A Note on Bilharz s Example Regarding Nonexistence of Natural Density

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1 Iteratioal Mathematical Forum, Vol. 7, 0, o. 38, A Note o Bilharz s Examle Regardig Noexistece of Natural Desity Cherg-tiao Perg Deartmet of Mathematics Norfolk State Uiversity 700 Park Aveue, Norfolk, VA 3504, USA cterg@su.edu Abstract I a 937 aer See [] or Chater 0 of [3] for a exositio, Bilharz roved Arti s cojecture for rimitive roots for the fuctio field case. I the same aer, it was metioed a examle cocerig a set of rimes M x i the ratioal fuctio field F = F x, where F is the fiite field of elemets ad M x is the set of rime divisors of F such that x is a rimitive root mod. The Bilharz claimed that M x does ot have a atural desity. However the roof rovided by Bilharz was ot clear at some oits ad it cotaied a small tyo. The goal of this ote is to fix the tyo, ad idicate a elemetary roof of the above claim for the cases whe satisfies some ieuality. Mathematics Subject Classificatio: A07, N05, N56 Keywords: atural desity, Bilharz s examle, Arti s cojecture for rimitive roots Exlaatio of the Examle Accordig to the footote of Bilharz s aer [], Bilharz ackowledged that the short roof of the examle came from H. Daveort by a writte commuicatio i Setember 936. Sice we have o kowledge whether this examle was from Daveort, or whether the roof was the exact roof that Daveort gave, we simly metio it as Bilharz s examle. Cosider the ratioal fuctio field F = F x, where is a ratioal rime. Take a F for the origial Arti s roblem to be x. Defie M x to be the set of rime divisors of F such that x is a rimitive root mod. The M x does

2 878 Cherg-tiao Perg ot have a atural desity. The atural desity is defied to be the limit which of course may ot exist of the uotiet πx as goes to ifiity, where π π x deotes the umber of rime divisors of F with orm such that M x, ad π is the umber of all rime divisors of F with orm. It was show by algebra i 4. of [] that π x = ν φν, where φ is the Euler fuctio. Ad by rime umber theorem for fuctio fields, it ca be show that π = ν ν + O +ε, which ca be roved to be + o Remark. I 4.3.I of [], the above sum was estimated as + o, which is obviously false. Though this small tyo does ot affect the validity of his argumets, we will iclude some elemetary argumets below to rove the correct estimate. To rove the oexistece of atural desity, it remais to estimate π x, comared to. The mai idea of the argumet give i [, 4.3.II.a, 4.3.II.b] is to fid two subsets S ad S of atural umbers N such that, o the oe had, for S ad, oe has ν φν = o., ad o the other had, for S ad, oe has ν φν = O, but o.

3 A ote o Bilharz s examle 879 Ufortuately the argumets for both of the above two cases do t seem to be clear. For examle, i the argumet for the first set S, we kow oly that the cotributio of the leadig term φ ca be made arbitrarily small comared to, but we do t kow much about the other summads i ν φν ; i the argumet for S, it was metioed that by alyig Romaoff s Theorem see e.g. [3] for the statemet ad roof, oe could get ν φν = O, but o, which is ot very clear either. We will give a elemetary argumet to justify the oexistece of atural desity of the above examle, valid for all rimes satisfyig some ieuality. First let s correct the tyo metioed above. Correctio of a Tyo Namely the estimate i 4.3.I of [] for π should read + o, istead of of sectio. Note that if ν ν = + o, 3 the by of sectio, π = + o. 4 Therefore to rove 4, it suffices to show 3, which is euivalet to the P statemet lim A = =, where A := ν. Note that A satisfies the recursive relatio A + = +A +. Settig B = A, B satisfies i tur the recursive relatio B + = + B +. It remais to show that lim B =0, which is a coseuece of the followig Lemma. Let {B } be a seuece satisfyig B + = α B + ε with The lim B =0. α r< ad lim ε =0.

4 880 Cherg-tiao Perg Proof. Ste. B is bouded. First choose ɛ>0 such that r + ɛ. By assumtio o ε, there exists 0 such that ε <ɛif 0. Case : B 0. The B 0 + r B 0 + ε 0 r + ɛ. Iductively, oe shows that B for all 0. Therefore B is bouded. Case : B 0 >. Cosider {C } defied by C = B. The C B 0 satisfies a recursive formula of the form C + = α C + ε, where α is as above ad ε = ε. Clearly lim B 0 ε = 0 ad ε <ɛfor all 0. By Case, C is bouded, hece B is bouded as well. Ste. lim B =0. If lim B = δ>0, choose ɛ>0such that δ 0 := rδ + ɛ+ɛ<δ,which is ossible sice r<. Sice lim ε = 0, there exists 0 such that ε <ɛ, 0. Similarly, by assumtio of δ, there exists 0 such that B <δ+ ɛ. Therefore, we have B + r B + ε rδ + ɛ+ɛ = δ 0 <δ, from which iductively B δ 0, +, a cotradictio. Ste 3. lim B =0. Give ɛ>0, there exists ɛ > 0 such that rɛ+ɛ <ɛ, sice r<. By assumtio o ε, there exists 0 such that ε <ɛ for 0. Similarly, by Ste, there exists 0 such that B < ɛ. Therefore B + r B + ε <rɛ+ ɛ <ɛ,from which iductively, we obtai B <ɛ,for +. This shows that lim B =0, as reuired. Remark. I the above lemma, the coditio α r< for all ca be relaced by α r< for all N. This is used to deal with the case =. 3 Mai Result We will show that, for some secific choices of rimes s, the atural desity of M x does ot exist. We assume from ow o that is odd. 3. Basic Strategy. We use the subsets S ad S described i Sectio. We will fid below a uer boud U for the suremum of πx whe we restrict to S ad let, ad also the lower boud L of the ifimum

5 A ote o Bilharz s examle 88 of πx whe we restrict to S ad let. For the atural desity to exist, the ecessary coditio is that U L. However for some choice of s, we have U <L. This shows that at least for these s, the atural desity i uestio does t exist. 3. The Case S. Let s record the followig result metioed i the first sectio without roof. Lemma 3. Daveort There exists a subset S of N such that φ = o if S ad. Now istead of showig ν φν = o for S ad, we will show that the ratio limit for S. For S, we have P ν φν has a uer ν φν = ν φν + o sice the + o usig highest term is o ν ν + o = the estimate of π with a correctio of tyo as above ad the fact that φ ν ν is eual to we have ν, sice we assume to be odd. Sice the last exressio lim ν φν 3.3 The Case S. + o =, := U. As i the aer, let ru through the rime umbers, which are the choice for S. Without usig Romaoff s Theorem, we will show that for S ν φν = O, but o, ad i fact we will give a more recise lower estimate. I the followig lemma, for a rime umber, we defie f to be the order of i the fiite field F.

6 88 Cherg-tiao Perg Lemma 3. Let > be a rime. The there exists a C>0such that C. f= +, Remark. The first ieuality of the above lemma is clear, sice by Fermat s lemma,, so that, hece +. I fact it is a euality if we assume that >. Proof of Remark. We eed to show that if + ad, the f =. But = f = f =orf =. If f =, the = + =, a cotradictio. Claim. +, log. Here, as i what follows, we follow + the covetio of umber theory to use log as atural logarithm. Proof of Claim. It suffices to ote that each is less tha or eual to + ad there are at most log log terms. By a easy estimate, the umber of odd rime factors of m is logm. Sublemma. log x logx for 0 x. Proof. E.g. use d dx log x < 0. Proof of Lemma. Combiig the sublemma ad the claim, we have e logp +, e log log +. +, Hece we may let C = e log log. By the remark followig Lemma 3. ad the roof of Lemma 3., if >isa rime ad >, the φ = = f= = φ f= +, +,

7 A ote o Bilharz s examle 883 Hece Therefore φ ν ν φ φ e log log. φ where ɛ 0as. Put i aother way, we have φ e log log. φ e log log ɛ, lim φ ν ν φ e log log = φ := L. log Coclusio. Combiig 3. ad 3.3, we ca fid two subseueces { α }, { β } resectively, such that α ν lim φν α α α ad lim β β ν φν β β φ log. But we ca secify may s such that U = Therefore by 3., for these s, the atural desity does t exist. < φ log = L. Addedum. Cotiuig from the above settig, i fact, we ca show the followig Stregtheed Statemet. Excet ossibly for = or = 3, the atural desity i uestio does ot exist. To rove the statemet, we ote that we have to show that for every odd rime other tha 3, the coditio φ > log is satisfied. For the roof, we eed the followig Lemma 3.3 φ > e γ 3 log log+ log log for >, where γ is Euler s costat.

8 884 Cherg-tiao Perg Proof. See the referece of Theorem of []. Proof of the Stregtheed Statemet. Let g = e γ log log + 3 log log log. The as a fuctio of a real variable, g ca be show to satisfy g > 0 for all 0. Therefore by Lemma 3.3, φ > e γ log log + 3 log log > log for all rimes 3. Moreover we ca verify directly that φ > log for every rime such that 3 <<3. Coseuetly, the coditio φ > log is satisfied for every rime >3. This shows that for all rimes other tha = ad =3, the atural desity i uestio does ot exist. ACKNOWLEDGEMENTS. This article forms art of the author s thesis. The research of this articular result was doe durig the author s 5-moth stay at NCTS of Taiwa i 004. The author would like to thak his advisor, Professor C.-L. Chai for his ecouragemet, ad Professor J. Yu of NCTS for his hositality. Refereces [] E. Bach ad J. Shallit, Algorithmic Number Theory Vol I: Efficiet Algorithms, The MIT Press, Cambridge, Massachusetts, 996. [] H. Bilharz, Primdivisore mit vorgegebeer Primitivwurzel, Math. A., 4 937, [3] M. Rose, Number theory i fuctio fields, Graduate Texts i Math. 0, Sriger-Verlag, 00. Received: February, 0