Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th k-highr Mahlr masur m k P ) of a Laurnt polynomial P as th intgral of log k P ovr th complx unit circl. In this papr w driv an xplicit formula for th valu of m k P ) /k! as k. Kywords: Mahlr masur, highr Mahlr masur. Introduction For a non-zro Laurnt polynomial P z) C[z, z ], th k-highr Mahlr masur of P is dfind [4] as m k P ) = log k P 2πit) dt. For k = this coincids with th classical log) Mahlr masur dfind as n n mp ) = log a + log max{, r j }), for P z) = a z r j ), j= j= sinc by Jnsn s formula mp ) = m P ) [3]. Though classical Mahlr masur was studid xtnsivly, highr Mahlr masur was introducd and studid vry rcntly by Kurokawa, Lalin and Ochiai [4] and Akatsuka []. It is vry difficult to valuat k-highr Mahlr masur for polynomials xcpt fw spcific xampls shown in [] and [4], but it is rlativly asy to find thir iting valus. In [5] Lalin and Sinha answrd Lhmr s qustion [3] for highr Mahlr masur by finding non-trivial lowr bounds for m k on Z[z] for k 2. Email addrss: arunabha.biswas@ttu.du Arunabha Biswas) Prprint submittd to Journal of Numbr Thory March 3, 24
In [2] it has bn shown using Akatsuka s zta function of [4] that for a =, m k z + a) /k! /π as k. In this papr w gnraliz this rsult by computing th sam it for an arbitrary Laurnt polynomial P z) C[z, z ] using a diffrnt tchniqu. Thorm.. Lt P z) C [ z, z ] b a Laurnt polynomial, possibly with rpatd roots. Lt z,..., z n b th distinct roots of P. Thn m k P ) = k k! π P z j ), z j S whr S is th complx unit circl z =, and th right-hand sid is takn as if P z j ) = for som z j S, i.., if P has a rpatd root on S. 2. Proof of th thorm W first prov svral lmmas which ssntially show that th intgrand may b linarly approximatd nar th roots of P on S. Lmma 2.. Lt P z) C [ z, z ] b a Laurnt polynomial and A [, ] b a closd st such that P 2πit) for all t A. Thn log k P 2πit ) dt = k k! A Proof. Sinc A is closd, du to th priodicity of 2πit and continuity of P 2πit ) thr xist constants b and B such that < b P 2πit ) B on A. Thn for ach positiv intgr k, log k P 2πit ) )/k! is boundd btwn log k b)/k! and log k B)/k!, and thrfor /k!) A logk P 2πit) dt is boundd btwn µa log k b)/k! and µa log k B)/k!, whr µa is th Lbsgu masur of A. Th rsult follows by ltting k tnd to infinity. Lmma 2.2. Lt P z) C [ z, z ] b a Laurnt polynomial with a root of ordr on at z = 2πit, and P z) b its drivativ with rspct to z. Thn for ach ε, ) thr xists δ > such that t t < δ implis 2π ε)t t )P 2πit) P 2πit) 2π + ε)t t )P 2πit). 2
Proof. St ft) = P 2πit). Thn f t ) = 2πiP 2πit ) and f t ) = t t ft) ft ) t t. Sinc f t ), it follows that for ach ε, ) thr xists δ > such that < t t < δ implis ε < ft) ft ) t t ) which provs th lmma sinc ft ) = P z ) =. f t ) < + ε, Lmma 2.3. Lt c, and t R. Thn for all ε >, t+ε k k! log k ct t ) dt = 2 c. t ε Proof. For k and x >, it follows from intgration by parts and induction that x log k u du = x log k x + x k j= ) j k! log k j x. k j)! Using th vn symmtry of th intgrand and substituting u = ct t ), w hav k! t+ε t ε log k ct t ) dt = 2 c k! cε log k u du, and it follows that t+ε k k! log k ct t ) dt 2 cε = log k u du t k ε c k! = 2ε log k cε k ) j log k j cε k + k! k j)! j= ) n log n cε = 2ε n! n= = 2ε log cε = 2/ c. Lmma 2.4. Lt P z) C [ z, z ] b a Laurnt polynomial with a root of ordr on at z = 2πit. Thn for all sufficintly small δ >, t+δ log k P 2πit ) dt = k k! π P 2πit ). 3
Proof. First notic that sinc z has ordr on, it cannot b a root of P z). Now lt ε, ). By Lmma 2.2 thr is a δ > such that t t < δ implis 2π ε)t t )P 2πit) P 2πit ) 2π + ε)t t )P 2πit). Stting c = 2π ε)p 2πit ) and d = 2π + ε)p 2πit ) it follows that for < t t < δ, and hnc log ct t ) log P 2πit ) log dt t ), log k ct t ) log k P 2πit ) log k dt t ), for all k N. Thrfor, t +δ log k ct t ) dt t +δ log k P 2πit ) t+δ dt log k dt t ) dt. But on t δ, t + δ), for ach fixd k, ithr all thr functions log k ct t ), log k P 2πit ) and log k dt t ) ar ngativ if k is odd), or positiv if k is vn). So th intgrals of thir absolut valus ar qual to th absolut valus of thir intgrals and thrfor w hav t +δ t +δ log k ct t ) dt log k P 2πit ) dt By Lmma 2.3 it follows that 2 c k k! t+δ t +δ log k P 2πit) 2 d. log k dt t ) dt. Sinc c = 2π ε)p 2πit ) and d = 2π+ε)P 2πit ) and ε > is arbitrary, w ar don. With ths lmmas, w now procd to prov th main thorm. Proof of Thorm.. First notic that m k P ) k! = k! log k P 2πit) dt. 4
If P z) dos not hav any roots on S thn choosing A = [, ] and applying Lmma 2. w s that m k P ) /k! as k and th thorm holds in this cas. Now lt t,..., t m [, ] such that 2πit,..., 2πitm of P on S. ar th distinct roots Lt δ > b sufficintly small so that P 2πitj ) < on ach intrval t j δ, t j + δ), j =,..., m, and ths intrvals ar disjoint and dfin A = [, ] m t j δ, t j + δ). j= Using Lmma 2., and th fact that log P 2πit ) < on [, ] \ A, w find that m k P ) = log k P 2πit ) dt + log k P 2πit ) dt k k! k k! A [,]\A m tj+δ = log k P 2πit ) dt 2.5) k k! j= t j δ If P has no rpatd roots on S, thn by Lmma 2.4, this final sum is qual π m j= P 2πitj ), and so th thorm is provn in this cas. Finally, if P has a rpatd root on S, w may assum without loss of gnrality that P z ) = P z ) = whr z = 2πit. With ft) = P 2πit ), w hav that ft ) = f t ) =. Thn for ach ε, ) thr is a δ ε, ) such that ft) t t = ft) ft ) t t ε, for all < t t < δ ε. It follows that log ft) log εt t ) < for all < t t < δ ε, and so log k ft) log k εt t ), for all < t t < δ ε. W may assum that δ ε < δ, and using 2.5) and Lmma 2.3 dduc that m k P ) t+δ log k P 2πit ) dt k k! k t δ t+δ = log k P 2πit ) dt k t δ t+δ ε log k εt t ) dt k = 2 ε. 5 t δ ε
Sinc ε, ) was arbitrary, th it in qustion divrgs to and th thorm is provn. [] H. Akatsuka, Zta Mahlr masurs, J. Numbr Thory 29 ) 29) 273 2734. [2] A. Biswas, Asymptotic natur of highr Mahlr masur, 23) Submittd). [3] G. Evrst, T. Ward, Hights of polynomials and ntropy in algbraic dynamics, Springr 29). [4] N. Kurokawa, M. Lalín, H. Ochiai, Highr Mahlr masurs and zta functions, Acta Arith. 35 3) 28) 269 297. [5] M. Lalín, K. Sinha, Highr Mahlr masur for cyclotomic polynomials and Lhmr s qustion, Ramanujan J. 26 2) 2) 257 294. 6