A First Digit Theorem for Square-Free Integer Powers

Transcription

1 Pure Matheatical Science, Vol. 3, 014, no. 3, HIKARI Ltd, A Firt Digit Theore or Square-Free Integer Power Werner Hürliann Feldtrae 145, CH-8004 Zürich, Switzerland Copyright 014 Werner Hürliann. Thi i an open acce article ditributed under the Creative Coon Attribution Licene, which perit unretricted ue, ditribution, and reproduction in any ediu, provided the original work i properly cited. Abtract For any ixed integer power, it i hown that the irt digit o quare-ree integer power ollow a generalized Benord law (L) with ize-dependent exponent that converge ayptotically to a L with invere power exponent. In particular, ayptotically a the power goe to ininity the equence o quareree integer power obey Benord law. Moreover, we how the exitence o a one-paraetric ize-dependent exponent unction that converge to thee L and deterine an optial value that iniize it deviation to two iniu etiator o the ize-dependent exponent over the inite range o quare-ree integer power le than 10, = 4,..., 10, where = 1,,3,4,5, 10 i a ixed integer power. Matheatic Subject Claiication: Priary 11A5, 11K36, 1137, 11Y55; Secondary 6E0, 6F1 Keyword: irt digit; quare-ree nuber; ayptotic counting unction; probabilitic nuber theory; generalized Benord law; ean abolute deviation; probability weighted leat quare 1. Introduction It i well-known that the irt digit o any nuerical data et are not uniorly ditributed. ewcob [14] and Benord [3] oberved that the irt digit o any erie o real nuber obey Benord law P B d) = log (1 + d) log, d 1,,...,9 (1.1) ( =

2 130 Werner Hürliann The increaing knowledge about Benord law and it application ha been collected in variou bibliographie, the ot recent being Beebe [] and Berger and Hill [4]. It i alo known that or any ixed power exponent 1, the irt digit o integer power, ollow ayptotically a Generalized Benord law (L) with exponent = 1 (0,1) uch that (ee Hürliann [7]) (1 + d) d =, d = 1,,...,9. (1.) 10 1 P Clearly, the liiting cae 0 repectively 1 o (1.) converge weakly to Benord law repectively the unior ditribution. We tudy the ditribution o irt digit o quare-ree integer power. The ethod conit to it the L to aple o irt digit uing two ize-dependent goodne-o-it eaure, naely the ETA eaure (derived ro the ean abolute deviation) and the WLS eaure (weighted leat quare eaure). In Section, we deterine the iniu ETA and WLS etiator o the L over inite range o quare-ree power up to 10, 4, 1 a ixed power exponent. Coputation illutrate the convergence o the ize-dependent L 1 with iniu ETA and WLS etiator to the L with exponent. Moreover, we how the exitence o a one-paraetric ize-dependent exponent unction that converge to thee L and deterine an optial value that iniize it deviation to the iniu ETA and WLS etiator. A atheatical proo o the ayptotic convergence o the inite equence to the L with invere power exponent ollow in Section 3.. Size-dependent L or quare-ree integer power To invetigate the optial itting o the L to irt digit equence o quareree integer power, it i neceary to peciy goodne-o-it (GoF) eaure according to which optiality hould hold. Firt o all, a reaonable GoF eaure or the itting o irt-digit ditribution hould be ize-dependent. Thi ha been oberved by Furlan [5], Section II.7.1, pp.70-71, who deine the ETA eaure, and by Hürliann [8], p.8, who applie the probability weighted leat quare (WLS) eaure ued earlier by Leei et al. [1] (chi-quare divided by aple ize). Let { x n } [1, ), n 1, be an integer equence, and let d n be the (irt) igniicant digit o x n. The nuber o x n, n = 1,...,, with igniicant digit d n = d i denoted by X (d). Then, Furlan ETA eaure or the L i deined to be

3 A irt digit theore or quare-ree integer power 131 ETA X ( d ) ( ) = MAD ( ), MAD ( ) = P, (.1) 9 d = 1 where MAD ( ) i the ean abolute deviation eaure. The latter eaure i alo ued to ae conority to Benord law by igrini [15] (ee alo igrini [16], Table 7.1, p.160). The WLS eaure or the L i deined by (e.g. [1]) WLS X ( d ) 1 9 ( P ) ( ) =. (.) d = 1 P Conider now the equence o quare-ree integer power { n }, n < 10, or a ixed power exponent = 1,,3,..., and arbitrary quare-ree nuber n below 10, 4. Denote by I k (d) the nuber o quare-ree power below k 10, k 1, with irt digit d. Thi nuber i deined recurively by the relationhip k k I k = S( ( d + 1) 10 ) S( d 10 ) + I, 1,,..., (.3) + 1 k k = where the counting unction S (n) i given by (e.g. Pawlewicz [18], Theore 1) n n S( n) = μ ( k), (.4) k = 1 k where μ (k) i the Möbiu unction uch that μ ( k) = 0 i p divide k and e μ ( k ) = ( 1) i k i a quare-ree nuber with e ditinct prie actor, and denote the integer-part unction. Recent algorith to eiciently copute thee arithetic unction are contained in Pawlewicz [18] and Auil [1]. Thereore, with (10 = S ) one ha X = I in (.1)-(.). A lit o the I, = 4,...,10, = 1,,3,4,5, 10, together with the aple ize = S(10 ), i provided in Table A.1 o the Appendix. Baed on thi we have calculated the optial paraeter which iniize the ETA (or equivalently MAD) and WLS eaure, the o-called iniu ETA (or iniu MAD) and iniu WLS etiator. Together with their GoF eaure, thee optial etiator are reported in Table.1 below. ote that the iniu WLS i a critical point o the equation

4 13 Werner Hürliann WLS ( ) = P (1 + d) = 1 9 P P d = 1 {ln( 1+ d 10 )10 ln(1 + d)} d (10 P 1) ( ) X ( d ) {ln( = 0, d 10 )10 ln}. (.5) For coparion, the ETA and WLS eaure or the ize-dependent L exponent 1 ( ) = {1 c 10 }, (.6) LL with c = 1, called LL etiator, are lited. Thi type o etiator i naed in honour o Luque and Lacaa [13] who introduced it in their L analyi or the prie nuber equence. Through calculation one oberve that the LL etiator iniize the abolute deviation between the LL etiator and the ETA (rep. WLS) etiator over the inite range o quare-ree power [1, 10 ], = 4,...,10, = 1,,3,4,5,10. In act, i one denote the ETA and WLS etiator o the equence { n }, n < 10, by ETA ( ) and WLS ( ), then one ha uniorly over the conidered inite range (conult the colun Δ ( 3) to LL etiate in Table.1 in unit o 10 ) ( 3) WLS ( ) LL ( ) , (.7) ( 3) ETA ( ) LL ( ) Table.1 diplay exact reult obtained on a coputer with ingle preciion, i.e. with 15 igniicant digit. The ETA (rep. WLS) eaure are given in unit o ( + 7) (+ 4) 10 (rep. 10 ). Taking into account the decreaing unit, one oberve that the optial ETA and WLS eaure decreae with increaing aple ize. Table.1: L it or irt digit o quare-ree power: ETA v. WLS criterion =1 paraeter Δ to LL etiate ETA GoF eaure WLS GoF eaure = WLS ETA WLS ETA LL WLS ETA LL WLS ETA = paraeter Δ to LL etiate ETA GoF eaure WLS GoF eaure = WLS ETA WLS ETA LL WLS ETA LL WLS ETA

5 A irt digit theore or quare-ree integer power 133 =3 paraeter Δ to LL etiate ETA GoF eaure WLS GoF eaure = WLS ETA WLS ETA LL WLS ETA LL WLS ETA =4 paraeter Δ to LL etiate ETA GoF eaure WLS GoF eaure = WLS ETA WLS ETA LL WLS ETA LL WLS ETA =5 paraeter Δ to LL etiate ETA GoF eaure WLS GoF eaure = WLS ETA WLS ETA LL WLS ETA LL WLS ETA =10 paraeter Δ to LL etiate ETA GoF eaure WLS GoF eaure = WLS ETA WLS ETA LL WLS ETA LL WLS ETA Ayptotic counting unction or quare-ree integer power The ollowing i a light extenion o the arguent by Luque and Lacaa [13], 1 Section 5(a). It i well-known that a rando proce with unior denity x generate data that are Benord ditributed. Siilarly, a equence o nuber

6 134 Werner Hürliann generated by a power-law denity x, (0,1), ha a L irt-digit ditribution P 1 with exponent 1. Fro uch a denity it i poible to derive a counting unction C () or that equence in the interval [1, ]. However, auing a local denity o the or uch that C( ) ~ x ( x ) dx x (x) i not appropriate in general. Indeed, the quare-ree power relation over an interval [1, ] that belong to (.6), naely 1 c + ( ) ( ) =, ( ) =, (3.1) doe not behave oothly in [1, ], which hould be the cae or uch an approxiation. Thi drawback can be overcoe. Denote by Q ( ) the counting unction or quare-ree power in [1, ]. Intead o x deine Q ( 6 ) = π x ( ) ( ) dx dx, (3.) where the integral pre-actor i choen to ulill the ayptotic liiting value or the quare-ree nuber counting unction, that i (note that n < i, and only i, one ha n < ) Q ( li ) 6 =. (3.3) π In act, two iproved ayptotic expanion o S () are known, naely 6 S ( ) = + O π ( ) ε 54, and ( ) = + O( ) S. (3.4) π The irt one i claical and proved in Hardy and Wright [6], p.69, and Jaeon [9], Section.5, or exaple. The econd iproved etiate i due to Jia [11] (ee alo Pappalardi [17]). However, it uice to ue the iple etiate (3.3), which i obtained a ollow. Fro (3.) one get or arbitrary = 1,,... 6 ( ) 6 1 (1 ( )) Q ( ) = x dx =. (3.5) π π (1 ( )) With (3.1) thi tranor to

7 A irt digit theore or quare-ree integer power ( ) 6 Q ( ) = = π 1 ( ) π ln( ) exp c, (3.6) c which i independent o and iply denoted by Q (). The equality Q ( ) = Q( ) relect the act that there are a any quare-ree power in [1, ] a there are quare-ree nuber in [1, ]. ow, what i a good value o c [1, )? Clearly, the actor c ln( ) ( c ) ( c) = exp (3.7) converge to 1 a atiie the property or any ixed c. It derivative with repect to c ln( ) 1 c ( c) < 0, c [1, ln( ) ) [1, ), 4, (3.8) which iplie the ollowing in-ax property o (3.7) at c = 1: in{ ax ( c)} = 4 (1) = (3.9) ln( ) c [1, ) ln( ) The ize-dependent exponent (3.1) with c = 1 not only iniize the abolute deviation between the LL etiator and the ETA (rep. WLS) etiator over the inite range o quare-ree power [1, 10 ], = 4,...,10, = 1,,3,4,5, 10, a hown in Section, but it turn out to be uniorly bet with axiu error le 3 than 10 4 againt the ayptotic etiate, at leat i 10. Moreover, the ollowing liiting ayptotic reult ha been obtained. Firt Digit Square-Free Integer Power Theore (L or quare-ree integer power). The ayptotic ditribution o the irt digit o quare-ree integer power equence n < 10, 4, or ixed = 1,,3,..., a, i given by I 1 1 li = li ( ) ( ) = P 1 ( ), = 1,...,9, ( ) = 1. d P d d (3.10) (10 ) S 10 Table 3.1 copare the new counting unction Q( ) = Q ( ), = 1,,..., with 6 the exact and ayptotic counting unction S () and S a ( ) =. π

8 136 Werner Hürliann Table 3.1: Coparion o quare-ree nuber counting unction or = 10 S() Q() 6/π² Q()/S() '083 6'074 6' '794 60'786 60' '96 607' ' '079'91 6'079'61 6'079' '79'694 60'79'699 60'79' '97'14 607'97' '97' '079'70'94 6'079'71'005 6'079'71' '79'710'80 60'79'710'170 60'79'710' '97'10'74 607'97'101' '97'101' '079'71'018'94 6'079'71'018'5 6'079'71'018' '79'710'185'947 60'79'710'185'383 60'79'710'185' '97'101'854' '97'101'854' '97'101'854' Concluding Reark 3.1. A proved by Jaeon [10] the proportion o odd quare-ree nuber i ayptotically equal to 4 / π, ro which it ollow that the ratio o odd to even quare-ree nuber i :1. The intereted reader ight invetigate the correponding equence o odd and even quare-ree integer power and derive iilar L reult. Appendix: Table o irt digit or quare-ree integer power Baed on the recurive relation (.3)-(.4), the coputation o I, = 4,...,10, i traightorward, at leat i a table o the Möbiu unction i available (e.g. equence A in OEIS ounded by Sloane [19]). Thee nuber are lited in Table A.1. The entry correpond to the liiting Benord law a the power goe to ininity.

9 A irt digit theore or quare-ree integer power 137 Table A.1: Firt digit ditribution o quare-ree power up to 10, = 4,..., 10 =1 / irt digit 6'083 60' '96 6'079'91 60'79' '97'14 6'079'70' '753 67' '491 6'754'775 67'547' '474' '759 67' '45 6'754'706 67'547' '474' '745 67'53 675'495 6'754'719 67'547' '474' '768 67' '458 6'754'749 67'547' '474' '743 67' '463 6'754'764 67'547' '474' '76 67' '513 6'754'770 67'547' '474' '749 67' '43 6'754'684 67'547' '474' '758 67' '486 6'754'746 67'547'4 675'474' '757 67'55 675'501 6'754'781 67'547' '474'509 = / irt digit 6'083 60' '96 6'079'91 60'79' '97'14 6'079'70'94 1 1'171 11'65 116'439 1'164'549 11'645' '456'69 1'164'566' '934 89' '61 8'936'003 89'360'03 893'603' '56 75' '361 7'533'536 75'334' '34' '640 66' '684 6'637'031 66'370'85 663'708' '997 59' '04 6'000'331 60'003' '037' '517 55' '776 5'517'885 55'179'01 551'791' '137 51' '610 5'135'979 51'359' '595' '835 48'49 48'41 4'83'838 48'37'994 48'379' '556 45'64 456'54 4'56'474 45'64' '45'56 =3 / irt digit 6'083 60' '96 6'079'91 60'79' '97'14 6'079'70'94 1 1'367 13' '857 1'368'703 13'687' '874'710 1'368'747' '615 96'05 960'08 9'601'504 96'014' '145' '647 76' '375 7'643'73 76'436' '369' '445 64'55 645'443 6'454'686 64'548'10 645'481' '63 56' '38 5'64'55 56'4' '5' '055 50' '53 5'045'43 50'454' '540' '590 45' '58 4'585'135 45'850' '505' '13 4'169 41'70 4'17'19 4'17'165 41'7' '97 39' '544 3'915'357 39'153'34 391'53'759 =4 / irt digit 6'083 60' '96 6'079'91 60'79' '97'14 6'079'70'94 1 1'477 14' '76 1'477'917 14'779'58 147'793'000 1'477'98' '90 99' '957 9'909'733 99'097'85 990'978' '677 76' '577 7'665'808 76'658' '584' '344 63'40 633'807 6'337'591 63'376'08 633'760' '449 54' '694 5'447'186 54'471' '717' '797 48' '351 4'803'64 48'03' '34' '319 43' '315 4'313'05 43'130'90 431'303' '94 39'58 39'585 3'95'70 39'57'307 39'573' '603 36' '088 3'611'08 36'109' '099'38

10 138 Werner Hürliann =5 / irt digit 6'083 60' '96 6'079'91 60'79' '97'14 6'079'70'94 1 1'541 15'46 154'569 1'545'536 15'455' '554'413 1'545'54'94 1'016 10' '847 1'008'55 10'085' '853'684 1'008'538' '664 76' '804 7'668'33 76'68' '84' '54 6'587 65'898 6'59'311 6'593'59 65'933' '35 53'71 53'603 5'35'760 53'57'49 53'573' '664 46' '699 4'656'89 46'568' '684' '153 41'50 415'167 4'151'661 41'516' '164' '75 37' '533 3'755'169 37'551' '518' '433 34' '499 3'434'913 34'349' '490'661 = / irt digit 6'083 60' '96 6'079'91 60'79' '97'14 6'079'70'94 1 1'831 18' '004 1'830'049 18'300'44 183'004'300 1'830'04'905 1'071 10' '050 1'070'510 10'705'06 107'050'653 1'070'506' '596 75' '539 7'595'36 75'953' '536' '89 58' '144 5'891'41 58'914'5 589'14' '814 48' '366 4'813'641 48'136'47 481'364' '070 40' '989 4'069'876 40'698' '987' '56 35'55 35'550 3'55'487 35'54'878 35'548' '110 31' '971 3'109'700 31'097' '970' '78 7'817 78'173 '781'71 7'817'0 78'17'183 Reerence [1] F. Auil, An algorith to generate quare-ree nuber and to copute the Möbiu unction, Journal o uber Theory 133, (013), [].H.F. Beebe, A bibliography o publication about Benord law, Heap law, and Zip law, (014), verion 1.60 at URL: tp://tp.ath.utah.edu/pub/tex/bib/benord-law.pd. [3] F. Benord, The law o anoalou nuber, Proc. Aer. Phil. Soc. 78, (1938), [4] A. Berger and T. Hill, Benord Online Bibliography, 009. URL: [5] L.V. Furlan, Da Haroniegeetz der Statitik : Eine Unteruchung über die etriche Interdependenz der ozialen Ercheinungen, Bael, Verlag ür Recht und Geellchat AG, Reviewed by H. Geiringer, J. Aer. Statit. Aoc. 43(4), (1948), [6] G.H. Hardy and E.M. Wright, An introduction to the theory o nuber, (5 th ed), Oxord Univerity Pre, 1979.

11 A irt digit theore or quare-ree integer power 139 [7] W. Hürliann, Integer power and Benord law, Int. J. Pure Appl. Math. 11(1), (004), [8] W. Hürliann, Generalizing Benord law uing power law: application to integer equence, Int. J. Math. and Math. Sci., (009), Article ID [9] G.J.O. Jaeon, The prie nuber theore, Cabridge Univerity Pre, Cabridge, 003. [10] G.J.O. Jaeon, Even and odd quare-ree nuber, Math. Gazette 94, (010), [11] C.H. Jia, The ditribution o quare-ree nuber, Sci. China Ser. A 36(), (1993), [1] L.M. Leei, B.W. Scheier and D.L. Evan, Survival ditribution atiying Benord law, The Aer. Statitician 54(3), (000), 1-6. [13] B. Luque and L. Lacaa, The irt-digit requencie o prie nuber and Rieann zeta zero, Proc. Royal Soc. A 465, (009), [14] S. ewcob, ote on the requency o ue o the dierent digit in natural nuber, Aer. J. Math. 4, (1881), [15], M.J. igrini, Digital analyi uing Benord law: tet tatitic or Auditor, Vancouver, Canada, Global Audit Publication, 000. [16] M.J. igrini, Benord Law, Application or orenic accounting, auditing, and raud detection, J. Wiley & Son, Hoboken, ew Jerey, 01. [17] F. Pappalardi, A urvey on k-reene, In: S.D. Adhikari, R. Balaubraania and K. Sriniva (Ed.), Raanujan Matheatical Society Lecture ote Serie 1, (004), [18] J. Pawlewicz, Counting quare-ree nuber, Preprint, (011), URL: arxiv: v1 [ath.t]. [19].J.A. Sloane, The On-Line Encyclopedia o Integer Sequence, (1964), URL: Received: June 1, 014