ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS

ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS NORBERT KAIBLINGER Abstract. Results of Lid o Lehmer s problem iclude the value of the Lehmer costat of the fiite cyclic group Z/Z, for 5 ad all odd. By complemetary observatios we determie the Lehmer costat of Z/Z, for all except for multiples of 420. 1. Itroductio Let be a positive iteger. Give a polyomial with iteger coefficiets, f Z[x], deote by m (f) its logarithmic Mahler measure over Z/Z, defied by m (f) = 1 1 log f(e 2πik/ ). k=0 By λ > 0 we deote the Lehmer costat of Z/Z, λ = mi m (f), f Z[x], m (f)>0 see [11]. We otice later that the miimum is ideed attaied, ad that it is the same if deg f 1 is assumed. Lid [11] has give a upper boud for λ, see below, ad he obtaied the values λ 1 = log 2, λ 2 = 1 2 log 3, λ 4 = 1 4 log 3, ad λ = 1 log 2 for all odd. We sharpe his result, complemet it by a lower boud, ad obtai the value of λ for all except for multiples of 420. The mai result is formulated i Sectio 2 ad it is proved i Sectio 3. 2. Mai result { } { } ρ() prime umber For a positive iteger, let ρ deote the smallest () positive iteger that does ot divide. We write p k whe p k is a pricipal divisor of, 2010 Mathematics Subject Classificatio. Primary 11R09; Secodary 11B83, 11C08, 11T22. Key words ad phrases. Lid s Lehmer costats, logarithmic Mahler measure, fiite cyclic group, cyclotomic polyomial. Supported by the Austria Sciece Fud FWF grat P 21339. 1

2 NORBERT KAIBLINGER that is, if p is a prime ad k is a positive iteger such that p k ad p k+1. Let ( ρ () = mi mi p, mi p pk) ( = mi ρ(), mi p pk). p p k p k Lid proved that λ 1 log ρ(), for all. Extedig his result we obtai the followig theorem, our mai result. Theorem 1. The Lehmer costat of Z/Z is of the form λ = 1 log Λ, with a iteger Λ 2 ot dividig ad i the rage ρ () Λ ρ (). For all = 1,..., 419 (mod 420), we have Λ = ρ () = ρ (). Example 1. Example ew values are λ 6 = 1 log 4, λ 6 8 = 1 log 3, or more 8 geerally, λ = 1 log 3 if, ad oly if, = 2k with 3 k, λ = 1 log 4 if, ad oly if, = 6k with odd k. Remark 1. (i) Theorem 1 yields the exact value of λ whe ρ () = ρ() or more geerally, whe ρ () = ρ (). Thus it also icludes certai multiples of 420. For example, let = 6 k 420 with 11 k. The ρ () = ρ() = 11 ad thus λ = 1 log 11. (ii) By Theorem 1 the kow upper boud λ 1 log ρ() is sharpeed strictly for all = 6 (mod 12), where it yields the exact value for λ, ad also for certai multiples of 420. For example, let = 11 13 420. The the theorem implies λ = 1 log Λ with Λ {8, 9, 16}, while ρ() = 17. Ope questio: Determie λ = 1 log Λ for = 420. By Theorem 1 we have Λ 420 {8, 9, 11}. We have, for f Z[x], 3. Proof of Theorem 1 (1) m (f) = 1 1 log (f) with (f) = f(e 2πik/ ). The umber (f) is always a iteger, ad there is a elemetary way to see that. To this ed we recall the determiatal relatio of [13], readily exteded here to f of arbitrary degree. If deg f 1, write f(x) = a 0 + a 1 x+ +a 1 x 1, with zero coefficiets where ecessary. If a polyomial of higher degree is give, with coefficiets a 0, a 1,..., replace it first with f as above by defiig a k = l=k (mod ) a l. Let C a deote the k=0

ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS 3 iteger circulat matrix with first row a = (a 0,..., a 1 ). The det C a = 1 k=0 f(e2πik/ ) ad it implies (2) (f) = det C a. Hece, (f) is ideed a iteger. Observe that expressig m (f) i terms of the iteger (f) justifies the defiitio of the Lehmer costat λ as a miimum, ot just a ifimum. We will also use the expressio of (f) as a resultat, for example see [2, 5, 11]. Ideed sice Res ( x 1, f(x) ) = 1 k=0 f(e2πik/ ), we have (3) (f) = Res ( x 1, f(x) ). The more commoly used expressio Res ( f(x), x 1 ), with iterchaged argumets, works as well, as log as oly absolute values are cosidered. Ideed the sig of the determiat i (2) or of the resultat i (3) is irrelevat for m (f). We remark that the opposite sig is obtaied for the polyomial f (x) = x 1 f(1/x), with coefficiet sequece ( a 1, a 2,..., a 0 ), the egative of the usual reciprocal polyomial. Remark 2. (i) Lehmer ad Pierce [10, 13] ivestigated the sequeces { 1 (f), 2 (f),... }, for f Z[x]. For example, f(x) = 2 x yields (f) = 2 1, the Mersee umbers; we refer to [6, 7, 8, 9]. For Lehmer s problem, formulated i [10], we refer to [3, 14] ad the spectacular solutio for odd coefficiets i [2]. Lid s Lehmer costats λ relate to the family { (f): f Z[x] }, for fixed. (ii) Our approach highlights ad makes use of the fact that fidig possible (or miimal) values of the logarithmic Mahler measure over Z/Z is equivalet to fidig possible (or miimal) values of iteger circular determiats, a ope problem attributed to Taussky-Todd [12]. Call f Z[x] cyclotomic if all its zeros lie o the complex uit circle. As a prelimiary observatio we determie, for all, the exact value of a cyclotomic variat of Lid s Lehmer costats. Lemma 1. For cyclotomic polyomials f Z[x], the miimal possible value of m (f) > 0 is determied by (4) mi m (f) = 1 f Z[x] cyclotomic log ρ (). m (f)>0 Proof. First, Kroecker s theorem implies that ay cyclotomic polyomial f Z[x] is the product of some of Φ 1, Φ 2,... ad a costat, if ecessary; here Φ m Z[x] deotes the m-th cyclotomic polyomial, i.e., the moic polyomial whose zeros are the primitive m-th roots of uity. Sice always (5) (f 1 f 2 ) = (f 1 ) (f 2 )

4 NORBERT KAIBLINGER ad cosequetly, m (f 1 f 2 ) m (f 1 ) + m (f 2 ), we thus obtai (6) mi m (f) = f Z[x] cyclotomic m (f)>0 mi m (Φ m ). m=1,2,... m (Φ m)>0 Let ϕ() deote Euler s totiet of. We poit out that (7) (Φ m ) = Res ( x 1, Φ m (x) ) 0 if m, 1 if at least two distict primes divide m/ gcd(m, ), p ϕ(q) if m/ gcd(m, ) is the power of a prime p = here we write gcd(m, ) = q, p ϕ(q)pk if m/ gcd(m, ) is the power of a prime p here we factorize gcd(m, ) = p k q with p k. We remark that by our approach o egative sig is eeded here, for ay m,. This formula is obtaied from [1, proof of Theorem 2], where it is used for a short proof of the formula for Res ( Φ m1 (x), Φ m2 (x) ) ; secodly, sice (8) Res ( x 1, Φ m (x) ) = Res ( Φ 1 (x ), Φ m (x) ), the formula (7) also follows from applyig [4, Propositio 14]; a third, coveiet ad direct source is [5, Theorem 3]. Notice that (7) implies for ay, m, that particularly ( (9) (Φ m ) = 0, 1, or (Φ m ) mi Sice (7) also yields (10) mi p (Φ p ) = p for p, ad (Φ p k+1) = p pk for p k, we coclude that the iequality i (9) is sharp, that is, (11) mi (Φ m ) = ρ (). m=1,2,... (Φ m) 2 p, mi p pk) = ρ (). p k Fially, sice m (Φ m ) = 1 log (Φ m ), the statemet of the lemma follows by combiig (6) ad (11). Lemma 2. Let satisfy 6 (mod 12) ad 0 (mod 420). The ρ() = ρ (), that is, the least o-divisor of is a prime (ad ot a prime power). Remark 3. The example give i Remark 1(i) shows that the implicatio of Lemma 2 caot be reversed.

ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS 5 Proof of Lemma 2. Suppose that 6 (mod 12) ad ρ () < ρ(); we verify that it implies 420. First, if 6, the either ρ () = ρ() = 2 or ρ () = ρ() = 3. This cotradicts the assumptio ρ () < ρ(). Hece, we have = 6k, for some k. The case k odd is excluded by the assumptio 6 (mod 12), so we obtai k eve. I other words, = 12k, for some k. If 5 k, the we have ρ () = ρ() = 5, i cotradictio to the assumptio ρ () < ρ(). Therefore, we have = 60k, for some k. Fially, if 7 k, the ρ () = ρ() = 7, agai i cotradictio to ρ () < ρ(). Thus we coclude that = 420k, for some k. Proof of Theorem 1. Step I: First otice that ideed λ = 1 log Λ for a iteger Λ 2; i fact, (12) Λ = mi (f). f Z[x] (f) 2 Therefore, Λ = (f 0 ), for some f 0 Z[x] with deg f 0 = 1. Upo replacig f 0 with f 0 defied above, if ecessary, we ca assume that Λ = (f 0 ). Step II: We show that Λ. Suppose that Λ divides. The there exists a prime p dividig both Λ ad. Let p m Λ ad p k. Sice Λ we otice that m k. O the other had, let C a be the iteger circulat matrix whose first row cosists of the coefficiets of f 0, so that (13) Λ = (f 0 ) = det C a. The we have p k ad p m det C a, ad a result by Newma [12, Theorem 2] thus implies that m k + 1, so we obtai a cotradictio. Step III: The previous step yields that the positive iteger Λ does ot divide. By defiitio, ρ () is the smallest umber with this property. We thus obtai the lower boud ρ () Λ. Step IV: The upper boud Λ ρ () is a cosequece of Lemma 1. Step V: Suppose that = 6 (mod 12). The 2 ad 3, while 4. Hece, ρ () = 4. O the other had, (14) mi p pk = 2 21 = 4, p k ad thus ρ () = 4; otice that ρ() 5. Therefore i Theorem 1 the lower ad upper boud coicide, ad we obtai Λ = ρ () = ρ () = 4. Step VI: Suppose that 6 (mod 12) ad 0 (mod 420). By Lemma 2 these coditios o imply that ρ() = ρ (). Sice always ρ () ρ () ρ(), we coclude that ρ () = ρ (), Thus the lower ad upper boud i Theorem 1 coicide ad we obtai Λ = ρ () = ρ ().

6 NORBERT KAIBLINGER Refereces [1] T. M. Apostol, Resultats of cyclotomic polyomials, Proc. Amer. Math. Soc. 24 (1970), 457 462. [2] P. Borwei, E. Dobrowolski, ad M. J. Mossighoff, Lehmer s problem for polyomials with odd coefficiets, A. of Math. 166 (2007), 347 366. [3] D. W. Boyd, Mahler s measure ad special values of L-fuctios, Experimet. Math. 7 (1998), 37 82. [4] C. C. Cheg, J. H. McKay, ad S. S.-S. Wag, Resultats of cyclotomic polyomials, Proc. Amer. Math. Soc. 123 (1995), 1053 1059. [5] J. E. Cremoa, Uimodular iteger circulats, Math. Comp. 77 (2008), 1639 1652. [6] M. Eisiedler, G. Everest, ad T. Ward, Primes i sequeces associated to polyomials (after Lehmer), LMS J. Comput. Math. 3 (2000), 125 139. [7] G. Everest, P. Rogers, ad T. Ward, A higher-rak Mersee problem, i: Algorithmic Number Theory, W. L. Mag, C. Fieker, ad D. R. Kohel (eds.), Spriger, Berli, 2002, 95 107. [8] A. Flatters, Primitive divisors of some Lehmer-Pierce sequeces, J. Number Theory 129 (2009), 209 219. [9] C. J. Hillar ad L. Levie, Polyomial recurreces ad cyclic resultats, Proc. Amer. Math. Soc. 135 (2007), 1607 1618. [10] D. H. Lehmer, Factorizatio of certai cyclotomic fuctios, A. of Math. 34 (1933), 461 479. [11] D. Lid, Lehmer s problem for compact abelia groups, Proc. Amer. Math. Soc. 133 (2005), 1411 1416. [12] M. Newma, O a problem suggested by Olga Taussky-Todd, Illiois J. Math. 24 (1980), 156 158. [13] T. A. Pierce, The umerical factors of the arithmetic forms i=1 (1 ± αm i ), A. of Math. 18 (1916), 53 64. [14] C. Smyth, The Mahler measure of algebraic umbers: a survey, i: Number Theory ad Polyomials, J. McKee ad C. Smyth (eds.), Cambridge Uiv. Press, 2008, 322 349. Faculty of Mathematics Uiversity of Viea Nordbergstraße 15 1090 Viea, Austria E-mail: orbert.kaibliger@uivie.ac.at