Restricted divisor sums

Transcription

1 ACTA ARITHMETICA ) Restrcted dvsor sums by Kevn A Broughan Hamlton) Introducton There s a body of work n the lterature on varous restrcted sums of the number of dvsors of an nteger functon ncludng that descrbed n [2 9, ] and summarsed n Secton 2 below In ths paper asymptotc epressons are derved for sums of the form d α fn)) n where the functon fn) s n, n 2 or n 2 + n + p + )/, where p 3 mod s a ratonal prme, and where d α n) = #{d : d n and d α} for real α Motvaton for consderng these sums comes from an epresson whch s derved for the class number of a quadratc feld wth dscrmnant p, n terms of a certan restrcted dvsor sum Ths sum s currently too dffcult to estmate, n that the restrctons on dvsors depend on the summaton varable n In dervng asymptotc epressons for the sum d α n 2 ) n t s natural to ntroduce two so-called nteger square root functons r + n) and r n) Both are multplcatve and take nteger values Ther Drchlet seres are epressble n a compact ratonal form n terms of the Remann zeta functon 2 The class number Defne, for postve ntegral n and real a, b wth a < b, the restrcted dvsor functon: ) dn, a, b)) = #{d : d n and a < d < b} 2000 Mathematcs Subject Classfcaton: A25, A5, M06, N25, N37 Key words and phrases: Drchlet seres, dvsor sum, class number, nteger square root [05]

2 06 K A Broughan Theorem 2 Let p be a ratonal prme wth p > 3 and p 3 mod Then the class number for quadratc forms wth dscrmnant p can be epressed n the form 2) h p) = n 2 d p 3 n 2 + n + p +, 2n +, n 2 + n + p + )) Proof If f, y) = a 2 + by + cy 2 n Z[, y] s a quadratc form then t s prmtve f ) 0 b a = c or ) a < b < a < c or ) 0 < b = a < c The dscrmnant s 3) p = b 2 ac, so b 0 Correspondng to each trple a, b, c) satsfyng ), ) or ) there s a form, and dfferent trples correspond to nequvalent forms In case ) p = 2a) 2 b 2 = 2a + b)2a b) so 2a + b = p and 2a b = Therefore a = p + )/ and b = p )/2 so there s one soluton at most But b a mples p 3, so there are no solutons n case ) In case ) assume b a < c, snce f b s a soluton so s b and vce versa By 3), b s odd Snce also p 3 mod, ac = b2 + p s an nteger Hence a b 2 +p)/ and a < c therefore a < b 2 + p)/ hence a 2 p < b 2 < a 2 so a < p/3 and b p/3 Smlarly a < c f and only f a < b 2 + p)/ In case ), b = a < c so p = ac a) The relaton a = p s mpossble, snce then c a = and c = p + )/ < p = a If a = and c a = p we obtan c = p + )/, leadng to the so-called prncpal soluton a, b, c) =,, p + )/) Conversely, f a p+b 2 )/ and b < a < p + b 2 )/ wth b p/3, then p = b 2 ac and b < a < c For each odd value of b satsfyng b p/3 we count the number of dvsors a of b 2 + p)/ satsfyng b < a < b 2 + p)/, double to account

3 Restrcted dvsor sums 07 for each soluton a, b, c), and add for the prncpal soluton to obtan )) p + b 2 p + b 2 h p) = + 2 d, b, b p/3 b odd Fnally, let b = 2n + to obtan formula 2) Corollary 2 The class number h p) s odd Corollary 22 For all prmes p 3 mod wth p > 3, ) + p h p) d snce + p)/ s never a square Corollary 23 The followng upper bound s an mmedate consequence: h p) ) p + d n 2 + n + p + ) 2 3 p 3 0 n 2 Eample 2 If p = 59 a set of nequvalent representatves s {,, 5), 3,, 5), 3,, 5)} and h 59) = 3 If p = 5 then d38,, 6]) =, d0, 3, 6]) = 2, d, 5, 6]) = 0, d50, 7, 7]) = 0, so h 5) = ) = 7 3 Estng results To begn wth there s Drchlet s famous dvsor sum theorem of 850, and ts mprovements due to Vorono n 903 and van der Corput n 922 We have D) = n dn) = log + 2γ ) + Of)) where a more recent mprovement s f) = a wth a = 2/37 + ε, due to Kolesnk [7], and f) = a log b wth a = 23/73 and b = 6/6, due to Huley [5], s the best known publshed result In 952 Erdős [2] showed that f f s a polynomal wth nteger coeffcents, then there are postve constants A and A 2 such that A log < n dfn)) < A 2 log where the constants A depend on the coeffcents and hence also the degree) of f

4 08 K A Broughan In [2] Scourfeld quotng a result of Bellman Shapro states that f f s an rreducble quadratc polynomal wth ntegral coeffcents, then dfn)) = A log + O log log ), n where the constant A depends on the coeffcents of f Ths result was mproved by McKee n [8 0] who derved an error bound of O) In 963 Hooley [] consdered the specal case of n dn2 + a) and found asymptotc epressons for the cases a = k 2 and a k 2 Other results for restrcted dvsor sums nclude dvsors n short ntervals [3, ] and a number of results for dvsors n arthmetc progressons The monograph [3] covers n depth a range of related concepts In ths artcle we begn the task of analysng the class number dvsor sum derved n Theorem 2 above by lookng at the sums where the dvsors are restrcted n sze, ndependent of the summaton range Ths s done frst for fn) = n then fn) = n 2 and fnally fn) = n 2 + n + p + )/ In each case asymptotc epressons are derved Sums restrcted by dvsor sze Theorem Let α be real numbers Then D, α) = d α n) = log α + γ + O/α) + Oα) n Proof Smply count the lattce ponts below the curve uv = and above the nterval [, α] see for eample []): D, α) = = { }) j j j where γ s Euler s constant j α j α = log α + γ + O/α)) + Oα) = log α + γ + O/α) + Oα) 5 Integer square roots In ths secton we wll derve an asymptotc epresson for the restrcted dvsor sum D 2, α) = d α n 2 ) n by frst epressng t n terms of the nteger square root functon r + n) defned as follows: If n s a postve nteger, r + n) n and n r + n) 2, and f d s such that d n wth n d 2 then r + n) d Ths defnes r + n) unquely

5 Let r n) = n/r + n) Then f we have r + n) = m = Restrcted dvsor sums 09 n = m = p α p α /2 and r n) = m = p α /2 Note that r + n) 2 = n f and only f n s a perfect square, that for all prmes p, f p n then p r + n), r + n)r n) = n and n r + n) n Also r + n), r + m)) = r + n, m)) where n, m) s the greatest common dvsor Fnally, both r + and r are multplcatve, but not completely multplcatve We wll develop four Drchlet seres for these functons: ψ ± s) = n= r ± n) n s, φ ± s) = n= /r ± n) n s Theorem 5 For σ = Rs) suffcently large, the Drchlet seres satsfy the followng: ζ2s + )ζs + ) φ + s) = ζ2s + 2) σ > 0), ζ2s + )ζs) φ s) = ζ2s) σ > ), ζ2s )ζs ) ψ + s) = ζ2s 2) σ > 2), ζ2s )ζs) ψ s) = ζ2s) σ > ) Proof If p s a prme then r + p α ) = p α/2 so φ + s) = ) + r p + p)p + s r + p 2 )p + 2s + pp s + pp 2s + p 2 p 3s + ) p 2 p s + = p = p = p = p ) + ps pp 2s ) + pp 2s + ps pp 2s ) 2 + /pp 2s ) + ps /pp 2s ) ) /pp 2s ) ) + ) p 2s+ p s+ ) 2 ) pp 2s +

6 0 K A Broughan But ζ2s) ζs) = p + p s ) Hence φ + s) = n= ζ2s + )ζs + ) ζ2s + 2) Fnally σ + >, 2σ + > and 2σ + 2 > f σ > 0 Net, to derve the epresson for ψ s) use r n) /r + n) ψ s) = n s = n s = φ + s ) n= The other two dervatons follow n a smlar manner Lemma 5 D 2, α) = j α r + j) Proof If j α and jm = n 2 for some n, let j 0 be such that jj 0 = n 2 0 s the smallest multple of j whch s a square: f m j = then j 0 = m = = p α p α mod 2 and α + α mod 2)/2 = α /2 so r + j) = n 0 Then D 2, α) = = j α j=n 2 n j α jj 0 m 2 =n 2 n But jj 0 m 2 = n 2 n 2 0 m2 = n 2 n 0 m = n, n m /n 0 Hence D 2, α) = r + j) j α Eample 5 An elementary dervaton leads to an asymptotc formula for the partal sums of the squarefree recprocals: F ) = n = log + O) n n squarefree

7 To see ths, observe that µn) 2 = µd) = n Restrcted dvsor sums n d 2 n d µd) = d 2 + O ) = 6 π 2 + O ) d= µd) d 2 = µd) d d 2 + O ) The result now follows by Abel s Theorem for partal summaton [] A more precse result was found by Suryanarayana [3] It s ths form whch we use n the restrcted dvsor dervaton below, so we state t as a lemma Lemma 52 Suryanarayana) F ) = log + γ 2 ζ 2) Theorem 52 For and α, D 2, α) = log2 α + log α [ 3γ 2 ζ 2) ) + O ) ] + O) + Oα) Proof Usng Lemma 5, we obtan D 2, α) = = r + n) r + n) + { } = Sα) + Oα), r + n) n α n α n α where S) = n r + n) Now let = y 2 and for d =, 2, let S d be the set of postve ntegers n wth largest squared factor d 2 Note that f Q) s defned to be the set of squarefree ntegers less than or equal to then S d has Qy 2 /d 2 ) elements Note also that f n s squarefree, then r + nd 2 ) = nd Therefore Now let Sy 2 ) = d y = d y n Qy 2 /d 2 ) d n Qy 2 /d 2 ) r + nd 2 ) = d y n = ) y 2 d F d 2 d y β = γ 2 ζ 2) n Qy 2 /d 2 ) nd

8 2 K A Broughan and apply Lemma 52: S) = ) n n F n 2 = [ log + β) = log + β) 2 = log2 + log n n 2 n log + γ + O )) log 2 log2 + A + O + O) ) 3γ 2 ζ 2) + O) for some constant A Therefore [ log 2 α D 2, α) = + log α 3γ 2 ζ 2) = log2 α + [ 3γ log α 2 ζ 2) )] log n + O n )) ) ] + O) + Oα) ] + O) + Oα) 6 Bounds for restrcted sums for quadratc forms Theorem 6 Let fn) be an rreducble polynomal wth fn) > 0 for n =, 2, If α are real then there est postve constants c and c 2 such that c log α d α fn)) c 2 log α n Proof Defne the three functons θ, ϱ and N by { f fj), θ, j) = 0 f fj), ϱ) = #{j : fj), j }, N, ) = #{j : fj), j } Then, snce fj) f + j) we have ) ϱ) N, ) and so ) + ϱ) 2 ϱ) N, ) 2 ϱ) )

9 Therefore and so or where α 2 α 2 ϱ) ϱ) Restrcted dvsor sums 3 α j j α 2 Rα) j R) = By Lemma 9 and Secton 5 of [2], θ, j) α θ, j) 2 d α fj)) 2Rα) ϱ) 2c log R) 2 c 2 log and the concluson of the theorem follows drectly α 2 ϱ) In case f s quadratc, the prevous result can be strengthened to gve an asymptotc formula, usng the results of McKee [8 0] Theorem 62 Let fn) = an 2 + bn + c be an rreducble quadratc polynomal wth fn) > 0 for n =, 2, Let = b 2 ac < 0 not be a perfect square If α are real then there ests a postve constant A f such that d α fn)) = A f log α + O) where n A f = 6H ) π p a ), p + H ) s the weghted class number namely, the number of prmtve and mprmtve forms A 2 + By + Cy 2 wth B 2 AC =, gvng weght one half to forms proportonal to 2 + y 2, and one thrd to those proportonal to 2 + y + y 2 ) Proof Usng the same notaton as n the prevous theorem, t follows drectly from ) that ϱ) N, ) = ϱ) + Oϱ)) The same argument as that used n the theorem shows that d α fn)) = ϱ) ) + O ϱ) n α α

10 K A Broughan Usng the results of [0] gven n Lemma 7 and Lemma 8 for the two sums n ths formula, we obtan the result of the theorem wth the gven value for the constant A f Acknowledgments Ths work was done n part whle the author was on study leave at Columba Unversty The support of the Department of Mathematcs at Columba Unversty and the valuable dscussons held wth Patrck Gallagher are warmly acknowledged Helpful comments from a referee, especally those resultng n Theorem 62, are also gratefully acknowledged References [] T M Apostol, Introducton to Analytc Number Theory, Sprnger, New York, 976 [2] P Erdős, On the sum k= dfk)), J London Math Soc ), 7 5 [3] R R Hall and G Tenenbaum, Dvsors, Cambrdge Unv Press, 988 [] C Hooley, On the number of dvsors of quadratc polynomals, Acta Math 0 963), 97 [5] M N Huley, Area, Lattce Ponts and Eponental Sums, Oford Unv Press, 996 [6] A E Ingham, Some asymptotc formulae n the theory of numbers, J London Math Soc 3) 2 927), [7] G A Kolesnk, An mprovement n the remander term n the dvsor problem, Mat Zametk 6 969), n Russan); Englsh transl: Math Notes 6 969), [8] J F McKee, On the average number of dvsors of quadratc polynomals, Math Proc Cambrdge Phlos Soc 7 995), [9], The average number of dvsors of an rreducble quadratc polynomal, bd ), 7 22 [0], A note on the number of dvsors of quadratc polynomals, n: Seve Methods, Eponental Sums, and ther Applcaton to Number Theory, London Math Soc Lecture Note Ser 237, Cambrdge Unv Press, 997, [] Y Motohash, On the sum of the number of dvsors n a short segment, Acta Arth ), [2] E J Scourfeld, The dvsors of a quadratc polynomal, Proc Glasgow Math Assoc 5 96), 8 2 [3] D Suryanarayana, Asymptotc formula for n µ2 n)/n, Indan J Math 9 967), Department of Mathematcs Unversty of Wakato Hamlton, New Zealand E-mal: kab@mathwakatoacnz Receved on 2000 and n revsed form on )