Negative values of truncations to L(1, χ)

Transcription

1 Clay Mathematics Proceeigs Volume 7, 2006 Negative values of trucatios to L, χ Arew Graville a K Souararaja Abstract For fie large we give uer a lower bous for the miimum of χ/ as we miimize over all real-value Dirichlet characters χ This follows as a cosequece of bous for f/ but ow miimizig over all comletely multilicative, real-value fuctios f for which f for all itegers Eaig our set to all multilicative, realvalue multilicative fuctios of absolute value, the miimum equals o, a i this case we ca classify the set of otimal fuctios Itrouctio Dirichlet s celebrate class umber formula establishe that L, χ is ositive for rimitive, quaratic Dirichlet characters χ Oe might attemt to rove this ositivity by tryig to establish that the artial sums χ/ are all o-egative However, such trucate sums ca get egative, a feature which we will elore i this ote By quaratic recirocity we may fi a arithmetic rogressio mo 4 such that ay rime q lyig i this rogressio satisfies q = for each Such rimes q eist by Dirichlet s theorem o rimes i arithmetic rogressios, a for such q we have q / = λ/ where λ = Ω is the Liouville fuctio Turá [6] suggeste that λ/ may be always ositive, otig that this woul imly the truth of the Riema Hyothesis a reviously Pólya ha cojecture that the relate λ is o-ositive for all 2, which also imlies the Riema Hyothesis I [Has58] Haselgrove showe that both the Turá a Pólya cojectures are false i fact = 72, 85, 376, 95, 205 is the smallest iteger for which λ/ < 0, as was recetly etermie i [BFM] We therefore kow that trucatios to L, χ may get egative Let F eote the set of all comletely multilicative fuctios f with f for all ositive itegers, let F be those for which each f = ±, a F 0 be those for which each f = 0 or ± Give ay a ay f F 0 we may fi a rimitive quaratic character χ with χ = f for all agai, by usig Le remier auteur est artiellemet souteu ar ue bourse u Coseil e recherches e scieces aturelles et e géie u Caaa The seco author is artially suorte by the Natioal Sciece Fouatio a the America Istitute of Mathematics AIM c 2006 Clay Mathematics Istitute

2 2 ANDREW GRANVILLE AND K SOUNDARARAJAN quaratic recirocity a Dirichlet s theorem o rimes i arithmetic rogressios so that, for ay, χ = δ f 0 := mi f F 0 mi χ a quaratic character Moreover, sice F F 0 F we have that f δ := mi δ 0 f F f δ := mi f F We eect that δ δ a eve, erhas, that δ = δ for sufficietly large Trivially δ / = log +γ+o/ Less trivially δ, as may be show by cosierig the o-egative multilicative fuctio g = f a otig that 0 g = [ ] f f + We will show that δ δ < 0 for all large values of, a that δ 0 as Theorem For all large a all f F we have f log log 3 5 Further, there eists a costat c > 0 such that for all large there eists a fuctio f= f F such that f c log I other wors, for all large, δ δ log log 3 0 δ c 5 log Note that Theorem imlies that there eists some absolute costat c 0 > 0 such that f/ c 0 for all a all f F, a that equality occurs oly for boue It woul be iterestig to etermie c 0 a all a f attaiig this value, which is a feasible goal eveloig the methos of this article It woul be iterestig to etermie more recisely the asymtotic ature of δ, δ 0 a δ, a to uersta the ature of the otimal fuctios Istea of comletely multilicative fuctios we may cosier the larger class F of multilicative fuctios, a aalogously efie δ f := mi f F Theorem 2 We have δ = 2 log + e + 4 e log t t + t log 2 + o = o

3 NEGATIVE VALUES 3 If f F a is large the uless Fially if a oly if k= + f 2 k 2 k log + k= f, log log f 2 k 2 k log 20 f 3 /+ e k= = δ + o f k k + /+ e 2 Costructig egative values Recall Haselgrove s result [Has58]: there eists a iteger N such that N λ = δ + f = o with δ > 0, where λ F with λ = for all rimes Let > N 2 be large a cosier the fuctio f = f F efie by f = if /N + < /N a f = for all other If the we see that f = λ uless = l for a uique rime /N +, /N] i which case f = λl = λ + 2λl Thus 2 f = = λ + 2 /N+< /N λ 2δ /N+< /N l / λl l A staar argumet, as i the roof of the rime umber theorem, shows that λ = 2πi 2+i 2 i ζ2s + 2 s ζs + s s e c log, for some c > 0 Further the rime umber theorem reaily gives that log log/n log/n + N log /N+< /N Isertig these estimates i 2 we obtai that δ c/ log for large here c δ/n, as claime i Theorem Remark 2 I [BFM] it is show that oe ca take δ = for N = a therefore we may take c

4 4 ANDREW GRANVILLE AND K SOUNDARARAJAN 3 The lower bou for δ Proositio 3 Let f be a comletely multilicative fuctio with f for all, a set g = f so that g is a o-egative multilicative fuctio The f = g + γ f + O log 5 Proof Defie F t = t t f We will make use of the fact that F t varies slowly with t From [GS03, Corollary 3],we fi that if w /0 the 3 F F /w We may easily euce that 32 F F /w log 2w 2 π log log + log log 2w log 2w 2 π log log log log 2w log log log 2 3 log log log 2w + log 2 3 log Iee, if F a F /w are of the same sig the 32 follows at oce from 3 If F a F /w are of oosite sigs the we may fi v w with /v f a the usig 3 first with F a F /v, a seco with F /v a F /w we obtai 32 We ow tur to the roof of the Proositio We start with 33 g = [ ] f = f { } f 4 Now { } f = j = j log /j+< /j /j /j+ t 2 f j /j+< t ft + O log From 32 we see that if j log, a /j + < t /j the /j+< t Usig this above we coclue that 34 { } f = f f = t logj + f + O j + jlog 4 j log log j + j + O j + log log 2 log 4 Sice j J log+/j /j+ = logj + j J+ /j+ = γ+o/j, whe we isert 34 ito 33 we obtai the Proositio

5 NEGATIVE VALUES 5 Set u = f/ By Theorem 2 of A Hilebra [Hil87] with f there beig our fuctio g, K = 2, K 2 =, a z = 2 we obtai that g + g +g Oe log β, σ e ma0, g where β is some ositive costat a σ ξ = ξρξ with ρ beig the Dickma fuctio Sice ma0, g f/2 we euce that g e u log e u/2 ρe u/2 + Oe log β 35 e ueu/2 log + Oe log β, sice ρξ = ξ ξ+oξ O the other ha, a secial case of the mai result i [HT9] imlies that 36 f e κu, where κ = Combiig Proositio 3 with 35 a 36 we immeiately get that δ c/log log ξ for ay ξ < 2κ This comletes the roof of Theorem Remark 32 The bou 35 is attaie oly i certai very secial cases, that is whe there are very few rimes > e u for which f = + o I this case oe ca get a far stroger bou tha 36 Sice the first art of Theorem ees o a iteractio betwee these two bous, this suggests that oe might be able to imrove Theorem sigificatly by etermiig how 35 a 36 ee uo oe aother 4 Proof of Theorem 2 Give f F we associate a comletely multilicative fuctio f F by settig f = f We write f = hf/ where h is the multilicative fuctio give by h k = f k ff k for k Now, 4 f = h m / = h log 6 fm m m / fm m + O log Sice h = 0 a h k 2 for k 2 we see that h 42 log 2 h log 2 2 >log 6 3 h >log 6 The Dickma fuctio is efie as ρu = for u, a ρu = /u u u ρtt for u

6 6 ANDREW GRANVILLE AND K SOUNDARARAJAN Further, for log 6, we have writig F t = t f = F F / + / / F t t = log t usig 32 Usig the above i 4 we euce that f = f h log 6 t f as i sectio 3 h log f log 6 f + O log 5 +O log 5 Arguig as i 42 we may ete the sums over above to all, icurrig a egligible error Thus we coclue that with f H 0 = f = H 0 + H = h, a H = f + O = log 5 h log Note that H 0 = + h/ + h2 / 2 + 0, a that H 0, H We ow use Proositio 3, keeig the otatio there We euce that 43 f = H 0 g + γh 0 + H, f + O log 5 If H 0 log 20 the we may argue as i sectio 3, usig 35 a 36 I that case, we see that f / /log log 3 5 Heceforth we suose that H 0 log 20 Sice H 0 + h2 2 we euce that ote h2 = 0 44 k=2 + h f 2 + f , 2 + h2 k 2 k k= This roves the mile assertio of Theorem 2 Writig = 2 k l with l o, H = l o hl l = log 2 = 3 log 2 3 k= k=0 + f 2 k 2 k log 20 h2 k 2 k k log 2 + log l kh2 k hl 2 k l l o + h + h Olog 20 log log + O, log 20,

7 NEGATIVE VALUES 7 where we have use 44 a that k= kh2k /2 k = 3 + Olog log /log Usig these observatios i 43 we obtai that 45 f = H 0 g + 3 log 2 + h + h log h + h2 2 + f + o 20 f + o Let r be the comletely multilicative fuctio with r = for log, a r = f otherwise The Proositio 44 of [GS0] shows that f = f r + O log 20 log Sice f2 = + OH 0 we euce from 45 a the above that 46 f log f + f r + o Oe of the mai results of [GS0] see Corollary there shows that 47 r 2 log+ e log t e+4 t+o = o, t + a that equality here hols if a oly if 48 /+ e r + /+ e + r = o Sice the rouct i 46 lies betwee 0 a we coclue that 49 f 2 log + e log t e + 4 t + t log 2 + o, a for equality to be ossible here we must have 48, a i aitio that the rouct i 46 is + o These coitios may be writte as 3 /+ e k= f k k + /+ e f = o If the above coitio hols the, by 35, g log a so for equality to hol i 45 we must have H 0 = o/ log Thus equality i 49 is oly ossible if k= + f 2 k 2 k log + 3 /+ e k= f k k + /+ e f = o Coversely, if the above is true the equality hols i 45, 46, a 47 givig equality i 49 This roves Theorem 2

8 8 ANDREW GRANVILLE AND K SOUNDARARAJAN Refereces [BFM] P Borwei, R Ferguso & M Mossighoff Sig chages i sums of the liouville fuctio, rerit [GS0] A Graville & K Souararaja The sectrum of multilicative fuctios, A of Math , o 2, [GS03], Decay of mea values of multilicative fuctios, Caa J Math , o 6, [Has58] C B Haselgrove A isroof of a cojecture of Pólya, Mathematika 5 958, 4 45 [Hil87] A Hilebra Quatitative mea value theorems for oegative multilicative fuctios II, Acta Arith , o 3, [HT9] R R Hall & G Teebaum Effective mea value estimates for comle multilicative fuctios, Math Proc Cambrige Philos Soc 0 99, o 2, Déartmet e Mathématiques et Statistique, Uiversité e Motréal, CP 628 succ Cetre-Ville, Motréal, QC H3C 3J7, Caaa aress: arew@msumotrealca Deartmet of Mathematics, Stafor Uiversity, Blg 380, 450 Serra Mall, Stafor, CA , USA aress: ksou@mathstaforeu