AVERAGING SPECIAL VALUES OF DIRICHLET L-SERIES.

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1 AVERAGING SPECIAL VALUES OF DIRICHLET L-SERIES KEVIN JAMES Abstrct In ths pper we derve estmtes or weghted verges o the specl vlues o Drchlet L-seres whch generlze smlr estmtes o Dvd nd Ppplrd [] Introducton Fx r, m, n Z wth m, n = Let d p = r 4p nd Br = mx5, r /4 Dene { } Br < p X : p s prme; p m mod n; Sm, r n, X := 4p r mod ; d p 0, mod 4 nd Ar, m, n, X := X p S r m,n,x L, χ dp log p Dvd nd Ppplrd see [] Theorem 3 nd Lemm 4 proved n estmte or Ar,,, X whch ws n ntegrl prt o ther proo tht the Lng-Trotter conjecture s true on verge In relted work on the Lng-Trotter conjecture or ellptc curves wth nontrvl rtonl torson subgroups see or exmple [] t ws neccessry to prove smlr estmtes on Ar, r, n, X or vrous sureree n In ths pper we gve n estmte or Ar, m, n, X or m, n = rbtrry In order to stte the mn result, we wll need bt more notton We wll let = r 4m nd put 3 Q < r,m,n = { >, prme : n; r; ord < ord n} Q r,m,n = { >, prme : n; r; ord ord n} For Q < r,m,n, we wll denote by γ, the gretest nteger whch s less thn ord /, tht s γ := ord / Also, we wll let { / ord r,m ord s even, postve nd nte, 4 Γ = 0 otherwse In ths pper we prove: Dte: August 3, Mthemtcs Subject Clsscton Prmry M06;Secondry G05 Key words nd phrses Drchlet L-seres, Specl vlues o L-seres The uthor s prtlly supported by NSF grnt DMS nd

2 KEVIN JAMES Theorem where C r,m,n = φn C r,m,n Q < r,m,n Q r,m,n, odd n r Ar, m, n, X C r,m,n X, ordn ordn, odd n r ord r,m / ordn ordn 3 n r m Γ Γ Γ ord ord r,m nd C r,m,n s dened by 3 r s odd 4 3 r s even; nd 4 n, r mod 4; ord n ord, 3 ord n 4 3 ord n ord n 3 ord n r mod 4; ord n = ord nd ord, r mod 4; ord n = ord nd ord, r mod 4; ord n = ord ; ord ; ord r,m mod 4, r mod 4; ord ord n n = ord ; ord OR r,m C r,m,n = ord r,m 3 mod 4, r mod 4; ord n > ord ; ord s even nd r,m ord r,m mod 8, 4 3 ord r,m r mod 4; ord n > ord ; ord s even nd r,m ord r,m 5 mod 8, ord r,m r mod 4; ord n > ord ; ord OR r,m ord r,m 3 mod 4, 5 r 0 mod 4; ord 3 n = ; m 3 mod 4, r 0 mod 4; 8 n; m 3 mod 4; r,m mod 8, 4 4 r 0 mod 4; 8 n; m 3 mod 4; r,m 5 mod 8, 3 4 r 0 mod 4; 4 n; m mod 4,

3 AVERAGING SPECIAL VALUES OF DIRICHLET L-SERIES 3 Proos We rst stte the ollowng result whch s essentlly due to Dvd nd Ppplrd, n the sense tht one cn obn proo by ollowng the sme lne o rgument gven n the proo o Theorem 3 n [] wth mnor modctons such s crryng the condton p m mod n throughout ther rgument Proposton Suppose tht r, m, n Z nd tht m, n = Then or ny c > 0, X Ar, m, n, X = K r,m,n X O log c, X where nd K r,m,n = k := = k= k kφ[n; k ] mod 4k 0, mod 4 r,4k =4 4m r mod 4n,4k k For the ske o brevty, we omt the proo o ths result nd reer the reder to [] The proo o the mn result now reures only reconclng o the constnts K r,m,n nd C r,m,n To tht end we begn wth n nvestgton o the k For convenence, we wll splt these nto two sums: 5,0 k := nd, k := k k mod 4k 0 mod 4 r,4k =4 4m r mod 4n,4k mod 4k mod 4 r,4k =4 4m r mod 4n,4k In order to descrbe the behvor o the, k s we hve the ollowng lemms The rst lemm ollows drectly rom the bove dentons We stte t or the ske o convenence only Lemm For,0 k to be nonzero, t s necessry tht we hve r, even; k, odd, r/, = nd n, r/ m For, k to be nonzero, one o the ollowng must hold r nd re both odd, r, = nd n, b r mod 4, r/, =, 4n, I ord n ord, then we reure tht ord mxord n, 4 I ord n = ord, then we reure tht ord = ord n I ord n ord, then we reure tht ord = ord nd mod 4 ord r,m

4 4 KEVIN JAMES c r 0 mod 4, mod 4, r, / = nd n, / r/ m I n 0 mod 4, then we lso need m 3 mod 4 Lemm, k =0, s multplctve uncton o k Proo I r s odd,,0 k = 0 nd the multplctvty o, k cn be shown s n [], lemm 33 So, we wll consder only the cse when r s even In ths cse, r/, =, n, r/ m nd k s odd, then we obtn 6,0 k = mod k r/,k= r/ m n, n, mod n n,,k k, nd zero otherwse Snce, runs through certn congruence clsses modulo k n the bove sum, the multplctvty o,0 k now ollows orm the Chnese remnder theorem nd the multplctve propertes o the Legendre symbol We need only tret the cses n whch, k s possbly nonzero see lemm For cse, k s odd, then we hve 7, k = Z/kZ r,k= n, n, mod n n,,k k In cses b nd c, when k s odd, we hve 8, k = Z/kZ r/ /,k= r/ m n,/ / n,/ mod n n,/,k k In ether o these cses, we see tht the sums vry over congruence clsses modulo k whch s odd The multplctvty o, now ollows rom the Chnese remnder theorem nd the multplctve propertes o the Legendre symbol

5 AVERAGING SPECIAL VALUES OF DIRICHLET L-SERIES 5 Lemm 3 Gven r, m nd n, let = 0 or nd dene τ s ollows r mod 4; = nd ord n ord nd ord n, ord n r mod 4; = nd ord n ord nd ord n >, ord n r mod 4; = ; ord τ = n = ord ; nd ord n s odd, ord r mod 4; = ; ord n ord ; ord s even nd r,m ord r,m mod 4, r 0 mod 4 nd =, 0 r s odd or = 0 I s chosen such tht r, m, n nd stsy one o the condtons n lemm or,, nd s n odd prme, then we hve, α = τ ord, α Also, r, m, n nd stsy one o condtons, b or c o lemm, then 9,0 α = mod α r/,=, α = ord, α Proo We wll rst tret the cse when = 0 nd r, m, n nd stsy condton o lemm Usng 6, we hve α ord n ord, mod α r/ m ord ord mod n ord,α α ord n > ord, Note tht s sure whch s coprme to Thus mkng the chnge o vrble = ord, the lst sum becomes ord α 0 Now combnng ths wth 9, we hve,0 α = mod α r/ m ord mod n ord,α mod α r/,= mod α r/ m ord mod n ord,α α ord n ord, α ord n > ord

6 6 KEVIN JAMES Usng ths expresson one cn esly see tht,0 α = ord,0 α, nd thus we hve proved tht the lemm holds n ths cse In ll other cses when s n odd prme, the proo s smlr For the lst sserton, we ssume tht r, m, n nd stsy ether o condtons b or c o lemm From 5, we hve We note tht, α = 3, α = mod α mod 4 r/ m n,/ / n,/ mod n n,/,α mod α mod 4 = r/ m ord ord α α ord n ord, α ord n ord mod α mod n n,/,α s n odd sure Thus, lettng ord = yelds ord α mod α ord n ord, mod 4 α mod α ord n ord r/ m ord mod n n,/,α Usng the lst expresson, one cn esly check tht, α = ord, α, s desred In the cse tht r, m, n nd stsy condton o lemm, the proo s smlr In order to evlute the α, = 0,, we hve the ollowng two lemms β, Lemm 4 Suppose tht s n odd prme nd α > 0 Lettng d = β,0 α when r s even; r, = nd n, β r/ m, or lettng d = τ β, α when r, m nd n stsy condtons, b or c o lemm, we hve r α β = 0; α, odd; n r α β = 0; α, even; n α α β = 0 nd n 4 d = n, α 0 β > 0; α, odd nd ord n β α β > 0; α, even nd ord n β α α / β β > 0 nd ord n > β n β,α Proo We wll prove the lemm or 4 β, α where s n odd prme nd where r, m nd n stsy condton b o lemm The proos or the other cses re smlr From 8 bove, we

7 hve 5 AVERAGING SPECIAL VALUES OF DIRICHLET L-SERIES 7 4 β, α = = Z/ α Z r/ 4 β,= r/ m 4β n, β n, β mod n n, β,α Z/ α Z r/ 4,= r/ m 4 mod n, α Z/ α Z r/ m 4β n, β n, β α α β = 0, α β > 0 mod n n, β,α Observe, tht when nd n, the second condton o our summton or the cse β = 0 s empty We lso note tht when n, we hve m, snce m, n = So, the second condton o the summton or the cse β = 0 mples the rst Wth these observtons, one cn now esly deduce the desred result The next lemm llows us to evlute the, t powers o The proo s smlr to tht o the prevous lemm nd or the ske o brevty we omt t Lemm 5 I r s odd, then,, α = { α 4 n, α n, α 4 n, I r s even nd r, = β, m nd n stsy ether o condtons b or c o lemm, then 0 ord n β nd α s odd, β, α = α ord n β nd α s even, α r/ m/ β α ord n β ord n β, α Now, let κn denote the multplctve uncton generted by { 6 κl α l α s odd, = α s even, or ny prme l nd ny α > 0 Then we hve the ollowng bound Lemm 6 For ll k,, k k/κk, where = 0, Proo From lemms 3, 4 nd 5, t ollows mmdetely tht or ny prme, {, α α α s even, 7 α α s odd = α /κ α

8 8 KEVIN JAMES The lemm now ollows rom the multplctvty o, nd κ We recll the ollowng ct rom [] Lemm 34 Lemm 7 Let c = l,prme converges l l Then, k U κkφk Thus rom lemms 6 nd 7, we see tht K r,m,n s nte constnt We rewrte K r,m,n s 8 K r,m,n = K 0 r,m,n K r,m,n, where 9 K 0 r,m,n = = k=,0 k kφ[n, k ] nd K r,m,n = = c U In prtculr, k= k=, k kφ[n, k ] Now we compute the constnts K r,m,n = 0, We recll the ollowng denttes A, B 0 φab = φaφb φa, B, nd thereore, we lso hve B A, φ A B = φaφ A B, B φb A B, B In prtculr, we cn wrte φ[n, k ] = φnk n, k κkφk Now, we recll or xed choce o r, m nd n, tht must be chosen such tht r, m, n nd stsy the condtons o lemm or, k to be non-zero We wll denote by S r,m,n the set o s whch stsy the condtons o lemm, nd we let τ be dened s n lemm 3 Then, we cn wrte 3 Kr,m,n = τ, kn, τ k φ τ n, k τ φ τ n kφk τ n, k = τ S r,m,n k= Usng lemm nd the multplctvty o φ nd lettng, b := ord,b, we cn rewrte the nner sum bove s, 4 τ, j n, τ j φ τ n, j j φ j τ n, j, prme Usng lemm 3, 4 cn be rewrtten s

9 AVERAGING SPECIAL VALUES OF DIRICHLET L-SERIES 9 5 =, prme τ, j n, τ j φ τ n, j j φ j τ n, j τ, j n, τ j φ τ n, j j φ j τ n, j τ ord, j n, τ j φ τ n, j j φ j τ n, j τ ord, j n, τ j φ τ n, j j φ j τ n, j τ, j n, τ j φ τ n, j j φ j τ n, j Now, substtutng ths lst expresson bck nto 3 nd usng 0, we obtn the ollowng expresson or K r,m,n 6 τ φ τ n, prme = τ S r,m,n τ, j n, τ j φ τ n, j j φ j τ n, j φ τ n, φ τ n, τ ord, j n, τ j φ τ n, j j φ j τ n, j τ, j n, τ j φ τ n, j j φ j τ n, j Now, S r,m,n =, then the bove expresson s just 0 So, we wll ssume or now tht S r,m,n, nd n ths cse we cn rewrte the sum rom 6 s product 7, prme β= τ β S r,m,n φ τ n, β β φ β τ n, β τ β, j n, τ βj φ τ β n, j j φ j τ β n, j τ, j n, τ j φ τ n, j j φ j τ n, j Ths llows us to rewrte 6 s

10 0 KEVIN JAMES 8 τ φ τ n, odd n, odd n τ, j j φ j β= τ β S r,m,n j n, τ j β φ β τ, j n, j j φ j β= τ β S r,m,n τ β, j φ β, j j φ j β, j β φ β τ, j n, τj φ τ n, j j φ j τ n, j β= τβ S r,m,n φ τ n, β β φ β τ n, β τ β, j n, βj φ β n, j j φ j β n, j τβ, j n, τβj φ τβ n, j j φ j τβ n, j Snce, n the rst product, n, nd snce we re ssumng tht S r,m,n, τ β S r,m,n β nd only r So usng lemm 4, the rst product n 8 becomes or ll 9, odd n r, odd n r Recllng 3 nd 4, nd usng lemm 4 the second product o 8 becomes 30 Q < r,m,n Q r,m,n ordn ordn ord r,m / Γ Γ ordn ordn 3 Γ ord n r,odd ord r,m m

11 AVERAGING SPECIAL VALUES OF DIRICHLET L-SERIES Next, we evlute the thrd ctor o 8,whch we wll denote by T r,m,n Usng lemms nd 5, we nd tht 3 ord n5 = ; r mod 4; ord n ord, ord n = ; r mod 4; ord 3 n = ord, ord n = ; r mod 4; ord 3 n = ord ; ord s even nd r,m r,m mod 4, ord = ; r mod 4; ord n > ord ; ord s even nd r,m r,m mod 8, ord r,m = ; r mod 4; ord 3 n > ord ; ord s even nd r,m r,m 5 mod 8, n = ; r 0 mod 4; 4 n, T r,m,n 3 = 6 = ; r 0 mod 4; ord 3 n = nd m 3 mod 4, 8 = ; r 0 mod 4; 8 n; m 3 mod 4; r,m 4 mod 8, 8 = ; r 0 mod 4; 8 n; m 3 mod 4; r,m mod 8, 3 = nd r s odd, 9 7 = 0; r mod 4 nd n s odd, 6 7 ord n = 0; r mod 4; 0 < ord n ord ; ord n s even, 5 = 0; r 7 ord n ord r,m = 0; r mod 4; ord n > ord, = 0 nd r 0 mod 4 mod 4; ord n ord ; ord n s odd, 3 Thus, K r,m,n = T r,m,n 0 φn T r,m,n τ φ τ n Q < r,m,n Q r,m,n, odd n r ordn ordn, odd n r ord r,m / ordn ordn 3 n r m Γ Γ Γ ord ord r,m

12 KEVIN JAMES Now one cn check tht C r,m,n = φn T r,m,n 0 when τ φ τ Sr,m,n n 0 S r,m,n nd 0 otherwse Thus Theorem now ollows rom Proposton nd rom 3 φn T r,m,n Reerences [] C Dvd nd F Ppplrd, Averge Frobenus dstrbutons o ellptc curves Internt Mth Res Notces [] K Jmes, Averge Frobenus dstrbutons or ellptc curves wth 3-torson, preprnt Deprtment o Mthemtcl Scences Clemson Unversty BOX Clemson, SC , USA kevj@clemsonedu Deprtment o Mthemtcl Scences, Clemson Unversty, BOX Clemson, SC , USA E-ml ddress: kevj@clemsonedu URL: kevj/