Explicit Bounds for the Sum of Reciprocals of Pseudoprimes and Carmichael Numbers

Transcription

1 Journal of Integer Sequences, Vol ), Article Exlicit Bounds for the Sum of Recirocals of Pseudorimes and Carmichael Numbers Jonathan Bayless and Paul Kinlaw Husson University College Circle Bangor, ME 0440 USA baylessj@husson.edu kinlaw@husson.edu Abstract From a 956 aer of Erdős, we know that the base-two seudorimes and the Carmichael numbers both have a convergent sum of recirocals. We rove that the values of these sums are less than 33 and 28, resectively. Introduction By Fermat s little theorem, if is a rime number then for all a Z we have a a mod ). However, a number satisfying this roerty need not be a rime. For all a Z, a base-a Fermat seudorime or briefly, an a-seudorime) is a comosite number n such that gcda,n) = and a n a mod n). A Carmichael number is an odd comosite number n for which a n a mod n) for all integers a, and so is an a-seudorime for all a with gcda,n) =. Let P 2 = {34,56,645,05,...} be the set of 2-seudorimes, also called Poulet or Sarrus numbers, andletp 2 x) = {n P 2 : n x} bethecorresondingcountingfunction. Let C = {56,05,729,...} be the set of Carmichael numbers and write its counting function as Cx) = {n C : n x}. Erdős [] roved that for sufficiently large x, we have P 2 x) < x/exc logxloglogx)

2 and ) c2 logxlogloglogx Cx) < x/ex loglogx for constants c,c 2 > 0. This imlies that both sets have asymtotic density zero, as well as the stronger statement that both sets have a bounded sum of recirocals. Pomerance [8] imroved these bounds, showing that ) logxlogloglogx P 2 x) < x/ex 2loglogx for all sufficiently large x, and that the constant c 2 in the above bound on Cx) may be taken as. In the other direction, it is well known that there are infinitely many seudorimes with resect to a given base. In 994 Alford, Granville and Pomerance [] roved that there are infinitely many Carmichael numbers. In articular, they roved that Cx) > x 2/7 for sufficiently large x. Their work has since been imroved by Harman [3] to show that for large x, the inequality Cx) > x α holds for some constant α > /3. In this aer we determine exlicit uer and lower bounds for the sum of recirocals of 2-seudorimes, as well as for the sum of recirocals of Carmichael numbers. This extends revious work [4, 5, 3, 4] on recirocal sums. See [5] for a discussion of recirocal sums and their imortance in number theory. 2 Preliminary lemmas We state several reliminary lemmas. Throughout the aer m, n and k denote ositive integers, denotes a rime number, πx) is the rime counting function, logx denotes the natural logarithm and Pn) denotes the largest rime factor of n. Put x 0 = ex00) and y 0 = ex0). We begin with Korselt s criterion [7, Thm ] which rovides a useful characterization of Carmichael numbers. Lemma Korselt s Criterion). A ositive integer n is a Carmichael number if and only if it is comosite, squarefree, and for each rime n we have n. It follows from Korselt s criterion that Carmichael numbers are odd and have at least three different rime factors. The following lemma involves certain divisibility roerties ossessed by all 2-seudorimes. In articular, it shows that if a 2-seudorime n is divisible by 2 for a rime, then must be a Wieferich rime, that is, 2 mod 2 ). It follows that is greater than or equal to 093, the smallest Wieferich rime, see [7,. 3]. Lemma 2. Let n be a 2-seudorime. If 2 n then is a Wieferich rime. Furthermore, if 7 n then n mod 3). 2

3 Proof. Suose that 2 n and let k be the order of 2 in Z/ 2 Z). Thus k ϕ 2 ) = ). Now 2 n mod n), so 2 n mod 2 ), and thus k n, so k does not divide. Thus k, so that 2 mod 2 ). This comletes the roof of the first assertion. For the second assertion, we have 2 n mod n), so if 7 n then 2 n mod 7). The order of 2 modulo 7 is 3, so 3 n. We will also use exlicit versions of several classical theorems. The following modification of [2, Thm. 3.2] gives fairly shar exlicit bounds on the artial sums of the harmonic series. Lemma 3. For all x, n logx+γ) < x n x where γ = denotes Euler s constant. Wewillalsousethefollowingmodificationof[3, Lem.6.2]toboundthesumofrecirocals of numbers in a certain interval and residue class. Lemma 4. Let a Z and d = 2 b<k c 9 2 i, where i denotes the i-th rime number. We have i=2 n ad) 0 9 <n x 0 n < d Proof. Let b = 0 9 d+)/d and c = x 0 /d. Without loss of generality, we may assume that a {0,...,d }. By Lemma 3 we have dk +a 00 log0 9 d+)+ d c + ) < b d We will use Dusart s bounds [0, Thm. 5.6] on the sum of recirocals of rime numbers u to x. Lemma 5. For all x , we have loglogx+b) < 0.2 log 3 x, x where B = denotes the Mertens constant. We will also use Dusart s bounds [0, Cor. 5.2 & Thm. 5.9]. 3

4 Lemma 6. For all x > we have and for all x 599 we have πx) x logx x + logx ) log 2, x πx) x + ). logx logx Lemma 7. For all x , we have ) ex γ) logx We will also use the following result [9, Lem. 9.6]. 0.2 ) log 3. x Lemma 8. Let f be a multilicative function such that fn) 0 for all n, and such that there exist constants A and B such that for all x >, we have f)log Ax x and α 2 f α ) α log α B. ) Then, for x >, we have x fn) A+B +) logx n x n x fn) n. The following lemma makes the imlied constants in [9, Lem. 9.7] exlicit and modifies [3, Lem. 2.4]. Lemma 9. Let f be a multilicative function such that 0 f α ) ex ) 2α 3 for all rimes and integers α, and such that f α ) = 2α/3logy) for all y. Then for all x x 0 and y y 0, we have n xfn) x ) ) f α ). α x α 0 Proof. We wish to aly Lemma 8, and so we first establish values of A and B to use in inequality ) above. We have f) log ex2/3)θx) ex2/3)x x, 2) x using the bound θx) < x for all x > 0 [7, Cor. 2]. We also have y y 0 and f α ) = 2α/3logy) 4

5 whenever y. For brevity, let gt) = ft) logt)/t. We have g α ) = g α )+ g α ). <y 0 >y 0 α 2 α 2 Starting to bound the sum with = 2, observe that Here we used the fact that for r <. By similar reasoning, αlog2) 2 α/5 α 2 α 2g2 α ) = log2 2 α α 2 α 2 < αr α = α 2 3 <y 0 α 2 2r r2 r) 2 g α ) < α 2 /5 ) α Let ht) = 2 e 2/3 /t)logt)/t e 2/3 ) 2. By artial summation we have g α ) e 4/3 2 e 2/3 /)log e 2/3 ) 2 >y 0 α 2 >y 0 ) = e hy 4/3 0 )πy 0 ) πt)h t) dt y 0 ) < e hy 4/3 πt) 22 e2/3 )logt dt t 0 )πy 0 )+ y 0 t e 2/3 ) 3 ) < e hy 4/3 2t e2/3 ) dt t 0 )πy 0 )+ y 0 t e 2/3 ) e 0 = e 4/3 3e 2/3) ) < e 2/3 e 0 ) 2 Here we used Lemma 6. Furthermore, for all x x 0 we have ) logx. x by Lemma 7. Alying Lemma 8 with A = and B = , and noting that A+B + = and < , we comlete the roof of the lemma. 5

6 We now rove the following exlicit uer bound on de Bruijn s function Ψx, y), defined as the number of numbers u to x that are y-smooth that is, free of rime factors exceeding y). Lemma 0. For all x x 0 and y y 0, we have Ψx,y) x /2logy). Proof. We follow [9, Thm. 9.5]. By Rankin s method, if β > 0, then Ψx,y) x 3/4 + n ) βχy n), 3) x 3/4 n x where χ y n) = if Pn) y and χ y n) = 0 if Pn) > y. Set β = 2. By Lemma 9, we 3logy have that n xn β χ y n) x ) ) αβ ). 4) y α 0 Now, for any given value of, ) αβ ) = α 0 ) = β = + β β β. For 3, theroductcontributesafactorlessthan totheroductininequality 4). Bounding this second term for > 3, we see that since for 7, β = β β β ) 7 β 7 7 /5 < β < For 0 < t 2 3 we have et.4260t and therefore we may bound y = β = y = ex log β) β y < β log y. log Here we used Rosser and Schoenfeld s bound 3.23), [9,. 70], y log < logy logy < logy 6

7 for y 32. We therefore have inequality 4) bounded above by e βlogy Using this in inequality 3) for x x 0 gives the bound Ψx,y) x 3/ x /2logy) x /2logy). 3 The sum of recirocals of base-two seudorimes We rove exlicit bounds on the sum of recirocals of 2-seudorimes by exanding Luca and De Koninck s roof [9, Pro. 9.] of the following result. Proosition. For all c < /2 2) we have P 2 x) xex c logx ). By artial summation, it follows that the sum of recirocals is convergent. Theorem 2. The sum of the recirocals of base-two seudorimes satisfies < n P 2 n < 33. To rove Theorem 2 we slit the 2-seudorimes into three ranges. 3. The small range We first comute the recirocal sum over n 0 9. Feitsma [2] has comuted an exhaustive list of all 2-seudorimes n 0 9. We use this information to directly comute the recirocal sum from the 2-seudorimes n 0 2 to seven decimal laces as n P 2 n 0 2 n = For each k = 2,...,8, the contribution to the recirocal sum from 2-seudorimes n such that 0 k < n 0 k+ is bounded above by the number of such n times 0 k. We thus obtain n P 2 n 0 9 n

8 3.2 The middle range We now bound the recirocal sum over n such that 0 9 < n x 0. By Lemma 2, all 2- seudorimes are odd and not divisible by the square of any rime < 093. Therefore, they must lie in one of 9 D = 2 i ) i=2 residue classes modulo d, where d is as defined in Lemma 4. Also by Lemma 2, no 2- seudorime is congruent to 0 or 4 modulo 2. Therefore we may rule out 30 more residue classes modulo Thus 2-seudorimes must lie in one of D = residue classes modulo d. Therefore by Lemma 4, we have n P 0 9 <n x 0 n < D d < We may also remove the contribution to the sum from rime numbers. By Lemma 5, we have > < x 0 The sum in the middle range is therefore bounded above by 3.3 The large range = Finally, we bound the recirocal sum over 2-seudorimes n > x 0. Let x > x 0 and define y = ex logx), with y 0 = yx 0 ) = ex0). For ease of notation, write = Pn) and define t as the multilicative order of 2 modulo. Define the set Q = { : t < /4 }, and let Qx) = { Q : x}. Each 2-seudorime n > x 0 falls into exactly one of the following categories:. y, 2. > y and Q, 3. > y and Q. For each i 3, write A i for the set of 2-seudorimes n x satisfying roerty i above and ut A i x) = {n A i : n x}. 8

9 Observe that A x) = Ψx,y). By Lemma 0 we have Ψx, y) x/ exu/2) where u = logx logy = logx logx = logx. We thus have for x > x 0. Therefore, by artial summation, n A n>x 0 n x 0 A x) x ex 0.5 logx ) dt tex 0.5 logt ) = dw ex w/4) [ = e w/2 ] 00 w +2) < Here we used substitution followed by integration by arts. We now consider the second set, A 2. We first show that Qx) < x for all x. A comuter check shows that the claim holds for x e 22, that Qe 22 ) = 26, and that > e Q x Q e 22 Assume that x > e 22. We have < 2 t = ex log2 t<x /4 t<x /4 t ex x /2), while also Q x = Q e 22 Q e 22 < x > e e 22) Qx) 26 = e 22Qx) , so that Qx) < x /2 for all x as claimed. We thus have A 2 x) < 2 m<x/y x ) /2 x/y x /2 dt x <. m t/2 y /2 9

10 By artial summation, we therefore have n A 2 n>x 0 n x dt tex logt)/4) = dw ex w/4) < Let n A 3 and let = Pn). Then t /4 > y /4. Following [] we show that for such n we have n mod t ). We clearly have n mod ). We also have n mod t ). To see this, note that n = n ) ). We have t by Fermat s little theorem, and t n since n is a 2-seudorime and n. Since t < we have gcd,t ) =, and thus also n mod t ) as claimed. Thus the number of such n is bounded above by x/t )+. Also n > since is not a seudorime), so the number of such n is in fact bounded above by x/t ). Thus the number of 2-seudorimes in A 3 satisfies x A 3 x) x t >y >y 5/4. By Lemma 6 we have t/logt) +/logt) πt).25386t/logt for t y 0. Thus by artial summation we have A 3 x) x πy) y + 5 ) πt) dt 5/4 4 y t9/4 x +/logy ) dt = x = x y /4 logy ) 4 +/logy y /4 logy y Ei logy 4 logt)t 5/4 )) +/ logx Ei logx ex logx)/6 ) ) logx, where e t Eix) = x t dt denotes the exonential integral function. By another alication of artial summation, we thus have n +/ logt Ei ) logt t logt ex logt)/6) + 4 dt t n A 3 n>x 0 x 0 +/ w) dw ) w = Ei dw 00 w ex w/6) 00 4 = 8e Ei 2.5) Ei 2.5)+56e 2.5) <

11 Here we substituted w = logt and used integration by arts. Adding the contributions from the small, middle and large ranges, we obtain comleting the roof of Theorem < , 4 The sum of recirocals of the Carmichael numbers Theorem 3. The sum of recirocals of Carmichael numbers satisfies < n C n < To rove Theorem 3, we modify the small, middle and large ranges from the roof of Theorem 2 above. 4. The small range: n 0 2 A table of all 0000 Carmichael numbers u to has been comuted by R. Pinch, [5, 6]. Their contribution to the recirocal sum is easily comuted to be Pinch also determined that there are additional Carmichael numbers u to 0 2. Therefore an uer bound for the sum of recirocals of all Carmichael numbers n 0 2 is given by / < The middle range: 0 2 < n x 0 = ex00) We will use the following slight modification of Lemma 4, whose roof is nearly identical. Lemma 4. Let a Z and d = 2 0 i=2 2 i, where i denotes the i-th rime number. We have n ad ) n < d. 0 2 <n x 0 Since Carmichael numbers are odd and squarefree by Lemma, they must lie in one of 0 E = 2 i ) i=2

12 residue classes modulo d. We may also remove the contribution to the sum from rime numbers. By Lemma 5 we have > < x 0 It follows from Lemma that Carmichael numbers have at least three rime factors, so we may also remove the contribution to the sum from odd, squarefree numbers in the middle range with exactly two rime factors. Let π 2 x) = {n = q x : < q} denote the counting function of squarefree numbers with exactly two rime factors. Similarly, let π 2x) = {n = q x : 2 < < q} denote the counting function of odd squarefree numbers with exactly two rime factors. By a slight modification of Bayless and Klyve s lower bound [6] on π 2 x), we are able to show that for all x 0 2, we have π2x) xloglogx ). logx Letting A denote the set of odd squarefree numbers with exactly two rime factors, we therefore obtain by artial summation that where It follows that by 0 2 <n x 0 n A x0 n loglogx loglog I, 00 log0 2 loglogt )dt I = = 0 tlogt [ 2 log 2 ] 00 w = logw <n x 0 n A log n > logw )dw log0 w 2 Therefore, the contribution to the recirocal sum from the middle range is bounded above E/d < The large range: n > x 0 Finally, we bound the recirocal sum over Carmichael numbers n > x 0. Let x > x 0 and recall our definitions y = ex logx) and y 0 = yx 0 ) = ex0). Write = Pn). We slit the Carmichael numbers n > x 0 into two categories. 2

13 . y 2. > y. For i =,2, write B i for the set of Carmichael numbers n x satisfying roerty i above and ut B i x) = {n B i : n x}. By the same argument in the roof of Theorem 2 above, we have by Lemma 0, and B x) = Ψx,y) x ex 0.5 logx ) n B n>x 0 n < We next determine an uer bound for B 2 x). Observe that if n for a Carmichael number n, then n mod )). To see this, we clearly have n. Furthermore, since n = n ) ) and since n by Lemma, we have n. It follows that n mod )), since gcd, ) =. Therefore, the number of Carmichael numbers n x which are divisible by a given rime is bounded above by x/ )). We therefore have By artial summation, we have >y Here we used Lemma 6 to bound B 2 x) ) = x >y >y x ) = πy) yy ) + t logt πy) + y 2 ylogy = y + logt y y + ylogy for all t y 0. Therefore, for all x x 0 we have B 2 x) x y ). πt)2t ) dt t 2 t ) t dt tt ) 2 ) dt + logy y t 2 + ). logy ) πt) t logy 3 + )). logy

14 It follows by another alication of artial summation that n B 2 n>x 0 n = x 0 tex logt ))dt logt logt ex ))dw < w w w Adding the contributions from the small, middle and large ranges, we obtain < , comleting the roof of Theorem 3. 5 Concluding remarks It will take more work to substantially sharen the uer bound on the sum of recirocals of 2-seudorimes. In fact, the result is close to otimal for the arguments used, in the following sense. Reworking the arguments for different choices of the cutoff x = x 0 for the large range, the coefficients aearing in the bounds for each case tend to vary relatively slowly. Therefore we may attemt to roughly otimize the bound by minimizing the function fx) = logx log0 9 d+)) I +I +I 2 +I 3, which reresents the major contributions from the small, middle and large ranges to the recirocal sum, where d is as defined in Lemma 4, and I 3 = logx I = loglogx loglog log 3 x 0.2 log 3 0 9, I = I 2 = logx logx dw ex w/4), dw ex w/4, +/ w w ex w/6) Ei ) ) w dw. Alying the fundamental theorem of calculus and again substituting w = log x, the derivative is given by f x) = x J J J 2 +J 3 ), 4 where J = w w 4, 4

15 and J 3 = J = ex w/4), J 2 = ex w/4), +/ w Ei w ex w/6) Solving grahically for w, we find that the critical oint is aroximately w 79 so that x ex79), and we have fex79)) 32. This indicates that our cutoff of x 0 = ex00) for the large range is close to otimal for the method of argument used, esecially keeing in mind that lowering the cutoff to x 0 = ex79) will in fact raise the constants aearing in the various uer bounds for the large range. Furthermore, adjusting the coefficient in our definition of y = e clogx and reconsidering the argument, our choice of c = used above gave better bounds than larger or smaller values of c that we also considered. However, it may be ossible to further otimize the choice of coefficients, including those used in the roof of Lemma 0 bounding the count of smooth numbers. Another ossible way to imrove this bound would be to find a way to effectively utilize more information about the 2-seudorimes. For instance, more secific residue classes in the middle range could be ruled out using an inclusion-exclusion argument. Perhas the bound on the sum of recirocals of Carmichael numbers could be further otimized by utilizing more information about them, as well as otimizing the cutoff x 0. Another ossibility for imroving the bound for the sum of recirocals of Carmichael numbers in the middle range is to use an inclusion-exclusion argument to remove not only the odd squarefree numbers with exactly two rime factors, but also the ones which fall in certain secific residue classes that can be ruled out. Furthermore, Damgård et al [8, Thm. 5] roved that for all x, the number Nx) of Carmichael numbers u to x with exactly three rime factors satisfies the uer bound Nx) 0.25 xlogx) /4. Obtaining an exlicit lower bound on the count of odd, squarefree numbers u to x having exactly three rime factors would therefore allow one to remove the contribution from all such numbers in the middle range and relace it using this tighter bound. w 4 ). 6 Acknowledgments The authors would like to thank the referees for suggestions imroving both the writing and the mathematics of this aer. 5

16 References [] W. R. Alford, A. Granville, and C. Pomerance, There are infinitely many Carmichael numbers. Ann. of Math ), [2] T. Aostol, Introduction to Analytic Number Theory, Sringer Science & Business Media, 203. [3] J. Bayless and P. Kinlaw, On consecutive coincidences of Euler s function. Int. J. Number Theory 2 205), [4] J. Bayless and D. Klyve, On the sum of recirocals of amicable numbers. Integers 20), [5] J. Bayless and D. Klyve, Recirocal sums as a knowledge metric: theory, comutation, and erfect numbers. Amer. Math. Monthly ), [6] J. Bayless and D. Klyve, Sums over rimitive sets with a fixed number of rime factors. Submitted. [7] R. Crandall and C. Pomerance, Prime Numbers: A Comutational Persective, Lecture Notes in Statistics, Vol. 82, Sringer Science & Business Media, [8] I. Damgård, P. Landrock, and C. Pomerance, Average case error estimates for the strong robable rime test. Math. Com ), [9] J. M. De Koninck and F. Luca, Analytic Number Theory: Exloring the Anatomy of Integers, Graduate Studies in Mathematics, Vol. 34, Amer. Math. Soc., 202. [0] P. Dusart, Exlicit estimates of some functions over rimes. Ramanujan J. 206), 25. [] P. Erdős, On seudorimes and Carmichael numbers. Publ. Math. Debrecen 4 956), [2] J. Feitsma, Pseudorimes, 203, htt:// [3] G. Harman, Watt s mean value theorem and Carmichael numbers. Int. J. Number Theory ), [4] H. Nguyen, The recirocal sum of the amicable numbers. Dartmouth College undergraduate honors thesis, Hanover, NH, 204. [5] R. Pinch, The Carmichael numbers u to 0 2, Proceedings of the Conference on Algorithmic Number Theory, Turku Centre for Comuter Science ), [6] R. Pinch, Tables relating to Carmichael numbers, 2005, htt:// 6

17 [7] D. Platt and T. Trudgian, On the first sign change of θx) x. Math. Com ), [8] C. Pomerance, On the distribution of seudorimes. Math. Com ), [9] J. B. Rosser and L. Schoenfeld, Aroximate formulas for some functions of rime numbers. Illinois J. Math ), [20] L. Schoenfeld, Sharer bounds for the Chebyshev functions θx) and ψx). II, Math. Com ), Mathematics Subject Classification: Primary N25; Secondary Y99. Keywords: Carmichael number, seudorime, recirocal sum, smooth number. Concerned with sequences A00567 and A ) Received Aril 7 207; revised versions received May 3 207; June Published in Journal of Integer Sequences, June Return to Journal of Integer Sequences home age. 7