Euler Re-summation of Multiplicative Series

Size: px

Start display at page:

Download "Euler Re-summation of Multiplicative Series"

Leon Little
5 years ago
Views:

1 Euler Re-ummatio of Multiplicative Serie Lia Vepta February 209 Abtract...doe t wor. Which i a urpriig reult. The Euler traformatio of alteratig erie i ow to improve umeric covergece. Sometime. Applied to the zeta-lie erie M f () = for f a completely multiplicative arithmetic fuctio, it fail. The iteded quetio to be poed i: what clae of f reult i M f () obeyig the Riema hypothei? A umerical urvey addreig thi quetio eem traightforward, if oly the ummatio ca be re-writte to coverge quicly i the critical trip. Euler re-ummatio i a baic, imple tric for achievig thi. It wor lie a charm, for f =, ad utterly fail otherwie. f Itroductio A completely multiplicative equece i a arithmetic fuctio f taig value o the atural umber ad beig a homomorphim preervig the factorizatio of the iteger: amely, f m = f f m hold. By covetio, f : N C. Famou example iclude f = ad f = χ () the Dirichlet character. Of coure, the divior fuctio ad may other claical fuctio from umber theory are ow. The Riema hypothei famouly cocer the zeta fuctio ad aalogouly the Dirichlet erie ζ () = L χ () = χ () How far ca the hypothei be exteded? What other clae of equece obey it? A atural cojecture i that it ha omethig to do with the completely multiplicative ature of the erie. Thu, a object worth coideratio i the aalogou erie M f () = f ()

2 give a completely multiplicative arithmetic fuctio f. Several quetio ca be poed: Where are the zero of M f ()? What ort of fuctio f reult i zero o the critical lie? How geeral i the ettig for the Riema hypothei? Recall that the homomorphim f m = f f m completely determie the value of f o compoite iteger; thu, a completely multiplicative fuctio i completely pecified by it value o the et of prime P, i.e. by f : P C. It i ot further cotraied; there are ucoutably may completely multiplicative fuctio. Numerical exploratio Ca we eve get off the groud, here? Such a geeral ettig i o broad that it hard to fid a place to tart. Numerical exploratio might provide quic, eay iight. Perhap a imple place to tart would be a perturbatio of the prime f p = p( + ε) (2) for ome mall (real or complex) ε. Numerical exploratio require umerically table coverget erie. How might oe fid oe? Give ome geeric equece f p, it would eem liely that eq. (depedig o the equece) ha a pole at =. Thi obtruct aive ummatio; to get tarted, oe eed ome form of aalytic cotiuatio, or ome re-ummatio that coverge for R <. The firt obviou, imple tric i to create a coditioally coverget alteratig erie, aalogou to the Dirichlet eta. It i eay to how the idetity: M f () = 2 f 2 ( ) + f where (depedig o the equece) the um o the right might be expected to be coditioally coverget for R > 0. A writte, it i alo clear that the rate of covergece i far too low for umerical exploratio. The covergece of alteratig erie ca ofte be improved by mea of Euler ummatio. I thi particular cae, it eem promiig to write E f () = ( ) + = f =0 2 + =0 ( ) f+ (3) with the right-had ide beig tame eough for umerical exploratio. Or o oe might hope. The reult urpried me. Whe f p = p the the um coverge quicly ad eaily. Uig arbitrary-preciio umeric, aig for varyig degree of preciio, explorig the critical trip 0 < R <, there o particular problem. Whe f p = p( + ε) for ε > 0 the ummatio tart out reaoably eough, ad the become poor, vergig o o-exitet, oo offerig o advatage at all over brute-force ummatio of the alteratig erie. 2

3 Thee tatemet ca be made more precie. Coider the idividual term t = 2 + =0 ( ) f+ For ε = 0, the re-ummatio i ow to yield a globally coverget erie for the Riema zeta, a prove by Helmut Hae i 930; a moder treatmet i give by Sodow[], howig uiform covergece o compact et. Numerically, what thi mea i that, for large eough, that t (ε = 0) O ( 2 ) a a umerical obervatio (ad ot a a aalytic claim; but the proof of uiform covergece ay about a much.). Each term get maller by almot a factor of two. At that rate, it doe ot tae particularly log to coverge well. Covergece i expoetial. Thi doe ot happe for ε > 0. For the firt few term, oe doe ee a imilar behavior: t ( ε > 0) t (ε = 0) ca be ee for a hadful or few doze of term, depedig o ε. Thi oo diappear, beig replaced by t ( ε > 0) f That i, the Euler erie traformatio tric provide o acceleratio at all. That very very iteretig. Thi require ome thiig... (4) Euler Traformatio A recap of the Euler traformatio of erie i i order. Give a coverget alteratig erie, the Euler re-ummatio i give by ( ) a = 2 + =0( ) a + A imple viual derivatio proceed by re-ummig with fiite differece. Oe begi imply by rewritig: a a 2 + a 3 = a ( a a 2 + a 3 ) where a m = a m a m+ i the differece betwee ucceive term. The expreio i parethei i agai a alteratig erie, o the re-ummatio i repeated. Oe defie the fiite differece a m recurively a a m = a m a m+ termiatig the recurio by 0 a m = a m. The re-ummatio i ow 3

4 ( ) a = =0 a 2 + ad it i traight-forward to ivoe the biomial theorem to obtai the fiite differece i term of biomial coefficiet a m = =0( ) a +m A rich cla of reult ca be obtaied from algebraic re-arragemet, particularly whe the a be be iterpolated by a fuctio, viz. a = f () for ome complex-aalytic fuctio f (z) o the complex plae. I uch a cae, the re-ummtio ugget a Newto erie, which i tur li to the Newto-Melli-Poio cycle, a oted by Flajolet ad Sedgewic.[2] Thi ope the path for the applicatio of tool from aalytic combiatoric. The Newto iterpolatio formula i f (z) = =0 ( ) ( z ) a but it i well-defied oly whe the fiite differece are well-behaved. Returig to the perturbed prime equece of eq 2, the umeric evidece from eq 4 idicate that a 2 f which promptly overwhelm the biomial coefficiet. The perturbed prime equece doe ot have a Newto iterpolatio. Thi i eaily ee for z = 0, a ( ) = or geerally ( ) = + applie to z =. For poitive z, the ituatio i more ubtle. I eece, the f are extremely jagged, a a equece whe i a compoite umber, havig may factor, it become combiatorially large: f = ( + ε) Ω() where Ω() i the umber of prime factor of, with multiplicity. Whe Ω() i large, o the f i far out of lie from a placid liear progreio, ad the iterpolat i forced to iterpolate ever-wilder, piier wig. Cojecture Thi ugget ew quetio: for which completely multiplicative fuctio f doe a well-behaved Newto erie exit? If the Newto erie i well-behaved, the doe it 4

5 follow that eq 3 i uiformly coverget o compact domai? Ad fially: if the re-ummatio i uiformly coverget o compact domai, doe it the follow that the zero lie i vertical trip, if ot o vertical lie? The ue of the plural i the lat cojecture follow from claical idetitie for the Liouville lambda λ (), which i completely multiplicative, ad ha a Dirichlet erie λ () = ζ (2) ζ () o that the correpodig critical lie lie at 2 = + iτ. Similarly, for the divior fuctio σ α () geerate the Dirichlet erie σ α () = ζ ()ζ ( α) o that the zero appear o two vertical lie. Note that the divior fuctio i multiplicative, but ot completely multiplicative. The firt cojecture i too trog: the Newto erie for the Liouville lambda ad the divior fuctio are effectively udefied: the fiite differece grow without boud. The ecod cojecture i much weaer tha the firt: for the Liouville lambda, the fiite differece grow a λ () 2 ad o it doe t have a well-behaved Euler re-ummatio. But it doe mae it coceivable that a ill-behaved Newto erie might till have a reaoable Euler reummatio: the ivere-power-of-two factor i the Euler re-ummatio ca hide ome amout of mi-behavior. The cocluio eem to be that ummability doe ot have a direct ifluece o the Riema hypothei: it eem to be a ice igrediet, but ot a eceary oe. Cocluio The Riema hypothei i a tough ut to crac, ad part of that i that it i uclear where to earch. The zeta fuctio ha a umber of remarable propertie; but which of thee, or which combiatio lead to a olutio? Oe fairly evidet cojecture i that it ha omethig to do with multiplicative equece, which i what i beig explored here. Eaier aid tha doe. There are a ucoutable ifiity of completely multiplicative fuctio, eve if, a i the cae of the Liouville lambda, oe limit oeelf to oly two value. 5

6 Referece [] Joatha Sodow, Aalytic cotiuatio of Riema zeta fuctio ad value at egative iteger via Euler traformatio of erie, Proceedig of the America Mathematical Society, 20, 994, pp , URL S /S pdf. [2] Philippe Flajolet ad Robert Sedgewic, Melli Traform ad Aymptotic: Fiite Differece ad Rice Itegral, Theoretical Computer Sciece, 44(2), 995, pp. 0 24, URL ciece/article/pii/ m. 6

a 1 = 1 a a a a n n s f() s = Σ log a 1 + a a n log n sup log a n+1 + a n+2 + a n+3 log n sup () s = an /n s s = + t i

a 1 = 1 a a a a n n s f() s = Σ log a 1 + a a n log n sup log a n+1 + a n+2 + a n+3 log n sup () s = an /n s s = + t i 0 Dirichlet Serie & Logarithmic Power Serie. Defiitio & Theorem Defiitio.. (Ordiary Dirichlet Serie) Whe,a,,3, are complex umber, we call the followig Ordiary Dirichlet Serie. f() a a a a 3 3 a 4 4 Note