χ is the principal character. This result seems to be new. We

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1 Chcte vege of Seco Foth Powe of Diichlet -eie t Uity VIVEK V RE Detmet of Mthemtic The Ititte of Sciece 5 Mme Cm Ro Mmi- 3 Ii v_v_e@yhoocoi tct : Fo Diichlet chcte molo itege 3 we e highly imle elemety metho to give ymtotic foml fo whee mo mo i the icil chcte Thi elt eem to e ew We lo oti ymtotic foml fo exio of t mo ig owe eie Keywo : Hwitz zet fctio Diichlet chcte Diichlet -eie

2 Chcte vege of Seco Foth Powe of Diichlet -eie t Uity VIVEK V RE Detmet of Mthemtic The Ititte of Sciece 5 Mme Cm Ro Mmi- 3 Ii v_v_e@yhoocoi Itoctio : et itege σ it e comlex vile whee σ t e el Fo let mo eote Diichlet chcte let mo eote the icil chcte et -eie o tht fo Re e the coeoig Diichlet ; it lytic cotitio Fo let e the Hwitz zet fctio efie y fo Re ; it lytic cotitio Fo let fo Re ; it lytic cotitio et eote the Riem zet fctio Th lo ote tht Fo itege we hll wite with v v fo the give itege 3 ote tht whee i the ivio fctio

3 : : I wht follow hll eote Ele totiet fctio µ hll t fo Moei fctio hll eote ime me lo will me Ele cott Fo el [] hll t fo it itegl t I wht follow o hll me the mmtio with the etictio The mmtio will hve imil meig I wht follow the cott will e olte le tte othewie Effectively we hll e ig oly thee fct mely Ele mmtio Foml the oely coveget eie exeio o the it itevl π e i πi with elity fo o-itegl 3 Powe eie i of O oject i to ove the followig theoem Theoem : Fo itege 3 we hve mo whee mo i the icil chcte i the me of ivio of Rem : O Theoem ove fo eem to e the et elt i thi iectio O metho i highly imle elemety efoe ttig o Theoem we hll ee emm emm : et The fo y comlex me : 3 :

4 we hve whee! i the oe ic of - le lo t Coolly : We hve! Rem : O lemm ove i Theoem of tho [] We hll wite whee fo c whee c i j et i j fo Icietlly it i to e ote tht fo fo i the it itevl [] fo y itege we hve fiite Tylo eie mely! θ θ whee θ o tht fo! fo ome θ whee the O-cott i I ticl lettig we hve whee the O-cott i Fom thi we get the fiite Tylo eie of g c g θ follow : Fo y itege! θ whee g g fo ome θ θ with Th g c whee the O-cott i eeet o ext we tte o Theoem : :

5 Theoem : We hve fo 3 mo µ c µ whee x i eolli olyomil of egee ; fo c i j i j with whe Rem : I the light of fiite Tylo eie with emie fo the lt two tem ivolvig ifiite eie i the ttemet of Theoem ove c e elce y ymtotic exio with eo tem y oitive itege lo i the ce fo the tem fo Thee e my e o the etimtio of elt i the ce of mo M Yohimoto W Zhg [] which tte tht mo mo Howeve the ltet h ee give y S Kemit Y Tigw R whee H R µ with [ ] : 5 :

6 eig eolli olyomil of egee ; H i the cott efie y Poof of Theoem : Fo Diichlet chcte mo fo Re we hve mo We hll wite mo Z fo Re ; it lytic cotitio Hecefoth i wht follow i y cogece eltio fo evity we hll wite i lce of mo et e itege with Th fo m m Z σ m m Th fo fo mo fo we hve We hll how tht fo fo mo Fom the exeio fo σ o ig Ele mmtio foml fo ity x fo σ fo we hve : 6 :

7 x x x x x x ] [ ] [ ote tht x Hee emty m i tete zeo Thi exeio i vli fo σ fo We chooe x Thi give fo Re ] [ Icietlly thi give ] [ I wht follow i el oitive le tte othewie Fo ] [ whee - cott i olte Th fo fo we hve whee Th Coie which i ieeet of : 7 :

8 Hece fo fo mo we hve Similly fo fo mo Similly we hve fo fo mo Th fo fo mo we hve y Thi give mo fo ext we evlte We hve ote tht fo fixe we hve Th we hve fo Thi give : 8 :

9 ote tht mo ext Th we ee It i eogh if we let thogh el vle of ote tht ext we oti ote tht thogh el vle thi it i of the fom Th ig Hoitl le we hve thi it Thi give Th : 9 :

10 ext we etimte Coie 3 Th mo mo 3 3 mo 3 ext we hve mo : :

11 ext we etimte mo O ig Schwtz ielity we hve mo m m 3 m m Th 5 ote 3 Hece y Schwtz lemm 3 5 Th Hece Thi comlete the oof of Theoem Poof of Theoem : i the oof of Theoem ice we hve fo mo o : :

12 i the oof of Theoem we hve mo ext we etimte ig the owe eie i of We hve whee fo ext c whee j j i i c fo Icietlly ote tht the exeio i the o cet i which i eviet i view of the fct tht with i the fiite Tylo eie evelomet of we hve o tht Th : :

13 c c µ µ µ c µ whee i eolli olyomil of egee Th mo o µ c µ ext : 3 :

14 Th mo o µ o c µ Icietlly ote tht the tem c e exee fiite ymtotic eie with eo tem fo y itege Thi comlete the oof of Theoem Refeece [] S Kemit Y Tigw M Yohimoto W Zhg O the icete me e of of Diichlet -fctio t Mth Z 8 - [] VV Re Diichlet -fctio owe eie fo Hwitz zet fctio Poc I c Sci Mth Sci vol 3 o il

3-Colorability. CSE 589 Applied Algorithms Spring The Gadget. 3-CNF-Sat < P 3-Color. Reduction by Example. Properties of the Gadget

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