √ log x abc 1 N (x) + χ (x), a,b,c ≈ 6 log(a) log(b) log( c) 2 Sa,b,c which is also a very good approximation to Na,b,c(x). In the following sections, we present and prove rapidly convergent formulas for the functions Na,b(x) and N2,3(x), having the above Ramanujan approximations as their first term. These two formulas are other versions of a more rapidly convergent formula already found by Hardy and Littlewood around 1920 [11, 12], as it was communicated to us by Emanuele Tron [13]. We also prove very rapidly convergent formulas for the distribution of the 5-smooth, 7- smooth and all other smooth numbers. At the end of the paper, we give an exact formula for the counting function of the natural numbers of the form ap2 bq2 . We have searched all resulting formulas (which are given in theorems and corollaries) in the literature and on the internet, but we could only find the Hardy-Littlewood formula [11, 12]. Therefore, we believe that all other results are new. 2 2 The Formulas for Na,b(x) and the 3-Smooth Numbers Counting Function N2,3(x) Let a, b N such that a < b and gcd(a, b) = 1. ∈ + For x R , we define the function N (x) by ∈ 0 a,b Na,b(x):= 1. apbq ≤x p∈NX0,q∈N0 p q Moreover, we denote the set of natural numbers of the form a b by Sa,b and its characteristic function by χSa,b (x), that is S := apbq : p N , q N , a,b { ∈ 0 ∈ 0} 1 if x Sa,b χS (x):= ∈ . a,b 0 if x / S ( ∈ a,b We have that ⌊log (x)⌋ a x N (x)=1+ log + log (x) . a,b b ak ⌊ a ⌋ k X=0 j k Theorem 1. (Our Formula for Na,b(x)) For every real number x> 1, we have that 1 log(ax) log(bx) log(a) log(b) 1 1 log(x) N (x)= + + B∗ a,b 2 log(a) log(b) 12 log(b) 12 log(a) − 4 − 2 1 log(a) 1 log(x) log(a) log(x) log(b) log(x) B∗ B B − 2 1 log(b) − 2 log(b) 2 log(a) − 2 log(a) 2 log(b) ∞ ∞ 2πn log(x) 2πm log(x) log(a) log(b) cos log(a) cos log(b) 1 + − + χ (x), π2 m2 log(a)2 n2 log( b)2 2 Sa,b n=1 m=1 X X − where x 1 if x / Z B∗( x ):= { } − 2 ∈ , 1 { } 0 if x Z ( ∈ 1 B ( x ):= x 2 x + x R. 2 { } { } −{ } 6 ∀ ∈ The above formula converges rapidly. As usual we denote by x the fractional part of x. { } This is just another version of the following 3 Theorem 2. (The Hardy-Littlewood formula for Na,b(x))[11, 12] For every real number x 1, we have that ≥ 1 log(ax) log(bx) log(a) log(b) 1 log(x) log(x) N (x)= + + B∗ B∗ a,b 2 log(a) log(b) 12 log(b) 12 log(a) − 4 − 1 log(a) − 1 log(b) log(x)− 1 log(b) log(x)− 1 log(a) ∞ 2 2 1 cos 2πk log(a) cos 2πk log(b) 1 + + χS (x), − 2π πk log(b) πk log(a) 2 a,b k k sin k sin X=1 log(a) log(b) where the series is to be interpreted as meaning [12] log(x)− 1 log(b) log(x)− 1 log(a) ∞ 2 2 cos 2πk log(a) cos 2πk log(b) + πk log(b) πk log(a) k k sin k sin X=1 log(a) log(b) log(x)− 1 log(b) log(x)− 1 log(a) ⌊R log(a)⌋ 2 ⌊R log(b)⌋ 2 cos 2πk log(a) cos 2πk log(b) = lim + , R→∞ πk log(b) πk log(a) k k sin k k sin X=1 log(a) X=1 log(b) when R in an appropriate manner. →∞ This formula converges very rapidly. Setting a = 2 and b = 3, we get immediately two formulas for the distribution of the 3-smooth numbers, namely Corollary 3. (Our Formula for the 3-Smooth Numbers Counting Function N2,3(x)) For every real number x> 1, we have that 1 log(2x) log(3x) log(2) log(3) 1 1 log(x) N (x)= + + B∗ 2,3 2 log(2) log(3) 12 log(3) 12 log(2) − 4 − 2 1 log(2) 1 log(x) log(2) log(x) log(3) log(x) B∗ B B − 2 1 log(3) − 2 log(3) 2 log(2) − 2 log(2) 2 log(3) ∞ ∞ 2πn log(x) 2πm log(x) log(2) log(3) cos log(2) cos log(3) 1 + − + χ (x). π2 m2 log(2)2 n2 log(3) 2 2 S2,3 n=1 m=1 X X − Corollary 4. (The Hardy-Littlewood formula for N2,3(x))[11, 12] For every real number x 1, we have that ≥ 1 log(2x) log(3x) log(2) log(3) 1 log(x) log(x) N (x)= + + B∗ B∗ 2,3 2 log(2) log(3) 12 log(3) 12 log(2) − 4 − 1 log(2) − 1 log(3) log(x)− 1 log(3) log(x)− 1 log(2) ∞ 2 2 1 cos 2πk log(2) cos 2πk log(3) 1 + + χS (x), − 2π πk log(3) πk log(2) 2 2,3 k k sin k sin X=1 log(2) log(3) 4 where the series is to be interpreted as meaning [12] log(x)− 1 log(3) log(x)− 1 log(2) ∞ 2 2 cos 2πk log(2) cos 2πk log(3) + πk log(3) πk log(2) k k sin k sin X=1 log(2) log(3) log(x)− 1 log(3) log(x)− 1 log(2) ⌊R log(2)⌋ 2 ⌊R log(3)⌋ 2 cos 2πk log(2) cos 2πk log(3) = lim + , R→∞ πk log(3) πk log(2) k k sin k k sin X=1 log(2) X=1 log(3) when R in an appropriate manner. →∞ Using the computationally more efficient formula ⌊log (x)⌋ 3 x N (x)=1+ log + log (x) , 2,3 2 3k ⌊ 3 ⌋ k X=0 j k we get the following two tables: x N2,3(x) Our Formula for N2,3(x) Number of terms (n, m) 1 1 1.0510201857955517 (n, m)=(4, 4) at x =1.1 10 7 7.0071497373839231 (n, m) = (22, 22) 102 20 20.0045160354084706 (n, m) = (10, 10) 103 40 40.0039084310672772 (n, m) = (12, 12) 104 67 67.0408408937206653 (n, m) = (20, 20) 105 101 101.05072154439969785 (n, m) = (28, 28) 106 142 142.01315000789587358 (n, m) = (70, 70) 107 190 190.00707389223323501 (n, m) = (110, 110) 108 244 244.00659912032029415 (n, m) = (140, 140) 109 306 306.00585869480145596 (n, m) = (160, 160) 1010 376 376.02126583465866742 (n, m) = (170, 170) 10102 35084 35084.0568926232894816675 (n, m) = (2000, 2000) 10103 3483931 3483931.035272714689991309386 (n, m) = (4000, 4000) Table 1: Values of N2,3(x) 5 x N2,3(x) The Hardy-Littlewood Formula for N2,3(x) Number of terms R 1 1 1.00408281281244794423184044310637662236 R =1 10 7 7.01039536792580652845911613960427072715 R =6 102 20 20.00554687989157075178992137362449803027 R = 10 103 40 40.00416733658863125098651349198857667561 R = 26 104 67 67.04067163854917851848072234444363738009 R = 32 105 101 101.00383710643693392983460661037688277109 R = 44 106 142 142.00519665851176957826409909346411626717 R = 60 107 190 190.00431172466646336030921684292744206625 R = 100 108 244 244.00043300366963526250817238561664826018 R = 122 109 306 306.00450681431786167717856515798365069396 R = 146 1010 376 376.02231447192801988487484982661961706561 R = 160 10102 35084 35084.03451234481158685735036751788214481906 R = 3000 10103 3483931 3483931.03067546896021243171738747049589388966 R = 3000 10104 348149087 348149087.05625852937187129720297862230958308491 R = 24000 10105 34812470748 34812470748.06400873722492550333469431071713138958 R = 200000 Table 2: Values of N2,3(x) Corollary 5.