Using Linnik's Identity to Approximate the Prime Counting Function with the Logarithmic Integral

Transcription

1 Usig Lii's Ideiy o Approimae he Prime Couig Fucio wih he Logarihmic Iegral Naha McKezie /26/2 aha@icecreambreafas.com Summary:This paper will show ha summig Lii's ideiy from 2 o ad arragig erms i a cerai way will immediaely sugges a mehod for approimaig he prime power couig fucio. I will he prove his approimaio is he Logarihmic Iegral. The relaioship bewee he wo will he be show o provide oe way of reasoig abou he differece bewee he Logarihmic Iegral ad he prime power couig fucio. The paper will he use a similar approach o approimae Meres fucio. This approimaio will lead o a furher approimaio for Chebyshev's fucio ψ(), he domia erm for which is simply. The paper will coclude by showig how his approimaio leads o oe way of reasoig abou he differece bewee ad ψ(). Noe: This paper will use variables,y,z for sums raher ha he usual i,j,. This is doe o sress visually he symmeries bewee sums ad heir iegral approimaios, ad he role chagig bewee he wo plays as he sole source of error erms.. Lii's Ideiy We begi wih Lii's ideiy which says ha log Λ() log = 2 + d = ' () (.) where Λ () is he Vo Magold fucio, ad d ' () is he sric umber of divisors fucio such ha d ' ()= { := : d ' ()= d ' ( j)d ' ( j j ) (.2) Combiaorially, d ' () is he cou of soluios o he epressio a a 2... a =, where a, a 2,..., a 2 ad are iegers, ad order maers. So, as a eample d 2 ' (6)=2 because i has soluios of boh 2 3 ad 3 2. Now, Λ() log is a fucio equal o a primes, a primes 2 squared, a primes cubed, ad so o, ad oherwise. 3 So, if we sum Lii's ideiy from 2 o some value, we have π()+ 2 π( 2 )+ 3 π ( 3 )= j=2 = where π () is he prime couig fucio. + Combiaorially, he sum d ' ( j) is he cou of j=2 d ' ( j) soluios o he epressio a a 2... a, where a, a 2,...,a 2 ad are all aural umbers, ad where order maers, so we ca rewrie hose sums as j= j= j= d ' ( j)= d 2 ' ( j)= d 3 ' ( j)= z=2 y (.3) ad so o. We ca use his o rewrie our prime couig ideiy as π()+ 2 π( 2 )+ 3 π ( 3 )= The saeme z=2 2. Approimaig he Prime Couig Fucio z =2 y 4... y 4... (.4)

2 immediaely suggess a mehod for approimaig he prime couig fucio. Our approimaio A Π () will replace he sums i our origial fucio wih he followig closely relaed iegrals A Π ()= d 2 dy d+ 3 y dz dyd 4... We would lie o see if we ca rasform our approimaio A Π io a more recogizable fucio. Tha process will occupy he res of his secio. If we ame each of hese esed iegrals as A (), so A Π ()= + A = () ad evaluae io closed form, we have A 2 ()= A 3 ()= A ()= d= dy d= d=log + y dz dy d= dy d= y log +d= 2 log2 log + ad, more geerally, (2.) A = ( + ( log ) j ) j= j! (2.2) If we specify ha a ew value m is equal o log, he we ca rewrie his as A ()= ( +e m m j j= j! ) (2.3) We ur ow o wo represeaios of he icomplee gamma fucio, Γ(,u)= e u Γ(, u)=( )!e u u j j= j! Seig boh represeaios equal o each oher ad rearragig erms, we arrive a he followig ideiy e u u j j= j! = e d ( )! u which les us rewrie (2.3) as A ()= ( + e d) ( )! Le's rewrie our leadig erm log A ()= ( ( )! ( )! + ( )! log e d) which we ca rewrie as he gamma fucio Γ() lie so A ()= ( Γ() ( )! + ( )! log The gamma fucio has he followig iegral represeaio Γ( )= e d e d) which we ca subsiue io our mai epressio A ()= ( ( )! e d+ e d) ( )! log This simplifies o give us oe fial represeaio for A (), + A ()= ( )! e d log Now le's iser his bac io (2.) A Π ()= = This simplifies o ( )! log + + A Π ()= =! log e d e d Swappig summaio ad iegraio, we have which becomes A Π ()= A()= log log log ( e e ( =! )e d )e d= d= log log e d The epoeial iegral Ei() is give by e d= (2.4)

3 Ei( )= e d+log +γ where γ= , he Euler-Mascheroi Cosa,so log A Π ()= e d=ei(log ) log log γ The logarihmic iegral is li()=ei(log ), so his meas, fially, ha A Π ()= d 2 dy d+ 3 y dz dy d 4... =li() log log γ (2.5) (As a somewha ieresig aside, i is easier o evaluae he looser fiig d 2 which is jus log.) dy d+ 3 y dz dy d Error i his Approimaio So his provides oe way of hiig abou he coecio bewee he logarihmic iegral ad he prime couig fucio. Lii's ideiy le's us say ha he prime couig fucio ca be epressed lie so, π ()+ 2 π ( 2 )+ 3 π( 3 )= z=2 ad his ideiy ca be approimaed as li() log log γ= d 2 dy d+ 3 y 4... y dz dy d 4... From his perspecive, he error erm i he Prime Number Theory ca be reasoed abou as he volumes uder hese various hyperbolic curves ha are' coaied wihi he ieger laice volumes compleely uder each curve. Which is o say li() π () 2 π ( 2 ) 3 π( 3 )...= log log γ 2 ( + 3 ( y dy d ) dz dy d y ) z= (3.) I migh be ieresig o coec his saeme o Riema's eplici prime couig formula π ()+ 2 π( 2 )+ 3 π ( 3 )= li() li( ) log 2+ d ( 2 )log where are he zeroes of he Riema Zea fucio. We ca obviously resae his as li() π() 2 π( 2 ) 3 π( 3 )...= li( )+log 2 d ( 2 )log If we defie he followig acillary fucio E ()=log 2 d +log log +γ ( 2 )log he, comparig his paper's resuls o Riema's fucio, we ca say ha li( )+E ()= 2 ( + 3 ( 4 ( y y yz dy d ) dz dy d y ) z=2 y yz ) z=2 w=2 dw dz dy d (3.2) Give ha E() is O(log log ), i ca be largely igored, leavig us wih he observaio, lef wihou ay furher comme, ha he Zea Zeroes perfecly describe hese o-laice differeces. Wih a bi of wor, his equaio ca be furher

4 rasformed io li( )+E ()= 2 ( + 3 ( 4 ( y y yz dy d + { }) dz dy d dw dz dy d y + + y z=2 z=2 { y }) yz y { yz }) (3.3) where {} is he fracioal par fucio. This represeaio is oeworhy i ha all of he familiar high frequecy / discoiuous aspecs of he prime couig fucio seem o be capured by he fracioal par sums; he oher esed sums represe fairly smooh, predicable values. 4. Meres Fucio There is he followig ideiy for he Möbius fucio i erms of he sric cou of divisor fucios from (.2), derived i a fashio similar o Lii's ideiy μ()= d ' () = If we sum his ideiy from 2 o, we have M ()= d ' ( j) = j=2 where M () is he Meres fucio. Followig similar echiques from our firs secio ha led from (.3) o (.4), his meas we ca wrie Meres fucio as M ()= + z =2 y (4.) This oce agai immediaely suggess a approimaio for Meres fucio of he form ()= d+ dy d y dz dyd (4.2) These esed iegrals are he same iegrals of he form A () ha we saw before i (2.2); i fac, ()=+ A () = Subsiuig i our fial iegral represeaio for A () from (2.4), we have + ()=+ = ( )! e d log which simplifies o ()= = ( )! e d log Rearragig summaio ad iegraio, we have which leads o So, ()= ()= log log = log ( = ( )! )e d e e d = d= log log d+ dy d y dz dy d (4.3) Thus, he differece bewee Meres fucio ad our approimaio ca be epressed as M () log += ( +( ( y y yz dy d ) dz dy d y ) z=2 dw dz dyd y yz ) z=2 w=2 (4.4) which, give he smalless ad smoohess of log, meas ha he o-laice volumes are largely jus he iverse of Meres fucio iself. As wih he prime couig fucio, his ca be furher developed io

5 M () log+= y y yz ( +( ( dy d + { }) dz dy d dw dz dy d y + + y z=2 z=2 { y }) yz y { yz }) where all of he discoiuiies ad high frequecy chage are coaied i he fracioal par sums. 5. Chebyshev's Psi Fucio Meres gave he followig formula for Chebyshev's fucio ψ( ) ψ( )= log j M ( j=2 j ) (5.) Now, we ow from he previous secio ha we ca wrie Mere's fucio as M ()= so we ca rewrie ψ() as + z =2 y y w=2 z=2 y z y y + y z ψ( )= log ( ) z =2 z=2 w=2 (5.2) This epressio for ψ() immediaely suggess is ow approimaio, amely A ψ ()= log ( dy+ y dz dy y y z dw dz dy)d The ier sum of iegrals is our approimaio for () (4.2) wih aig he value (5.3). We ow from which we ca rewrie as So A ψ ()= A ψ ()= log ( log +log )d log d log log d+ log log d A ψ ()=(log +) ( log 2 log +log ) +( log 2 2 log +2 2) log = log ( dy+ A ψ ()= log y dz dy y y z dwdz dy)d (5.4) For wha follows, i is useful o muliply he log value hrough for boh our epressio for ψ() ad is approimaio. So, we have ψ( )=( log ) ( log y) ad +( z=2 log =( +( y y log z) ( log d) ( log z dz dy d) ( z=2 y y w=2 log y dy d) y z y z log w) (5.5) log w dw dz dy d) (5.6) Our real ieres here is relaig ψ() o. Thus, if we subrac our represeaio for ψ() from is approimaio, we have (4.3) ha his approimaio is jus log replace i wih, so we ca A ψ ()= log ( log )d

6 ψ()= log +( ( +( ( log d log ) y y y z log y dy d log y ) log z dz dy d z=2 log w dwdz dy d y log z) y z=2 w=2 y z log w) (5.7) The fucio ψ() ca be wrie i erms of he zeroes of he Riema Zea fucio as ψ()= Obviously, we ca rewrie his as ψ()= ' () ζ ζ() 2 log ( 2 ) ' () +ζ ζ() + 2 log ( 2 ) If we compare his resul o (5.7), ad we defie he followig O(log ) fucio he ha meas E ()=log 2π+ 2 log( 2 )+log + ( ( +( ( +E ()= log d log ) y y y z log y dy d log z dz dy d log y) z =2 log w dw dz dy d y log z) y z=2 w=2 6. Odds ad Eds y z log w) (5.8) Alhough his paper wo' cover he opic, a wide rage of oeworhy arihmeic summaory fucios ca be epressed i a fashio similar o (5.) ad (5.2). Cosequely, he approach ae i secio 5 ca also be applied o hem. Special has o J. Magalda from mah.sacechage.com for providig he eplici mehod for rasformig (2.2) io (2.5); I ew (2.2) seemed o be he Logarihmic Iegral, bu I was' quie sure why. Special has o Professor Mar Coffey, who provided helpful feedbac ad a few oes abou (3.3) a hp:// %2McKezie%2sums.pdf, which was a respose o a previous hreadbare wrie up of mie a hp:// mah.old.pdf