THE AVERAGE ORDER OF ELEMENTS IN THE MULTIPLICATIVE GROUP OF A FINITE FIELD. α(n) = 1

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1 THE AVERAGE ORDER OF ELEMENTS IN THE MULTIPLICATIVE GROUP OF A FINITE FIELD YILAN HU AND CARL POMERANCE ABSTRACT. We cosider the average multiplicative order of a ozero elemet i a fiite field ad compute the mea of this statistic for all fiite fields of a give degree over their prime fields.. INTRODUCTION For a cyclic group of order, let α) deote the average order of a elemet. For each d, there are exactlyϕd) elemets of orderdi the group where ϕ is Euler s fuctio), so α) = dϕd). It is kow vo zur Gathe, et al. [2]) that α) = 3ζ3) x+o logx) 2/3 loglogx) 4/3). x π 2 x We are iterested here i obtaiig a aalogous result where rus over the orders of the multiplicative groups of fiite fields. Let p deote a prime umber. We kow that up to isomorphism, for each positive iteger k, there is a uique fiite field of p k elemets. The multiplicative group for this field is cyclic of sizep k. We are cocered with the average order of a elemet i this cyclic group as p varies. We show the followig results. Theorem. For each positive iteger k there is a positive costat K k such that the followig holds. For each umber A > 0, each umber x 2, ad each positive iteger k with k log x)/2 log log x), we have πx) αp k ) p k d ) = K k +O A. This theorem i the case k = appears i Luca [3]. Usig Theorem ad a partial summatio argumet we are able to show the followig cosequece. Corollary 2. For all umbersa > 0,x 2, ad for ay positive itegerk logx)/2loglogx), we have ) αp k lix k+ ) x k ) = K k +O A, πx) lix) where K k is the costat from Theorem adlix) := 2 dt/logt Mathematics Subject Classificatio. B05, B75. This paper is based o the first author s 200 Dartmouth hoors thesis, writte uder the directio of the secod author. They gratefully ackowledge Floria Luca, who suggested the problem. The secod author was supported i part by NSF grat DMS-0080.

2 2 YILAN HU AND CARL POMERANCE Sicelix k+ )/lix) x k /k +) as x, Corollary 2 implies that αp k ) K k πx) k + xk, as x. We idetify the costats K k as follows. Let N k ) deote the umber of solutios to the cogrueces k mod ). Propositio 3. For each primepad positive itegerk let N k p j ) S k p) = p. 3j The S k p) < ad is a real umber with 0 < K k <. K k := p j= S k p)) 2. PRELIMINARY RESULTS I this sectio we prove Propositio 3 ad we also prove a lemma cocerig the fuctio N k ). Proof of Propositio 3. We clearly haven k ) ϕ) for every, sicen k ) couts the umber of elemets i the group Z/Z) with order dividig k ad there are ϕ) elemets i all i this group. Thus, we have S k p) j= ϕp j ) p = ) 3j p j= p p = ) p 2j p p 2 = p+. This proves the first assertio, but it is ot sufficiet for the secod assertio. For p a odd prime, the groupz/p j Z) is cyclic so that the umber of elemets i this group of order dividigk is N k p j ) = gcdk,ϕp j )). ) The same holds for p j = 2 or 4, or ifp = 2 ad k is odd. Suppose ow that p = 2,j 3, ad k is eve. SiceZ/2 j Z) is the direct product of a cyclic group of order 2 ad a cyclic group of order 2 j 2, we have N k 2 j ) = 2 gcdk,2 j 2 ) = gcd2k,ϕ2 j )). 2) Thus, we always haven k p j ) 2k, ad so 2k 2kp S k p) = p3j p 3. j= I particular, we have S k p) = O k /p 2 ), which with our first assertio implies that the product fork k coverges to a positive real umber that is less tha. This completes the proof. Lemma 4. For every positive iteger k ad each real umberx we have N k ) 2+logx) k. x

3 THE AVERAGE ORDER OF ELEMENTS IN THE MULTIPLICATIVE GROUP OF A FINITE FIELD 3 Proof. Letω) deote the umber of distict primes that divide ad letτ k ) deote the umber of ordered factorizatios ofito k positive itegral factors. Sicek ω) is the umber of ordered factorizatios ofito k pairwise coprime factors, we havek ω) τ k ) for all. Further, from ), 2) ad the fact thatn k ) is multiplicative i the variable, we haven k ) 2k ω), so that N k ) 2τ k ). Thus, it suffices to show that τ k ) +logx) k. 3) x We prove 3) by iductio ok. It holds fork = siceτ ) = for all, so that N ) = x + dt x x t = +logx. Assume ow thatk ad that 3) holds fork. Siceτ k+ ) = d τ k), x τ k+ ) = x d x τ k d) = d x d τ k d) d m x/d m τ k d) d +logx) +logx)k+, by the iductio hypothesis. This completes the proof. Corollary 5. Fork a positive iteger ady a positive real withk +logy, we have N k ) 2k +) +logy)k. 2 y >y Proof. By partial summatio, Lemma 4, ad itegratio by parts, we have N k ) N k ) +logt) k = dt 2 dt 2 >y y t 2 y< t y t 2 = 2 +logy) k +k+logy) k +kk )+logy) k 2 + +k! ) y 2k +) +logy)k, y usig k + log y. This completes the proof. Proof of Theorem. The fuctio 3. THE MAIN THEOREM αm) m = ϕ) m 2 m is multiplicative ad so by Möbius iversio, we may write αm) m = γ), m

4 4 YILAN HU AND CARL POMERANCE whereγ is a multiplicative fuctio. It is easy to compute that γp j ) = p p 2j 4) for every prime p ad positive iteger j. If rad) deotes the largest squarefree divisor of, we thus have γ) = ) ω)ϕrad)) 2 5) for each positive iteger. Note that 4), 5) are also i [3]. Forapositive ieger, label then k ) roots to the cogrueces k mod ) ass k,,s k,2,..., s k,nk ). We have αp k ) p k = = x k p k γ) = x k γ) p k N k ) γ) πx;,s k,i ), where πx; q, a) deotes the umber of primes p x with p a mod q). If q is ot too large i compariso to x ad if a is coprime to q, we expect πx;q,a) to be approximately ϕq) πx). With this thought i mid, lete q,ax) be defied by the equatio i= πx;q,a) = ϕq) πx)+e q,ax). Further, let y = x /2 /log A+4 x, where A is as i the statemet of Theorem. From the above, we thus have αp k ) p k = = y x k N k ) γ) πx;,s k,i ) i= γ)n k ) πx)+ N k ) E,ski x)+ ϕ) yγ) = T +T 2 +T 3, say. We further refie the mai term T as γ)n k ) T = πx) πx) ϕ) = >y The first sum here has a Euler product as γ)n k ) = ) γp j )N k p j ) + = ϕ) ϕp j ) = p j= p i= y< x k γ)n k ). ϕ) j= N k ) γ) πx;,s k,i ) i= ) N k p j ) = K p 3j k, where we used 4). For the secod sum i the expressio fort, we have by 5) ad Corollary 5, γ)n k ) ϕ) N k ) 2k +) +logy)k. 2 y >y >y

5 THE AVERAGE ORDER OF ELEMENTS IN THE MULTIPLICATIVE GROUP OF A FINITE FIELD 5 Here we have used k logx)/2loglogx) ad y = x /2 /log A+4 x, so that k + logy for all sufficietly large x depedig o the choice of A. Further, with these choices for k,y we have +logy) k < x /2 forxsufficietly large, so that πx) >y γ)n k ) ϕ) πx)2k +)+logy)k y πx) for all sufficietly large values ofxdepedig oa. Thus,T = K k πx)+o A πx)/). Thus, it remais to show that both T 2 ad T 3 are O A πx)/). Usig the elemetary estimateπx;q,a) +x/q, we have T 3 γ) N k ) + x ) N k ) N k ) +x, 2 y< x k y< x k y< x k by 5). We have see that the secod sum here is egligible, ad the first sum is bouded by 2+klogx) k usig Lemma 4. This last expressio is smaller tha log 2 ) k ) x x πx) = loglogx explogxlogloglogx/2loglogx)) = O A for ay fixed choice ofa. To estimatet 2, ote that T 2 y γ) N k ) max a,)= πx;,a) ϕ) πx) y max a,)= πx;,a) ϕ) πx), sice γ) ϕ)/ 2 / ad N k ) ϕ). Thus, by the Bombieri Viogradov theorem, see [, Ch. 28], we have T 2 = O A πx)/), by our choice of y. These estimates coclude our proof of Theorem. 4. PROOF OF COROLLARY 2 AND MORE ON THE CONSTANTS K k I this sectio we prove Corollary 2 ad we umerically compute a few of the costatsk k. Proof of Corollary 2. By partial summatio, we have αp k ) = αp k ) p k pk ) = x k ) αp k ) p k kt k 2 p t αp k ) p k dt. Thus, by Theorem, the prime umber theorem, ad itegratio by parts, we have ) πx)x αp k ) = x k )K k πx) kt k k K k πt)dt+o 2 ) πx)x = x k )K k lix) kt k k K k lit)dt+o 2 ) t k πx)x k = K k logt dt+o. 2

6 6 YILAN HU AND CARL POMERANCE This last itegral isk k lix k+ ) K k li2 k+ ), so the corollary ow follows via oe additioal call to the prime umber theorem. We ow examie the costatsk k fork 4. Sice N p j ) = for all p j, we have K = ) p = p ) = p 3j p 3 p j p This costat is also worked out i [3].) For K 2 we ote that N 2 p j ) = 2 for all prime powers p j except that N 2 2) = ad N 2 2 j ) = 4 forj 3. Thus, ad so p mod 3) j K 2 = 75 2 N 2 2 j ) 2 3j = = 37 2, p>2 2p ) = p 3 ForK 3, we haven 3 p j ) = 3 forp mod 3) ad forp = 3 adj 2. Otherwise,N 3 p j ) =. Thus, K 3 = 205 3p ) p ) = p 3 p 3 p 2 mod 3) For K 4, we have N 4 p j ) = 4 for p mod 4), N 4 p j ) = 2 for p 3 mod 4), N 4 2) =, N ) = 2,N ) = 4, ad N 4 2 j ) = 8 forj 4. Thus, K 4 = 299 4p ) 2p ) = p 3 p 3 p mod 4) p 3 mod 4) These calculatios were doe with the aid of Mathematica. With a little effort other costats K k may be computed, but ifk has may divisors, the the calculatio gets a bit more tedious. We close with the observatio that there is a ifiite sequece of umbersk o whichk k 0. I particular, if k = k m is the least commo multiple of all umbers up to m, the N k p) = p for every primep m+, so that K k < p N kp) p 2 ) < p m+ p ). p 2 Sice p )/p 2 = +, it follows that as m, K km 0. Usig the theorem of Mertes, we i fact havelimifk k loglogk < +. REFERENCES [] H. Daveport, Multiplicative umber theory, third editio, Spriger, New York, [2] J. vo zur Gathe, A. Kopmmacher, F. Luca, L. G. Lucht, ad I. E. Shparliski, Average order i cyclic groups, J. Théor. Nombres Bordeaux ), [3] F. Luca, Some mea values related to average multiplicative orders of elemets i fiite fields, Ramauja J ),

7 THE AVERAGE ORDER OF ELEMENTS IN THE MULTIPLICATIVE GROUP OF A FINITE FIELD 7 Yila Hu Yila.Hu.0@Alum.Dartmouth.org Carl Pomerace Departmet of Mathematics Dartmouth College Haover, NH 03755, USA carl.pomerace@dartmouth.edu