Andrew D. Loveless Department of Mathematics, University of Washington, Box , Seattle, WA , USA.

Transcription

1 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006, #A19 THE GENERAL GCD-PRODUCT FUNCTION Andrew D Loveless Department of Mathematcs, Unversty of Washngton, Box , Seattle, WA , USA aloveles@mathwashngtonedu Receved: 2/28/06, Revsed: 4/25/06, Accepted: 6/19/06, Publshed: 7/6/06 Abstract For a gven arthmetc functon h(x, we consder the functon g(n; h n h((j, n, where (j, n s the greatest common dvsor of j and n We gve evaluatons n terms of prme powers, Drchlet seres, and asymptotcs nvolvng g(n; h The Drchlet seres lead to several denttes nvolvng the Remann Zeta functon One such result s the followng: For real numbers α and β, we have n (j,nα n s n (j,nβ n s ζ(s α for Re(s > max{2, α+1, β+1} ζ(s β We fnsh by dscussng bounds along wth asymptotcs for the specal case g(n g(n; e where e(x : x Mathematcal Subject Classfcatons: 11A05, 11A25, 11M06, 11N37, 11Y60, 11Y11 1 Introducton For an arthmetc functon, h(x, we defne the General gcd-product Functon as follows: g(n; h : n h((j, n, where (j, n : gcd(j, n s the greatest common dvsor of j and n We also defne g(n : n (j, n Defnng e(x : x, note that g(n g(n; e In ths study, we nvestgate varous propertes of these functons Evaluatons and asymptotcs for the expresson n (j, n are dscussed by Broughan [2] A general development for sums of the form n, h((, j s gven everal publcatons

2 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006, #A19 2 by Cohen [3], [4], [5] Snce log[g(n; exp h] n h((j, n, many of our results wll generalze the prevous studes For clarfcaton, (exp h(x e h(x We begn by brefly mentonng two applcatons related to these functons For a postve nteger n, the number of dstnct solutons modulo n to the congruence x n 1 1 (mod n s gven by the formula p n (p 1, n 1, where the product s over all dstnct prmes dvdng n (a proof of ths formula s gven by Balle and Wagstaff [1] Formulas lke ths one are commonly used n the study of probablstc prmalty testng For nstance, ths author [6] gave a probablstc prmalty test whch never fals for Carmchael numbers by studyng formulas nvolvng products of the form p n [(p±1, n±1 2] These formulas do not easly submt to a general study Thus, we hope the general theory of the smpler functon g(n; h wll shed some lght on the growth and behavor of other formulas nvolvng gcd products Another applcaton arses n the study of lattce ponts on lnes n the plane Gven two nteger lattce ponts P (a, b and Q(c, d, the number of lattce ponts on the segment connectng P and Q s gven by (a c, b d + 1 Gveeveral dfferent lne segments, the formula for the number of ways to choose one lattce pont from each segment wll nvolve products of gcd values Iuch nstances, nformaton about the functon g(n can be useful The current study ncludes: 1 Evaluatons, denttes, and propertes of g(n; h 2 Drchlet seres and asymptotcs for log[g(n; h] 3 Bounds and asymptotcs for the specal case g(n 2 Evaluatons, Identtes and Propertes In the followng dscusson we assume that h(x s an arthmetc functon When we want h(x to have more propertes, lke the multplcatve property, we wll be explct Frst we consder the values of g(n; h at prme powers Observe, f p s a prme, then g(p; h p h((j, p h(ph(1 p 1 Theorem 1 If p s a prme and a s a postve nteger, then g(p a ; h h(pa h(p a 1 g(pa 1 ; h p

3 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006, #A19 3 Proof g(p a ; h p a h((j, pa p a 1 h((j, pa 2p a 1 jp a 1 +1 h((j, pa p a j(p 1p a 1 +1 h((j, pa h(pa p a 1 h(p a 1 h((j, pa 1 p a 1 h((j, pa 1 p a 1 h((j, pa 1 h(pa h(p a 1 g(pa 1 ; h p We can now gve the evaluaton at prme powers by nducton Theorem 2 If p s a prme and a s a postve nteger, then a 1 g(p a ; h h(p a h(p j (p 1pa j 1 Proof When a 1, we get the approprate value h(ph(1 p 1 Assume the result s true for k Then j0 g(p k+1 ; h h(pk+1 h(p k g(pk ; h p h(pk+1 h(p k ( h(p k k 1 j0 h(pj (p 1pk j 1 p h(p k+1 h(p k p 1 k 1 j0 h(pj (p 1pk j h(p k+1 k j0 h(pj (p 1pk j The result holds by nducton h(p k+1 k j0 h(pj (p 1p(k+1 j 1 Ths s not a mraculous result, but note that t holds for h(x n general For example, 17 2 (j, (j, ( ( (

4 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006, #A19 4 In the theory of arthmetc functons, the typcal next step s to prove that the functon s multplcatve However, g(n; h and g(n are not multplcatve n general Eveo, we can prove the followng relatonshp for relatvely prme ntegers Theorem 3 If h(x s multplcatve and m and n are ntegers wth (m, n 1, then g(mn; h [g(m; h] n [g(n; h] m Proof For (m, n 1 and j any postve nteger, observe that h((j, mn h((j, n(j, m h((j, nh((j, m Then g(mn; h mn h((j, mn [ mn h((j, m ] [ mn h((j, n ] [ m h((j, m ] n [ n h((j, n ] m [g(m; h] n [g(n; h] m Now defne f(n; h : g(n; h 1/n We can mmedately gve analogous results for f Notce that ths defnton of f(n; h was chosen to yeld the multplcatve property when h(x s multplcatve Theorem 4 If h(x s multplcatve, then the functon f(n; h g(n; h 1/atsfes the followng propertes: 1 If p s a prme and a s a postve nteger, then a 1 f(p a ; h h(p a p a h(p j (p 1p j 1 j0 2 If m and n are postve ntegers wth (m, n 1, then f(mn; h f(m; hf(n; h (f s multplcatve Proof The frst statement s a drect applcaton of Theorem 2 and the defnton of f For the second statement, usng Theorem 3, observe that f(mn; h g(mn; h 1/(mn [g(m; h n ] 1/(mn [g(n; h m ] 1/(mn f(m; hf(n; h

5 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006, #A19 5 Fnally, we have the general evaluatons of g(n; h and f(n; h based on the prme factorzaton of an nteger n when h(x s multplcatve Corollary 1 (Evaluatons of f(n; h and g(n; h If n k 1 pa s the prme factorzaton of n and h(x s multplcatve, then [ ] k a f(n; h h(p a 1 p a h(p j (p 1p j 1 and g(n; h 1 ( k [ 1 j0 h(p a p a a 1 j0 h(p j (p 1p j 1 ] n Takng h(x : x, we get all the same results for g(n and f(n as corollares Corollary 2 propertes: The functons f(n and g(n, where f(n g(n 1/n, satsfy the followng 1 If p s a prme and a s a postve nteger, then g(p a p pa 1 p 1 and f(p a p 1 p a p 1 2 If m and n are postve ntegers wth (m, n 1, then f(mn f(mf(n (f s multplcatve Proof Lettng h(x e(x : x n Theorem 4, we get g(p a g(p a ; e p a a 1 j0 (pj (p 1pa j 1 p a p (p 1pa 1 a 1 j0 j( 1 p j p a p (p 1pa 1 (a 1/p a a/p a 1 +1 p(1 1/p 2 p a p The result for f(p a s now mmedate (p 1 a 1 ap+p a (p 1 2 a 1 ap+pa a+ p p 1 p pa 1 p 1 Corollary 3 n, then (Evaluatons of f(n and g(n If n k 1 pa f(n [ 1 p k a k p p 1 and g(n p p a p 1 s the prme factorzaton of ] n

6 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006, #A Identtes nvolvng φ(n Here we wll gve denttes that relate g(n; h to the functon φ(n Not only are these denttes nterestng n ther own rght, but they wll be essental to our dervaton of Drchlet seres For a postve nteger k, the Euler φ-functon, φ(k, s defned to be the number of ntegers j wth 1 j uch that (j, k 1 For a dvsor e of n, we have (j, n e f and only f e j and ( j, n 1 Thus, the number of ntegers j wth 1 j uch that (j, n e, e e s gven by the formula φ(n/e Usng ths basc observaton we can evaluate the functon g(n; h n terms of φ(d, where d ranges over the dvsors of n, as follows: g(n; h n h((j, n e n h(eφ(n/e d n h(n/dφ(d Takng the logarthm of these equatons yelds the dentty below Theorem 5 If n s a postve nteger, then log[g(n; h] e n φ(n/e log[h(e] d n φ(d log[h(n/d] For h(x : x, we can conclude a lttle more Theorem 6 If n s a postve nteger, then g(n n n d n 1 d φ(d Proof Usng the relatonshp from Theorem 5 and the property d n φ(d n, we have g(n d n ( n d φ(d n d n φ(d d n 1 d φ(d nn d n 1 d φ(d 4 Drchlet Seres Defne the Drchlet seres for log[g(n; h] by G(s; h log[g(n; h] n log[h((j, n] for s A h, where A h C s the set of values for whch the sum converges dependng on h Now we gve the evaluaton of G(s; h n terms of the Remann zeta functon and the seres log[h(n]

7 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006, #A19 7 Before we proceed, we gve a bref dscusson of convergence of the seres log[h(n] For a gven functon, h(x, we defne the set S h and the constant c h by S h {α R log[h(n] O(n α as n } and c h max{2, 1 + nf S h }, where nf denotes the greatest lower bound We restrct our attenton to those functons h(x such that S h s non-empty The 2 n the maxmum part of the defnton of c h s requred so that ζ(s 1 converges as a sum (ths expresson s mportant n the followng theorems Theorem 7 The Drchlet seres for G(s; h converges for Re(s > c h and s gven by ( ( ζ(s 1 log[h(n], where s the Remann zeta functon Proof From Theorem 5, log[g(n; h] d n φ(d log[h(n/d] (φ (log h(n, where the product s the Drchlet convoluton Hence, gven the requred convergence, we have G(s; h log[g(n;h] ( ( φ(n log[h(n] ( ( ζ(s 1 log[h(n] ζ(s 1 Observe that defnton of c h s necessary to ensure convergence for the sum and the sum log[h(n] φ(n For the followng dscusson, we defne the class of functons h (α (x x α Consderng the two specal cases G(s; h (α and G(s; exp h, respectvely, we mmedately conclude n log[(j, nα ] n h((j, n αζ(s 1ζ (s ζ(s 1 ( h(n for s > 2 and (1 for s > c exp(h(x (2

8 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006, #A19 8 Usng (1 and (2, we can obtan a multtude of ntrgung denttes We collect several such results n the followng corollary Corollary 4 The followng relatonshps hold for all α, β R: ( n (j, nα ζ(s 1ζ(s α for Re(s > max{2, α + 1} ( n h((j,n n s h(n ζ(s 1 for Re(s > c exp(h(x ( n h 1((j,n n s n h 2((j,n h 1 (n n s h 2 (n for Re(s > max{c exp(h1 (x, c exp(h2 (x} (v (v n (j,nα n s n (j,nβ ζ(s α ζ(s β n φ((j, n ζ2 (s 1 ζ 2 (s for Re(s > max{2, α + 1, β + 1} for Re(s > 2 Proof Identty ( s equaton (2 wth h (α (x x α Identty ( s a restatement of Theorem 7 Settng up two equatons of the form n equaton (2 wth h 1 (x and h 2 (x, and then dvdng the equatons, yelds dentty ( Identty (v s a specal case of (, wth h 1 (x x α and h 2 (x x β Fnally, lettng h 1 (x φ(x n equaton (2 and usng the well-known dentty, we get dentty (v φ(n ζ(s 1 Note that denttes ( and (v suggest possble ways to evaluate the zeta functon For example, f we could choose a functon h(x uch a way that evaluated n closed form, then we would have an evaluaton for ζ(s 1 n h((j,n n s h(n could be That s, we would have a way to nductvely evaluate the functon at nteger values Such a solutons could gve evaluatons for ζ(3, ζ(5, etc Snce we have closed form evaluatons for ζ(2k when k s a postve nteger, we can gve evaluatons for certan ratos of the form n dentty (v For example, wth s 8, α 2, and β 4 we have: n (j,n2 n 8 n (j,n4 n 8 ζ(6 ζ(4 2π2 21 We can use the results of Theorem 7 and Corollary 4 to gve denttes nvolvng nfnte seres of zeta functon values

9 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006, #A19 9 Theorem 8 If h(x s a functouch that log(h(x admts an nfnte seres representaton log[h(x] k1 a ( 1 k k x that converges unformly for x 1 and for Re(s > ch, we have n ( log[h((j, n] ( ζ(s 1 a k ζ(s + k Proof Usng the nfnte seres expansons, we have n ( log[h((j, n] ( ζ(s 1 k1 ( ( ζ(s 1 k1 a ( 1 k ( k n ( ζ(s 1 k1 ( ( ζ(s 1 a k ζ(s + k k1 log[h(n] a k 1 +k Corollary 5 If Re(s > 2, then n log[1 1 ] ( ( 2(j,n ζ(s 1 k1 ζ(s + k 2 k k Proof Apply Theorem 8 wth h(x 1 ( 1 1 k 2 k k x for x > 1/2, and ak 1 2 k k k x, so that log[h(x] log(1 1 ( 1 1 k 2x k1 k 2x Now we gve asymptotc formulas for n x log[g(n; h] For the followng development, we only consder the case h(n O(n α for uffcently large and α a fxed non-negatve real number Includng a larger class of functons would consderably ncrease the length of ths study We frst need a standard estmate from analytc number theory Tenenbaum [7] gves the followng estmate n hs book: Φ 0 (x : n x φ(n x2 + O(x log(x 2ζ(2 We also need the followng lemma Lemma 1 If x and y are real numbers such that x y > 1, then ( x log < log(x y log(y

10 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006, #A19 10 Proof Fx x > 1 and consder the functon a x (y log(x log( x on the doman (1, x] log(y y Takng the dervatve wth respect to y gves a x(y log(x y log 2 (y 1 y 1 y [ ] 1 log 2 (y + 1 Thus, a x (y s monotoncally decreasng on the doman If y x, then the a x (x log(x log(x log( x x 1 Hence a x(y 1 for all y (1, x] Snce x was arbtrary, the result holds Theorem 9 If h(x satsfes the relatonshp h(n O(n α for uffcently large and α a fxed non-negatve real number, then the followng asymptotc relatonshp holds log[g(n; h] n x where H h log[h(k] k1 k 2 H h 2ζ(2 x2 + O(x log 2 (x for x suffcently large, Proof From the well-known bound, we have Φ 0 (x constant c and x suffcently large x2 2ζ(2 + cx log(x for some postve Therefore, usng Theorem 6 and Lemma 1, we have n x log[g(n; h] n x d n log[h(d]φ(n/d d x log[h(d]φ 0(x/d [ ] d x log[h(d] x 2 /d 2 + 2ζ(2 d x log[h(d][cx/d log(x/d] [ x2 2ζ(2 d x [ x2 2ζ(2 d x H h 2ζ(2 x2 + O(x log 2 (x log[h(d] ] + cx log(x log[h(d] d 2 d x d log(d log[h(d] ] + O(x log(x α log(d d 2 d x d log(d Corollary 6 If h(x satsfes the relatonshp h(n O(n α for uffcently large and α a fxed non-negatve real number, then the followng asymptotc relatonshp holds n x where H h log[h(k] k1 k 2 g(n; h O(x x log(x e H h 2ζ(2 x2 for x suffcently large,

11 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006, #A The Specal Case g(n In ths secton, we focus only on the expresson g(n n (j, n and ts multplcatve counterpart f(n g(n 1/n Frst, we gve some bounds For postve ntegers x 1, x 2,, x n, we have the well-known arthmetc-geometrc mean nequalty (x 1 x 2 x n 1/n x 1+x 2 + +x n In terms of our functon f, we have f(n n ( n 1/n (j, n n (j,n n Theorem 10 The functon f satsfes the nequaltes max(n 1/υ(n, n τ(n/(2n f(n 27 ( log(n ω(n ω(n, where n s any postve nteger, ω(n s the number of dstnct prme dvsors of n, τ(n s the number of dvsors of n, and υ(n s the largest prme power dvsor of n Proof The upper bound s a drect applcaton of the arthmetc-geometrc mean and Theorem 31 of Broughan [2] For the frst part of the lower bound, we frst note that 1 p a p 1 p a 1 > 1 (p a 1 + p a p + 1 a Thus, we have p a (p 1 p a p a f(n p a n p 1 p a p 1 p a/pa p a/υ(n n 1/υ(n p a n p a n For the second part of the lower bound, note ( 2 ( ( ( ( d n d d n d e n e d n d d n d n d n d d n n nτ(n n d So we have ( n 1/n f(n (j, n d d n 1/n n τ(n/(2n Corollary 7 The functon g satsfes the nequaltes n max(n n/υ(n, n τ(n/2 g(n 27 ( nω(n log(n, ω(n wth g(n n f and only f n s a prme Therefore, for all ɛ > 0, we have log[g(n] O(n 1+ɛ for uffcently large

12 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006, #A19 12 Proof Ths corollary drectly follows from prevous theorems Note that f n s composte, then n (j, n > nce (n, n n and (j, n > 1 for at least one j value less than n The Drchlet seres for log[g(n] s gven as a corollary of Theorem 7 wth h(x : x So we defne G(s log[g(n] Note that log[f(n] G(s + 1 Corollary 8 The Drchlet seres for G(s converges absolutely for Re(s > 2 and s gven by the formula ζ(s 1ζ (s Proof Takng h(x : x n Theorem 7 yelds G(s; h ζ(s 1 H h, where H h log(n ζ (s The asymptotc development s smlar to the general case n x log[g(n] However, we can do a lttle more n the case of log[f(n] Along wth the estmate for Φ 0 (x : x2 n x φ(n + O(x log(x, we wll also need the estmate Φ 2ζ(2 1(x : φ(n n x n x + O(log(x ζ(2 Theorem 11 The followng asymptotc relatonshps hold n x log[g(n] ζ (2 2ζ(2 x2 + O(x log 2 (x for x suffcently large, and n x log[f(n] ζ (2 ζ(2 x + O(log2 (x for x suffcently large Proof Usng Theorem 9, we have H x that log[f(n] 1 n d n φ(n/d log(d d n log(k k1 k 2 φ(n/d n/d ζ (2 For f(n g(n 1/n, we note log(d d Therefore, for x suffcently large and some constant c, the known bounds gve n x log[f(n] n x d n d x log(d d [ x ζ(2 d x log(d φ(n/d d n/d [ ] x/d + ζ(2 d x log(d [c log(x/d] d log(d ] + c log(x 1 d 2 d x d [ x log(d ζ(2 d x ] + O(log 2 (x d 2

13 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006, #A19 13 Corollary 9 The followng asymptotc relatonshps hold n x g(n O(xx log(x e ζ (2 2ζ(2 x2 for x suffcently large, and n x f(n O(xlog(x e ζ (2 ζ(2 x for x suffcently large 6 Concluson We hope that the functon g(n; h wll be a useful tool n the study of lattce ponts and solutons to equatons n fnte felds The author was ntrgued by the fact that the rato h((j,n n s h(n s always the same as the rato ζ(s 1 Hopefully ths dentty, along wth the others of ths study, wll be useful n the theory of the Remann zeta functon The functon g(n; h seems to be worthy of study, not only because of ts connectons wth known concepts n number theory, but also for the elegance of the formulas and denttes n whch t appears Acknowledgements My thanks to the revewer of ths artcle whose suggestons and comments greatly mproved the clarty of ths exposton References [1] R Balle, SS Wagstaff, Jr, Lucas Pseudoprmes, Math Comp 35 (1980, 152, [2] KA Broughan, The gcd-sum functon, Journal of Integer Sequences, Vol 4 (2001, Artcle 0122 [3] E Cohen, Arthmetcal functons of greatest common dvsor I, Proc Amer Math Soc 11 (1960 [4] E Cohen, Arthmetcal functons of greatest common dvsor II, An alternatve approach, Boll Un Mat Ital (3 17, (1962, [5] E Cohen, Arthmetcal functons of greatest common dvsor III Cesáro s dvsor problem, Proc Glasgow Math Assoc 5 (1961, [6] AD Loveless, A Composteness Test that Never Fals for Carmchael Numbers, Math of Comp, preprnt [7] G Tenenbaum, Introducton to Analytc and Probablstc Number Theory, Great Brtan: Cambrdge Unversty Press, 1995