GEOMETRIC PROOFS OF POLYNOMIAL RECIPROCITY LAWS OF CARLITZ, BERNDT, AND DIETER

Transcription

1 GEOMETRIC PROOFS OF POLYNOMIAL RECIPROCITY LAWS OF CARLITZ, BERNDT, AND DIETER MATTHIAS BECK Dedicted to Iet Shiow on the occsion of his 65th irthdy nd retirement Astrct We study higher-dimensionl nlogs of the Crlitz polynomils 1 X c (u, v;, ) := u 1 v, where u nd v re indeterminnts, nd re positive integers, nd x denotes the gretest integer x These polynomils stisfy reciprocity lw, from which one esily concludes mny clssicl reciprocity theorems for the Dedeind sum nd its generliztions, most notly y Hrdy nd Berndt Dieter We give new proofs of some generl reciprocity theorems of Berndt nd Dieter, using lttice points in polyhedrl regions 1 Introduction Our gol is to study higher-dimensionl nlogs of the polynomils 1 (1) c (u, v;, ) := u 1 v, where u nd v re indeterminnts, nd re positive integers, nd x denotes the gretest integer x Polynomils of the form of c (u, v;, ) were introduced y Crlitz [5], who proved the following reciprocity lw: Theorem 11 (Crlitz) If nd re reltively prime positive integers, then (u 1) c(u, v;, ) + (v 1) c(v, u;, ) = u 1 v 1 1 (Crlitz s reciprocity theorem stted in [5] is of slightly different form, which is esily seen to e equivlent to Theorem 11, which in this form ws first pulished y Berndt nd Dieter [3]) Theorem 11 implies the clssicl reciprocity lw for the Dedeind sum [6, 11] 1 s (, ) := (( )) (( )), where ((x)) := x x 1 2 if x / Z, 0 if x Z Dte: 12 July Mthemtics Suject Clssifiction 11F20 Key words nd phrses Dedeind sum, Hrdy sum, Berndt sum, Crlitz polynomil, reciprocity lw 11

2 12 MATTHIAS BECK By pplying the opertors u u nd v v to the identity of Theorem 11 nd susequently setting u = v = 1, one otins (fter some esy lgeric mnipultion) the reciprocity lw (2) s (, ) + s (, ) = ( ), rguly the most importnt property of s (, ) This reciprocity lw is the essentil ingredient to the trnsformtion formul for the Dedeind η-function [6], nd (2) implies Guss s theorem of qudrtic reciprocity [11] Together with the modulr property s (, ) = s ( mod, ), Dedeind s reciprocity lw (2) lso implies tht s (, ) cn e efficiently computed y mens of Eucliden-type lgorithm Other Dedeind-lie sums, due to Hrdy [8] nd Berndt [2] nd ppering in trnsformtion formuls for thet functions, re 1 hj (( )) S(h, ) = ( 1) j+1+ s 1 (h, ) = ( 1) hj j s 2 (h, ) = (( )) (( )) j hj ( 1) j s 3 (h, ) = 1 s 4 (h, ) = ( 1) hj s 5 (h, ) = ( 1) j (( hj ( 1) hj j+ )) (( )) j These Hrdy Berndt sums lso stisfy certin reciprocity reltions, originlly proved y Berndt [2] (Hrdy stted in [8] some of the reciprocity lws without proof) Apostol Vu [1] nd Berndt Dieter [3] showed how one cn use Crlitz s Theorem 11 to give simple proofs of the reciprocity reltions of the Hrdy Berndt sums Our purpose in this pper is to study the following higher-dimensionl generliztion of the polynomils of the form (1) Definition The Crlitz polynomil c (u 1, u 2,, u n ; 1, 2,, n ), where u 1, u 2,, u n re indeterminnts nd 1, 2,, n re positive integers, is defined s the polynomil c (u 1, u 2,, u n ; 1, 2,, n ) := 1 1 j j 2 3 u 1 1 u 1 2 u 1 3 u j n 1 n The min contriution of this note is geometric proof of the following reciprocity lw, sed on doule lttice-point enumertion in certin polytopes (Section 2) Theorem 12 (Berndt Dieter) If 1, 2,, n re pirwise reltively prime positive integers, then (u 1 1) c (u 1, u 2,, u n ; 1, 2,, n ) + (u 2 1) c (u 2, u 3,, u n, u 1 ; 2, 3, n, 1 ) + + (u n 1) c (u n, u 1,, u ; n, 1,, ) = u u u n 1 Crlitz s Theorem 11 is the cse n = 2 of this identity The generl cse of Theorem 12 ws given y Berndt nd Dieter [3, Theorem 51] In fct, they gve slightly more generl polynomil reciprocity lw, which we could lso prove with our geometric methods; however, for the se of simplicity of exposition, we chose to concentrte on Theorem 12 Pettet nd Ro [9] used the

3 GEOMETRIC PROOFS OF POLYNOMIAL RECIPROCITY LAWS 13 cse n = 3 of Theorem 12 to give simple proofs of three-term reciprocity lws of Golderg for the Hrdy Berndt sums [7] nd of Rdemcher for the Dedeind sum [10] Just lie Theorem 11 implies reciprocity lws for the Dedeind nd Hrdy Berndt sums, Theorem 12 hs numerous corollries, some of which we descrie in Section 3 2 Proof of the Reciprocity Lw for Crlitz Polynomils For ese of exposition, we first outline our pproch to proving Theorem 12 for the two-term cse, tht is, Theorem 11 After proving this cse in detil, we will discuss how to show the generl cse Proof of Theorem 11 We rewrite the sum on the left-hnd side of the identity s 1 (u 1) c(u, v;, ) + (v 1) c(v, u;, ) = u v 1 + v j u 1 j u 1 v 1 v j 1 u j nd pir up the monomils with positive nd with negtive sign More precisely, we collect the exponents (x, y) of the monomils u x v y in the four sets A + := (, ) : 1 1, ( ) B + := j, j : 1 j 1, A := ( 1, ) : 1 1, ( ) B := j, j 1 : 1 j 1 We will show tht A + B + nd A B re disjoint unions tht differ only in the elements (0, 0) nd ( 1, 1), which give rise the right-hnd side of the identity in Theorem 11 There is simple geometric picture underlying our proof, indicted in Figure 1 Nmely, we consider ll Z 2 -lttice points in the hlf-open polyhedron P := (x, y) R 2 : 0 x 1 0 y 1, x > y 1 y > x 1 which is setched in Figure 1 The sets A ±, B ± simply contin lttice points in certin prts of this polyhedron, nd the Crlitz polynomils encode these lttice points, ech point (, j) s the monomil u v j From Figure 1 one cn red off the disjoint unions A + B + = ( P Z 2) \ (0, 0) nd A B = ( P Z 2) \ ( 1, 1), which is our clim nd gives Theorem 11 Now for the non-picturesque proof We strt y proving tht A + nd B + re disjoint; the fct tht A nd B re disjoint follows nlogously Assume tht = j nd j = ; then j = nd hence = j But since nd re coprime, this cn only hppen if ; however, 1 1 Next, let S := P Z 2 = (x, y) Z 2 :, 0 x 1 0 y 1, x > y 1 y > x 1

4 14 MATTHIAS BECK y =5 x+1 = y/ x = y/ y+1=x/ B+ A =8 x y =5 x+1 = y/ x+1 = (y+1)/ y+1=x/ A- B =8 x Figure 1 The geometry ehind the proof We clim tht A + B + = S \ (0, 0) nd A B = S \ ( 1, 1), which proves Theorem 11 To prove our clim, we first note tht A ± S (nd similrly B ± S) ecuse > 1, nd tht (0, 0) / A + B + nd ( 1, 1) / A B It remins to show (i) S \ (0, 0) A + B + ; (ii) S \ ( 1, 1) A B

5 GEOMETRIC PROOFS OF POLYNOMIAL RECIPROCITY LAWS 15 To prove (i), ssume (x, y) S \ (0, 0) By construction of S, y 1 < x If x y, then x = y nd (x, y) B+ Otherwise, x > y x, so tht y < nd, since x 1 < y, we conclude tht y = x nd (x, y) A+ To prove (ii), suppose (x, y) S \ ( 1, 1), nd let := x + 1 nd j := y + 1 By construction of S, x < (y + 1) = j If j 1 < x, then x = j, tht is, (x, y) = ( ) j, j 1 B Otherwise, j 1 x, nd hence j = (y + 1), tht is, 1 y Since / Z (note tht < if = then j =, cse tht we excluded), we cn me this inequlity strict: 1 < y Furthermore, y construction of S, y < (x + 1) =, whence we hve y = (, tht is, (x, y) = 1, ) A The proof for n = 2 contins lredy the essentil ingredients needed for the generl Theorem 12 For generl n, gin we will construct (hlf-open) polytope, nd the Crlitz polynomils in the reciprocity lw encode different sets of lttice points in this polytope Proof of Theorem 12 As in the ove proof for n = 2, we group the terms on the left-hnd side of the identity ccording to sign s 1 1 j j j 2 3 n u 1 1u2 u 1 3 u j j j 2 3 n u 1 1 u 1 2 u 1 3 u 1 n n + + u nu j j 1 2 n n 1 u u 1 n u 2 u j n j j 1 2 n n 1 u 2 u nd collect the respective exponents in the sets ( ) A n + :=,,,, : ( ) A n + :=,,,, : ( ) A n :=,,,, : 1 n 1 n n n ( ) A n := 1,,,, : ( ) A n :=, 1,,, : ( A n 1 :=, n 2 n 2,, 2 n 2 j n ), 1 : 1 n 1

6 16 MATTHIAS BECK Define S := (x 1, x 2,, x n ) Z n : 0 x j j 1, x > x j 1 for ll 1 j, n ; j we will show tht n Aj + = S \ (0, 0,, 0) nd n Aj = S \ ( 1 1, 2 1,, n 1) Both unions re pirwise disjoint, which follows exctly lie in our ove proof for n = 2 Also just lie in the cse n = 2, one esily sees tht A j ± S nd tht (0, 0,, 0) / n ( 1 1, 2 1,, n 1) / n Aj, so tht it remins to prove: (i) S \ (0, 0,, 0) n Aj + ; (ii) S \ ( 1 1, 2 1,, n 1) n Aj To prove (i), ssume x := (x 1, x 2,, x n ) S \ (0, 0,, 0) x1 1, x 2 2,, xn Tht is, x j = n x j Hence x j x j for ll j, nd x A + Aj + nd Let x e the mximum of for ll j, nd, y construction of S, x j 1 < x j for ll j To prove (ii), ssume x := (x 1, x 2,, x n ) S \ ( 1 1, 2 1,, n 1) Let x +1 minimum of x1 +1 1, x 2+1 2,, xn+1 n y strict inequlity since (x + 1) j of S, x j < (x + 1) j for ll j Tht is, x j = e the Hence (x + 1) j 1 x j for ll j, which cn e replced / Z (note tht x < 1) Futhermore, y construction (x +1) j for ll j, nd x A 3 A Few Applictions The rchetype of corollries tht re immeditely implied y Theorem 12 is the following identity Corollry 31 If 1, 2,, n re pirwise reltively prime positive integers, then n ( ( )) ( ( Proof Apply the opertor u 1 u 1 u 2 Theorem 12 nd set u 1 = u 2 = = u n = 1 n n u 2 )) n = ( 1 1) ( 2 1) ( n 1) ( ( )) u n u n to oth sides of the identity of The cse n = 3 of this identity yields, fter replcing the gretest integer function with (( )) nd doing little rithmetic, the following three-term generliztion of Dedeind s reciprocity lw (2) due to Rdemcher [10]: Corollry 32 (Rdemcher) If,, nd c re pirwise reltively prime positive integers, then 1 (( )) (( )) c 1 (( c + = ( 12 c + c + c ) )) (( )) c 1 + (( c )) (( )) c

7 GEOMETRIC PROOFS OF POLYNOMIAL RECIPROCITY LAWS 17 Dedeind s reciprocity lw (2) follows from Corollry 32 y setting c = 1 It is well nown (see, eg, [11]) tht the Dedeind sum hs the trigonometric representtion s (, ) = cot π cot π, nd hence Dedeind s reciprocity lw nd its vrious generliztions gives rise to mny, sometimes curiously looing, trigonometric identities (see, eg, [4]) We mentioned lredy in the introduction tht the following reciprocity lws of the Hrdy Berndt sums [2] re strightforwrd consequences of Crlitz s Theorem 11, s shown in [1, 3], so we will not repet their proofs here Corollry 33 (Berndt) Suppose nd re reltively prime positive integers If + is odd then S(, ) + S(, ) = 1 If nd re odd then If is even then If is odd then s 5 (, ) + s 5 (, ) = s 1 (, ) 2 s 2 (, ) = ( 1 + ) 2 s 3 (, ) s 4 (, ) = 1 Insted, we give higher-dimensionl nlog of the first Hrdy Berndt sum 1 S(, ) = ( 1) j+1+ j = c ( 1, 1;, ), nmely the higher-dimensionl Hrdy Berndt sum It stisfies the following reciprocity lw 1 1 j h ( 1 ; 2,, n ) := ( 1) j+1+ j2 j + + jn 1 Corollry 34 If 1, 2,, n re pirwise reltively prime positive integers, then h ( 1 ; 2,, n ) + h ( 2 ; 3,, n, 1 ) + + h ( n ; 1,, ) = 1 ( 1) n+n Proof Sustitute u 1 = u 2 = = u n = 1 in Theorem 12 Similr higher-dimensionl nlogs of the other Hrdy-Berndt sums nd their reciprocity lws cn e deduced from Theorem 12 We finish with the remr tht we could lso introduce liner shifts in the rguments of the gretest integer functions in the definition of the Crlitz sum Our geometric proof remins essentilly the sme, nd it yields the generl polynomil reciprocity theorem of Berndt nd Crlitz [3, Theorem 51] 1 2

8 18 MATTHIAS BECK 4 Acnowledgments Mny thns to Msnori Ktsurd nd To Komtsu for the invittion to the Diophntine Anlysis nd Relted Fields conference in honor of Iet Shiow, nd especilly to To Komtsu for generously supporting my trvel through Grnt-in-Aid for Scientific Reserch (C) from the Jpn Society for the Promotion of Science References 1 Tom M Apostol nd Thiennu H Vu, Elementry proofs of Berndt s reciprocity lws, Pcific J Mth 98 (1982), no 1, Bruce C Berndt, Anlytic Eisenstein series, thet-functions, nd series reltions in the spirit of Rmnujn, J Reine Angew Mth 303/304 (1978), Bruce C Berndt nd Ulrich Dieter, Sums involving the gretest integer function nd Riemnn-Stieltjes integrtion, J Reine Angew Mth 337 (1982), Bruce C Berndt nd Boon Pin Yep, Explicit evlutions nd reciprocity theorems for finite trigonometric sums, Adv in Appl Mth 29 (2002), no 3, Leonrd Crlitz, Some polynomils ssocited with Dedeind sums, Act Mth Acd Sci Hungr 26 (1975), no 3-4, Richrd Dedeind, Erläuterungen zu den Frgmenten xxviii, Collected Wors of Bernhrd Riemnn, Dover Pul, New Yor, 1953, pp L A Golderg, Trnsformtion of thet-functions nd nlogues of dedeind sums, PhD thesis, University of Illinois, G H Hrdy, On certin series of discontinuous functions connected with the modulr functions, Qurt J Mth 36 (1905), Mrtin R Pettet nd R Sitrmchndr Ro, Three-term reltions for Hrdy sums, J Numer Theory 25 (1987), no 3, Hns Rdemcher, Generliztion of the reciprocity formul for Dedeind sums, Due Mth J 21 (1954), Hns Rdemcher nd Emil Grosswld, Dedeind Sums, The Mthemticl Assocition of Americ, Wshington, DC, 1972 Deprtment of Mthemtics, Sn Frncisco Stte University, Sn Frncisco, CA 94132, USA E-mil ddress: ec@mthsfsuedu