BENFORD S LAW AND ARITHMETIC SEQUENCES

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1 Jourl of Mtheticl Scieces: Avces Applictios Volue 3, 05, Pges -6 NFOR S LAW AN ARITHMTIC SQUNCS ZORAN JASAK NL Tuzl osi Herzegovi e-il: zorjs@lbb js_z@bihetb Abstrct efor s lw gives expecte ptters of the igits i uericl t It c be use s tool to etect outliers, for exple, s test for the utheticity relibility of trsctio level ccoutig t se o efor s lw tests for first two igits, first three igits, lst igits, lst igit, lst two igits hve bee erive s itiol lyticl tool efor s lw is ow s first igit lw, igit lysis or efor-newcob pheoeo Leig first igits we c tret s ivisio of itervl [ ; 0) i itervls; leig first two igits we c tret s ivisio of itervl [ 0 ; 00) i 0 itervls I this text, we elborte cse whe itervls re ivie i rbitrrily chose uber > of subitervls For such cse lyticl for for expecttio vrice re evelope Specil iterest is i cse whe Itrouctio I 88, Sio Newcob i the rticle Note o the Frequecy of use of ifferet igits i turl ubers [], stte tht the first sigifict igit is ofteer th y other igit, the frequecy iiishes up to This ie ce fro Newcob s observtio of the use of librry rith tbles where he otice tht the first pges of 00 Mthetics Subject Clssifictio: 60, 6Q Keywors phrses: efor s lw, rithetic sequeces, expecttio, vrice Receive eceber 3, Scietific Avces Publishers

2 ZORAN JASAK these tbles were ctully irtier th the lst Fro this, he coclue tht people re ore liely to use ubers strtig with sller igits th with lrger oes This pheoeo ws re-iscovere by Fr Albert efor I 38, he publishe pper etitle The Lw of Aolous Nubers [] He ivestigte 0 ifferet sets of turl ubers ivolvig over 0,000 sples, fro ewspper pges to hoe resses This theory ws the teste o lrge uber of ifferet sttisticl t sets prove to hol true for ost of the, with siilr lws erive for other orer igits It ws lso observe tht the ore ixe the t, the closer the istributio of igits ws to the rithic oe This is ow cooly clle efor s lw Other theticis (Gousith, Furry, Hurwitz, Pih, Rii, especilly Hill) gve the theoreticl bsis for this lw efor s Lw Arithetic Sequeces Arithetic sequeces Forultio of efor s lw is well-ow If is ro vrible which escribes first leig igit of uber, the Here c p [ ] c P c () c p is probbility for c to be the leig igit fro the set A {,, 3, 4, 5, 6, 7, 8, } Geerlly looig, set A is rithetic sequece with strtig eber, ifferece ebers Sigifics x hvig leig igit re fro itervl [ ) we c write i geerl for ; wht The sigific (lso coefficiet or tiss) is prt of uber i scietific ottio or flotig-poit uber, cosistig of its sigifict igits epeig o the iterprettio of the expoet, the sigific y represet iteger or frctio The wor tiss sees to hve bee itrouce by Arthur urs i 46 writig for the Istitute for Avce Stuy t Priceto, lthough this use of the wor is iscourge by the I flotig-poit str coittee s well s soe professiols such s the ivetor of flotig poit ottio Willi Kh (source :

3 NFOR S LAW AN ARITHMTIC SQUNCS 3 ( ) x <,,,, which iplies oe wy to geerlize efor s lw by use of forul [ ] p P x <, () ( ) where is strtig vlue, > 0 ifferece of rithetic sequece,, I the other wors, eoitor i () is -th eber of rithetic sequece Motive for this text is to give swers o soe questio which rise: is it possible to use rithetic sequece with ore th ebers o itervl [ ; 0)? hppes if? Is efor s lw pplicble i such situtio? Wht Let > 0 > 0 Sice ( ) for is icresig sequece, we hve > ( ) ( ) ( ) > ( ), by () ootoe ecresig sequece is crete The coitio for p to represet probbilities is p I this forul, > is the uber of clsses, ie, uber of isjoit sets of cses Fro this, we hve p ( ) ( ) p ( ) 0 (3)

4 4 ZORAN JASAK y this rithetic sequece { },,, is crete, where, ( ), > For uber of clsses is equl to uber of igits we hve, ecil igits, p represets the probbility of first leig igits efie by () If 0 0, the p represets the probbility of first two leig igits etc If is the bse, the i (3), we hve ( ) (4) For uber of clsses, >, we c e our ow choice, iepeetly of bse If we use (3) i (), we hve p (5) ( ) ( ) ( ) This es tht probbility p, for strtig vlue, epes oly o, uber of isjoit clsses, which eotes iex of eber i rithetic sequece Probbility p, give by (5) is, ctully, probbility for -th eber of rithetic sequece It is esy to chec tht for, we hve ow reltios for efor s lw As it is visible, we wor with ubers hvig ( ) s strtig vlues of subitervls Sigifics for the re i itervl [ ; ) I bse, this c be escribe by Z [ ) [ ; ( ) ) ; Probbilities re efie by Z [ [ ; )] P x ( ) ( ) (6)

5 NFOR S LAW AN ARITHMTIC SQUNCS 5 Let e soe choice of, for exple, 3 I this cse, right e of itervl [ ; ) is ; leig igits of ubers i this itervl re 3, 4, 5, 6, 7, 8,,, I the other wors, we gi wor ; for first igits I cse whe 0 we wor with itervl itervl [ ) [ ; 0 ) 0 etc Accorig to this, it is possible to tl bout pretric efor s istributio ef ( ; ), which epes o two preters, s strtig vlue, uber of clsses we wt to use Possible s vlues for re, s N Here s sts for uber of leig igits we wor with so it s possible to tl bout ef ( s; ) istributio, where s 0 s it is ssue tht is or geerlly xpecttio vrice, geerl cse I this sectio, we evelop forul for clcultio of expecte sigific vrice i geerl cse expresse i ters of Cosier rithetic sequece ( ),, where is strtig vlue, is uber of subitervls, is ifferece of rithetic sequece (ot igit) We ee ext reltios ( ) 0 () Let is ro vrible eotig leig igits xpecttio is ( ) ( ( ) ) ( ) ( ) ( ) ( )

6 ZORAN JASAK 6 First eber o right sie is 0 0 Seco eber o right sie is ( ) ( ) 0 () Arguet of lst rith i this forul is ow s Pochher sybol efie by x x x x This vlue c be expresse s x x x

7 NFOR S LAW AN ARITHMTIC SQUNCS 7 Sice, 0 x x we hve, 0 (3) 0, (4) (5) This c be siplifie i ext wy (6) I soe pplictios GAMMA fuctio is ipleete for prcticl purposes I MS xcel 00, it is oe by GAMMALN fuctio by use of turl rith so we c use forul:

8 8 ZORAN JASAK 0 0 ( ) l, l 0 (7) 0 0 ( ) 0 e l (8) Foruls (5), (6), (7) llow us to clculte expecttio for rbitrry For, we hve ( ) 34404, expecte sigific for first leig igits i str forultio of efor s lw For 0 0, we hve ( ) For 0, we hve ( ) 34404, s it is expecte It is possible to tbulte those vlues for vrious Iportt questio is: wht hppes if or if we ivie itervl [ ; 0 ) i ifiite uber of itervls? Oe wy to get swer is to clculte vlue of forul (6) by use of soe big ubers For 8 exple, if we te 0 (by wors: oe hure illios) 8 we hve tht ( ) ; for 0 0 we hve ( ) etc Geerlly, liitig vlue for is ( ) This is obvious fro (5) 0 whe It is esy to geerlize foruls () to (6) for rbitrry bse, (3) ( ) ( ) ( ), (4) ( ) ( ) ( ), (6)

9 NFOR S LAW AN ARITHMTIC SQUNCS, l l (7) l e (8) Clcultio of vrice i geerl cse is ore ifficult As first step, we ee to clculte First, we hve Fro this, we hve, 3 where,,

10 ZORAN JASAK 0 3 For, we hve For, we hve 8 For 3, we hve

11 NFOR S LAW AN ARITHMTIC SQUNCS We write this su s [ ], () where, 3, 3 33

12 ZORAN JASAK First, we clculte For 3, we hve Filly, for 33, we hve, 33 (0) 33 () This forul is suitble for clcultios Fro (0), we hve Now, forul () becoes

13 NFOR S LAW AN ARITHMTIC SQUNCS 3 Accorig to this, we hve First prt of this is 8 8 P [ ] 8 8 P [ ] 8 P 8 P 8 P After this, we hve

14 4 ZORAN JASAK ( ) ( ( ) ) 8 ( 0 ) ( ) 0 Now, vrice c be clculte by reltio Vr( ) ( ) ( ( ) ) 8 Vr ( 0 ) ( ) 0 3 Nuericl exples Vlues for expecttio vrice for 0 re clculte for vrious fro to 000 by step of 0, by use of Wolfr Mthetic 700 Vlues for soe vlues of re i ext tble 0 xp Vr xp Vr xp Vr

15 NFOR S LAW AN ARITHMTIC SQUNCS 5 Vlues of expecttio vrice for for 0 0 re ephsize i this tble It is visible tht i cse, whe, liitig vlue for expecttio is 3064 for vrice is 6003; i cse 0 liitig vlue for expecttio is 3064 for vrice is Coclusio I this text, lyticl for of geerl cse for expecttio ; 0 is ivie i vrice is erive i cse whe itervl [ ) subitervls, where > is chose rbitrrily Liitig vlues for expecttio, 30640, vrice, 6003, for re clculte whe Accorig to this, we re ble to clculte other vlues of expecttio vrice This pproch c be iportt fro prcticl poit of view Coo pproch i testig efor s lw is to e test for first, ; 0 by bigger uber of seco, igits iviig itervl [ ) subitervls, it is possible to e ore sesitive pproch to possible olies, for exple I cse for, for exple, we ow tht there re soe olies betwee [ ; ), where is soe igit iviig itervl by 0, we c ivestigte olies i itervls of size 05 etc Kowig theoreticl vlues of expecttio vrice, it is possible to couct sttisticl tests bout verges vrices Clcultio of this type c be coucte for seco, thir, igits groups of igits too Refereces [] Sio Newcob, Note o the frequecy of use of ifferet igits i turl ubers, Aeric Jourl of Mthetics 4(/4) (88), 3-40 [] Fr A efor, The lw of olous ubers, Proceeigs of the Aeric Philosophicl Society 78(4) (38), 55-57

16 6 ZORAN JASAK [3] Aro erger Theoore P Hill, A bsic theory of efor s lw, Probbility Surveys 8 (0), -6; ISSN: , OI: 04/-PS75 [4] Theoore P Hill, A Sttisticl erivtio of the Sigifict-igit Lw, School of Mthetics Ceter for Applie Probbility Georgi Istitute of Techoy, Atlt, Mrch 0, 6 [5] S Gousith W Furry, Sigifict Figures of Nubers i Sttisticl Tbles, Hrwr Uiversity, August, 45 [6] Rlph A Rii, The first igit proble, The Aeric Mothly 83(7) (76), g