KANSAS STATE UNIVERSITY Manhattan, Kansas

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1 PRIME POWER EXPONENTIAL AND CHARACTER SUMS WITH EXPLICIT EVALUATIONS by VINCENT PIGNO B.S., University of Kansas, 2008 M.S., Kansas State University, 2012 AN ABSTRACT OF A DISSERTATION subitted in artial fulfillent of the requireents for the degree DOCTOR OF PHILOSOPHY Deartent of Matheatics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 2014

2 Abstract Exonential and character sus occur frequently in nuber theory. In ost alications one is only interested in estiating such sus. Exlicit evaluations of such sus are rare. In this thesis we succeed in evaluating three tyes of od sus when is a rie and is sufficiently large. The twisted onoial su, S 1 = χxe 2πinxk /, the binoial character su, S 2 = χ 1 xχ 2 Ax k + B, and the generalized Jacobi su, S 3 = J nχ 1,..., χ k, = x 1 =1 where the χ are od Dirichlet characters. χ 1 x 1 χ k x k, > n, x k =1 x 1 + +x k = n We additionally show that these are all sus which can be exressed in ters of classical Gauss sus.

3 PRIME POWER EXPONENTIAL AND CHARACTER SUMS WITH EXPLICIT EVALUATIONS by Vincent Pigno B.S., University of Kansas, 2008 M.S., Kansas State University, 2012 A DISSERTATION subitted in artial fulfillent of the requireents for the degree DOCTOR OF PHILOSOPHY Deartent of Matheatics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 2014 Aroved by: Major Professor Christoher Pinner

4 Abstract Exonential and character sus occur frequently in nuber theory. In ost alications one is only interested in estiating such sus. Exlicit evaluations of such sus are rare. In this thesis we succeed in evaluating three tyes of od sus when is a rie and is sufficiently large. The twisted onoial su, S 1 = χxe 2πinxk /, the binoial character su, S 2 = χ 1 xχ 2 Ax k + B, and the generalized Jacobi su, S 3 = J nχ 1,..., χ k, = x 1 =1 where the χ are od Dirichlet characters. χ 1 x 1 χ k x k, > n, x k =1 x 1 + +x k = n We additionally show that these are all sus which can be exressed in ters of classical Gauss sus.

5 Table of Contents Table of Contents v Acknowledgeents vi Dedication vii 1 Introduction 1 2 Preliinaries Dirichlet Characters Reducing to the case of rie odulus The Power-Full Lea Definitions and Congruence Relationshis The case is odd The case = 2 and Gauss Sus Character Sus and The Duality of Gauss Sus Reduction Method of Cochrane and Zheng Evaluation of Gauss Sus Evaluation of the Gauss Su Rewriting Sus in Ters of Gauss Sus to Obtain Weil and Weil Tye Bounds 35 v

6 4.1 Gauss Sus and Weil tye bounds Twisted onoial sus as Gauss Sus Binoial Character Sus as Gauss Sus The Generalized Jacobi Su as Gauss Sus Evaluating the Twisted Monoial Sus Modulo Prie Powers Stateent of the Main Theore Proof of Theore Proof of Corollary When = 2, Evaluating the Binoial Character Sus Modulo Prie Powers Evaluation of the Sus for Odd Evaluating the Binoial Character Su for = Proof of Theore Initial decoosition Large values: > n + 2t Sall values: t + 2 n 2t Evaluating Jacobi Sus Proof of Theore A ore direct aroach Bibliograhy 94 vi

7 Acknowledgents The entirety of thesis reresents joint work with y friend and advisor Professor Chris Pinner. Additionally Chater 6 is joint work with Joe Sheard, Chater 7 is joint work with Misty Long and Chater 3 contains joint work with both Joe and Misty. I would like to exress y gratitude to y wife and faily for suorting e through the coletion of this thesis. Additionally I would like to thank the entire faculty and staff of the Kansas State University Deartent of Matheatics. I could not have asked for better role odels and entors. Secifically, the Nuber Theory grou: Professor Todd Cochrane, Professor Craig Sencer, and Chris Pinner. Thanks as well to Professors Virgina Naibo, Andrew Bennett, Andy Cherak, Bob Burckel, Marianne Korten, Dr. Wor, and Gown Swift. A secial thank you is deserved by Reta McDerott and Kathy Roeser for all their assistance and atience. I would also like to thank the additional ebers of y coittee, Professor Gary Gadbury and Professor Mitchell Neilsen for their coents and tie. A big shout out to Buggin Out Crew, for continually ushing e to work hard and think creatively. vii

8 Dedication This thesis is dedicated to y uncle, Francis Pigno. viii

9 Chater 1 Introduction We are concerned here with the exlicit evaluations of certain exonential sus odulo where is a rie, naely the twisted onoial su, S 1 = S 1 χ, nx k, = χxe nx k, 1.1 where e q x := e 2πix/q, 1.2 the binoial character su, S 2 = S 2 χ 1, χ 2, Ax k + B, = χ 1 xχ 2 Ax k + B, 1.3 and the generalized Jacobi su, S 3 = J nχ 1,..., χ k, = x 1 =1 χ 1 x 1 χ k x k, > n, 1.4 x k =1 x 1 + +x k = n 1

10 where the χ are ultilicative characters. We will show all of these sus can be exlicitly evaluated when is sufficiently large. Cases where exonential sus can be evaluated are rare, aking the sus which can be secifically evaluated standouts. As we shall see in Chater 4 these are all sus which can be exressed in ters of classical Gauss sus. In order to obtain our evaluations we aly the reduction ethod of T. Cochrane and Z. Zheng [4], thereby reducing our sus to the consideration of a articular characteristic equation. In Chater 5 we reduce the twisted onoial su, S 1 = χxe nx γt, with γ, to the equation c 1 + R t+1 nx γt 0 od t when t + 1 < < 2t + 2, and to c 1 + R t+s+1 nx γt 0 od t+s+1, 1.6 when 2t + 2, where the R j are araeters deendent on our choice of a riitive root od and the c i s are araeters deending on the character these araeters will be discussed in detail in Chater 2. Deending on the range, if 1.5 or 1.6 has no solution the su is zero; however if there is a solution we are able to directly evaluate the su as shown in the following theore, which aears as Theore in Chater 5. Theore For an odd rie, t Z, t 0, let fx = nx γt, nγ. Case I: Suose that t + 1 < 2t + 2. If χ is a d t -th ower of a riitive character 2

11 and the characteristic equation 1.5 has a solution α then S 1 χ, fx, = d 1 χαe fα. Otherwise, Sχ, fx, = 0. Case II: Suose that 2t + 2 <. If χ is a d t -th ower ower of a riitive character and 1.6 has a solution then S 1 χ, fx, = d 2 +t 2rc1 χαe fα ε, 1.7 where α is a solution of 1.6, γ, 1. is the Jacobi sybol, ε is as in 2.29 and d = It should be ointed out that in absolute value this result silifies to S 1 = 0 or d 1, if t + 1 < 2t + 2, S 1 χ, fx, = d 2 +t, if 2t + 2 <. We get a siilar evaluation when = 2, deendent again on solutions to certain characteristic equations. In Chater 6 we evaluate the ure character su, S 2 = χ 1 xχ 2 Ax k + B. Again using the ethods of Cochrane and Zheng we obtain a characteristic equation, g +n in{ 1, [ x 0 od 2 ]+t} 1.8 where g x coes fro writing χ 1 xχ 2 Ax k + B = χgx for soe od character χ, 3

12 leading to the following evaluation; see Theore Theore Suose that is an odd rie and χ 1, χ 2 are od characters with χ 2 riitive. If χ 1 = χ k 3, and 1.8 has a solution x 0 with x 0 Ax k 0 + B, then 1, if t + n + 1 < 2t + n + 2, χ 1 xχ 2 Ax k + B = dχ 1 x 0 χ 2 Ax k 0 + B +n 2 +t, if > 2t + n + 2, n even, +n 2 +t ε 1, if > 2t + n + 2, n odd, where ε 1 is as in 6.13 If χ 1 does not satisfy χ 1 = χ k 3, or 1.8 has no solution satisfying x 0 Ax k 0 + B, then the su is zero. For = 2 siilar results are obtained in Chater 6; see Theore In order to evaluate the ulti-variable general Jacobi su, 1.4, we use a general result fro Chater 4, that exresses S 1, S 2 and S 3 in ters of the classical Gauss su, Gχ, = χxe x. For exale, when the characters are riitive, J nχ 1,..., χ k, = k i=1 Gχ i, Gχ 1... χ k, n. 1.9 In Chater 3 we use the Cochrane and Zheng reduction ethod to get the following evaluation of the od Gauss su; see Theore 3.1. Theore Suose that χ is a od character with 2. If χ is iriitive, 4

13 then Gχ, = 0. If χ is riitive, then Gχ, = 2 χ cr 1 j e cr 1 j 2 c 2rc ε, if 2, and 3 3, ω c, if = 2 and 5, 1.10 for any j when is odd and any j + 2 when = 2. Here 2 2 x 1 denotes the inverse of x od, and ω := e πi/4. Using 1.9 and Theore we are then able to evaluate S 3 exlicitly; see Theore we write all three of our sus in ters of Gauss sus in Chater 4, however we only use this for direct evaluation in the case of our ulti-variable Jacobi su. Theore Let be a rie and n + 2. Suose that χ 1,..., χ k are k 2 characters od with at least one of the riitive. If the χ 1,..., χ k are not all riitive od or χ 1... χ k is not induced by a riitive od n character, then Jχ 1,..., χ k, = 0. If the χ 1,..., χ k are riitive od and χ 1 χ k is riitive od n, then J nχ 1,..., χ k, = 1 2 k 1+n χ 1c 1 χ k c k χ 1 χ k v δ, 1.11 for odd, 2r δ = k 1+n v n c1 c k ε k ε 1, n and v := n c c k, ω := e πi/ Evaluations of 1.4 are also given when = 2 in Chater 7; see Theore

14 Chater 2 Preliinaries Let q be a ositive integer. For a ultilicative Dirichlet character χ od q and fx, gx Z[x] we define a ixed exonential su where q Sχ, gx, fx, q := χgxe q fx 2.1 e q x = e 2πix/q. 2.2 Note it is soeties useful to use the equivalent notation e x = e 2πix/q. 2.3 q We are concerned here with the exlicit evaluations of these and closely related sus when q = with a rie and f, g are secific olynoials with integer coefficients, naely 1.1, 1.3 and 1.4. We show in Section 2.2 why it is enough to reduce to the rie ower case, but we first start with soe definitions and reliinaries. 6

15 2.1 Dirichlet Characters We begin by defining a secial class of ultilicative hooorhiss called grou characters. Definition Given a finite grou G, a character χ is a function χ : G C such that for a, b G, χa b = χaχb. For the identity eleent, e, and soe a in G we have χa = χe a = χeχa and since χ is nonzero on G, we have χe = 1. Further, χa G = χe = 1, and thus χa is a G -th root of unity. For any finite abelian grou, G, we know that G = Z n1... Z nr where the Z ni are additive cyclic grous. Let the generators of G be a 1 = 1, 0,..., 0,..., a r = 0,..., 0, 1. For a generator a i we have that χa i n i = χn i a i = χ0 = 1; thus χa i is an n i th root of unity. Since we know χa i = e ni c i for soe integer 0 c i n i 1 we have exactly n i distinct laces to send each a i resulting in r i=1 n i = G different characters. We also note that for finitely generated grous the characters are defined by their actions on the generators. In this thesis we let G = Z q where we write Z q for the ultilicative grou of units in Z/qZ. The characters on Z q are then extended to all of Z q by defining the characters to be zero on eleents of Z q not in Z q. Definition A Dirichlet character χ od q is a non-identically zero function χ : Z q C with χab = χaχb for all a, b Z q and χc = 0 if q, c > 1. One of the ost well known exales of a Dirichlet character is the Legendre sybol. We define the Legendre sybol odulo a rie,, by a = 1, if a, = 1, and a is a square od, 1, if a, = 1, and a is not a square od, 0, if a, >

16 Alternately we will denote the Legendre sybol as the quadratic character χ Q. For q 1 q we say that a od q character χ is induced by a od q 1 character, χ q1, if χ q1 n, if n, q = 1, χn = 0, otherwise. We call a character riitive if it cannot be induced by a lower odulus character. One can exaine the structure Dirichlet characters by exaining the characters on rie owers. Letting the rie factorization of q = k i=1 α i i where i are ries, we clai that there exists a corresonding riitive root a i such that Z α i = a i unless i = 2, with i α i 3 in which case Z 2 α i = 1, 5 see Section 2.8 in [18]. The order of each a i od α i i is exactly φ α i i, and thus χa i is a φ α i i root of unity unless i = 2 with α i 3 in which case 5 has order 2 α 2 and 1 has order 2 and thus χ 1 is a 2-nd root of unity and χ5 is a 2 αi 2 -nd root of unity. The characters of Z α i i that is unless i = 2 with α i 3 when we have are defined by their action on the a i s, χa i = e φ α i i c i, 1 c i φ α i i 2.5 χ 1 = ±1, χ5 = e 2 α i 2c i, 1 c i φ2 α i For any od α i i character χ we can extend χ to be a od q character by defining it to be 0 for all a Z q with a, q > 1. We additionally note that for two od q characters, χ 1 and χ 2, we have that χ 1 χ 2 is also a od q character. We clai that every od q character can be be written as the roduct k od q characters induced by od α i i thus we have k i=1 φα i i where the χ i characters, i = 1,..., k. The nuber of characters for Z α i is φ α i i ; i = φq choices for a od q character χ of the for χ = χ 1... χ k are od q characters induced by od α i i characters. As there are φq 8

17 characters for Z q if each of the choices of χ i gives a different od q character then every od q character ay be constructed in this way. Let χ and χ be two od q characters with χ = χ, and χ = χ 1... χ k, χ = χ 1... χ k, where the χ i and χ i α i i α i i are induced by od characters. By the Chinese Reainder Theore for each a i there exists an A i a i od with A i 1 od α j j, for j i. When i = 2 with α i 3 we also need an A i 5 od 2 α i and A 0 1 od 2 α i, A i A 0 1 od α j, i j. For 2 3 α j j, χ A j = χ A j, and thus χ ja j = χ j a j. When j = 2 with α j 3 we have that χ A j = χ A j and χ A 0 = χ A 0, ilying that χ j5 = χ j 5 and χ j 1 = χ j 1. Since a α i i character is deterined by it action on the generators of Z α i, χ j and χ j ust be the sae character i for each j. Therefore, any od q character, χ can be exressed as χ = χ 1... χ k 2.7 where the χ i are od q characters induced by od α i i characters. The character which sends all the eleents of the ultilicative grou to 1 is called the rincial character, defined by 1, if q, b = 1, χ 0 b = 0 else. 2.2 Reducing to the case of rie odulus Let q = k α i i with i rie. For a od q ixed exonential su we will use the fact that any od q character, χ = χ 1... χ k, where the χ i are od α i i characters extended to Z q to break our coosite sus u into sus odulo rie owers. Define i = q/ α i i let h i be integers such that k j=1 h j j = 1. Note that k j=1 x jh j j x i h i i x i od α i i, thus gx 1 h x k h k k gx i od α i i. and 9

18 Further, x 1 h x k h k k j x 1 h 1 1 j + + x k h k k j od q giving e q x1 h x k h k k j = e q x1 h 1 1 j + + x k h k k j = k i=1 e α i i h j i j 1 i x j i = k i=1 e α i h i i x j i. Thus e q fx = k i=1 e α i h i i fx i. We ay assue that f0 = 0, for if not we ay write e q fx = e q fx f0e q f0 and the e q f0 can be ulled out of the su straightaway. Additionally x 1 h x k h k k runs over a colete set of residues odulo q as the x i s run fro 1,..., α i i. So for the general od q ixed exonential su, 2.1, we have Sχ, gx, fx, q = = = q χgxe q fx α 1 1 x 1 =1 k i=1 χ 1 gx 1 e α 1 1 h 1fx 1 Sχ i, gx, h i fx, α i i. α k k x k =1 χ k gx k e α k h k fx k k Thus, for the general ixed exonential su is suffices to deal only with the case q = α which includes our S 1,

19 Siilarly if χ, and χ are od q characters then our S 2 su, S 2 χ, χ, Ax k + B, q = = = q χ xχ Ax k + B k α i i i=1 x i =1 k i=1 χ ix i χ i Ax k i + B S 2 χ i, χ i, Ax k + B, α i i, for χ i, χ i induced by od α i i Likewise, for the generalized Jacobi su, J B χ 1,..., χ k, q = characters. For S 2 it suffices to exaine only rie owers. q q x 1 =1 x k =1 x 1 + +x k B od q χ 1 x 1 χ k x k, if the χ i are od rs characters with r, s = 1 then, writing χ i = χ iχ i where χ i and χ i are od r and od s characters resectively, writing x i = u i rr 1 + v i ss 1, where u i = 1,..., s, v i = 1,..., r, and rr 1 + ss 1 = 1, gives J B χ 1,..., χ k, rs = = = s r s r χ 1 rr 1 u 1 + ss 1 v 1 χ k rr 1 u k + ss 1 v k, u 1 =1 v 1 =1 u k =1 v k =1 rr 1 u 1 +ss 1 v 1 + +rr 1 u k +ss 1 v k B od rs s r s r χ 1rr 1 u 1 χ 1ss 1 v 1 u 1 =1 v 1 =1 u k =1 v k =1 rr 1 u 1 +ss 1 v 1 + +rr 1 u k +ss 1 v k B od s rr 1 u 1 +ss 1 v 1 + +rr 1 u k +ss 1 v k B od r s s u 1 =1 u k =1 u 1 + +u k B od s χ 1u 1... χ ku k = J B χ 1,..., χ k, rj B χ 1,..., χ k, s. r v 1 =1 v k =1 v 1 + +v k B od r... χ krr 1 u k χ kss 1 v k r χ 1v 1... χ kv k. Hence, it suffices to consider the case of rie ower oduli, q =, for all three of our sus. 11

20 2.3 The Power-Full Lea A ost useful result for aniulating ixed and ure exonential sus is the connection between the degree of the olynoials f, g and the tye of character necessary for the su to not be zero. For instance in order to write our sus 1.1, 1.3 and 1.4 in ters of Gauss sus which will be discussed in Chater 4 we ust first rove the following owerful lea Lea For any odd rie, ultilicative characters χ 1, χ 2 od, and f 1, f 2 in Z[x], the su S = character χ 3. χ 1 xχ 2 f 1 x k e f 2 x k is zero unless χ 1 = χ k 3 for soe od While it should be noted this condition srings u naturally when evaluating the sus, it is useful to rove it here. Proof. Taking z = a φ /k,φ, a a riitive root od, we have z k = 1 and S = χ 1 xzχ 2 f 1 xz k e f 2 xz k = χ 1 zs. Hence if S 0 we ust have 1 = χ 1 z = χ 1 a φ /k,φ and consequently χ 1 a is a colex φ /k, φ -th root of unity and so χ 1 a = e φ c k, φ for soe integer c. Letting c 1 be any integer satisfying c k, φ c 1 k od φ, c 1 is unique od φ /k, φ we equivalently have χ 1 = χ k 3 where χ 3 a = e φ c 1. 12

21 2.4 Definitions and Congruence Relationshis The case is odd Let a be a riitive root od, the existence of which is an eleentary result in nuber theory see Section 2.8 in [18]. Further fro Theore 2.40 in [18] we know if a is a riitive root od 2 it is a riitive root for all higher owers of as well. For a od riitive root, a, we ust have a is also a riitive root od 2 to get that a is a riitive root od j for all j. By using that a 1 = 1 + r for soe r we take a + λ 1 a 1 + 1λa 2 od r λa 2 od r λa 2 od 2, and take λ such that r λa 2 giving us just the riitive root we are looking for. For this thesis we assue r and λ = 0, thus a is a riitive root for all owers of. We now define the integers r and R j by a φ = 1 + r, a φj = 1 + Rj j. 2.8 Note, r and for any j 2 a φj = 1 + R j j = a φj 1 = 1 + R j 1 j R j 1 j + R 2 2 j 1 2j 1 od 3j 1 giving R j R j 1 od j 1, 13

22 and thus for any j i we have R j R i od i. 2.9 For a character χ od we ilicitly define c by χa = e φ c, 2.10 with 1 c φ. Note, c exactly when χ is riitive. Lea For the -adic integer R := 1 log1 + r = 1 i=1 r i 1 i 1 i 2.11 we have R R j od j. Proof. a φj = 1 + r j 1 = 1 + R j j, so log1 + r j 1 = log1 + R j j. By taking the Taylor series exansion of log1 + x we get j 1 log1 + r = j R = R j j i 1 i 1, i i=1 and thus we have R = R j i i 1j 1 i 1. i i=1 14

23 If ν i for any i, then lainly ν < i 1 for all odd, giving R R j od j The case = 2 and 3 When is not odd and 3 we need two generators 1 and a = 5 for Z 2 again see [18], Chater 2, Section 8, and define R j, j 2, and c by a 2j 2 = 1 + R j 2 j, χa = e 2 2c, 2.12 with χ a od 2 character, riitive exactly when 2 c. Noting that R 2 i 1 od 8, we get R i+1 = R i + 2 i 1 R 2 i R i + 2 i 1 od 2 i For j i + 2 this gives the relationshis, R j R i+2 R i i R i + 2 i i R i 2 i 1 od 2 i and R j R i i 2 2 i 1 R i 1 2 i 2 od 2 i Gauss Sus We can now define our first and ost well known exonential su, the Gauss su Gχ, q = q χxe q x

24 where χ is a od q character. Letting q = where is rie, exlicit evaluations of these sus exist for > 1 which we will derive in detail in Chater 3. The cases when the = 1 su has exact evaluation are few, the ost faous exale being the quadratic Gauss su, if 1 od 4, e x 2 = i, if 3 od 4, the evaluation of which is involved for a nice treatent see Chater 1.3 of [2]. One can equivalently write the quadratic Gauss su in the for 2.16 using the Legendre sybol, e x 2 = = 1 + x e x = e x + x e x = Gχ Q,, x e x where recall we use χ Q to denote the od character that coincides with the Legendre sybol. Here we use that e x = 0, a central notion in the roofs of the ain theores in this dissertation. It is worth stating that, ore generally, suing a linear exonential su odulo q over a colete set of residues is either q or 0. That is q q, if q A, e q Ax = 0, otherwise Plainly if q divides A each ter in the su is 1 giving the total su to be q, if not then q e q Ax = e q A e q A q+1 1 e q A = 0. The bulk of this thesis deals with evaluating ixed sus odulo α with α 2 using 16

25 ethods of T. Cochrane and Z. Zheng as detailed in [4], where we are able to reduce certain ixed exonential sus and ure character sus to cases siilar to Character Sus and The Duality of Gauss Sus A ure character su has the following roerties φ, if χ is the rincial character, χx = 0, otherwise When χ is the rincial character it lain the su is φ. To see the su is zero otherwise we take a to be a riitive root od when is odd and write the su χx = φ γ=1 χa γ = φ γ=1 e φ cγ, giving an exonential su over a colete set of residues as in 2.17, giving the result. Siilarly for = 2 we write χx = 2 2 γ=1 2 2 χ5 γ + χ 1 γ=1 χ5 γ = 2 2 γ=1 2 2 e 2 2cγ + χ 1 γ=1 e 2 2cγ the rest follows fro Suing over characters gives a siilar result. Letting χ 1,..., χ φ be all the characters od we have that φ i=1 0, if b 1 od, χ i b = φ, if b 1 od By the definition of a Dirichlet character if b, > 1, χb = 0. Otherwise b = a β for soe 17

26 0 β φ when is odd, so we can write φ i=1 χ i b = φ i=1 χ i a β = φ i=1 e φ c i β. Since each character sends the riitive root a to a different 1 c i φ we have a su over a colete set of residues. By 2.17 the su is zero unless φ β, in which case b = a β = a φ β 1 od for β such that β = β φ, and the su is φ. When = 2, 3, b = 1 w 5 β, with 0 w 1 and 1 β 2 2 and we can write, 2 1 i=1 χ i b = 2 2 j=1 χ j 5 β j=1 χ j 1 w χ j 5 β = 2 2 j=1 e 2 2c j β j=1 e 2 we 2 2c j β. Since each character sends 5 to a different 1 c i 2 2 we have a su over a colete set of residues. Again by 2.17 the su is zero unless 2 2 β, and w = 0, giving that b 1 od 2. This brings us to a rather useful lea for icking out owers od. Lea For b such that b, = 1, if b is a kth ower od D, if is odd, χb = χ k =χ 0 2, kd, if = 2, 3, where k, φ, for odd, D = k, 2 2, for = 2, 3. If b is not a kth ower od χ k =χ 0 χb = 0. Using this Lea and observing that the nuber of x s that give the sae value as x k 18

27 is D or k, 2D if = 2, we can ick off kth owers in the following anner: χgx k e fx k = χ D 1 =χ 0 u=1 = u=1 χ D 1 =χ 0 χ 1 u χgue fu χ 1 uχgue fu, which will becoe very useful for writing our sus in ters of Gauss sus in Chater 4. Proof. We have seen there are exactly φ characters od. We will show that D of these characters are kth owers. For odd we have a riitive root a od and we can write any character χa = e φ c, 1 c φ. Thus if χ is a kth ower of soe character χ we have e φ c k = χ a k = χa = e φ c for soe c. Thus we are solving for c in the congruence c c k od φ which has D = k, φ solutions when D c. Therefore there are exactly D characters such that χ k = χ D = χ 0, naely the characters with c such that c = yφ /D for y = 1,..., D. If b is a kth ower od χ D =χ 0 χb = D. If b is not a kth ower od then b = a β where D β, giving χb = χ D =χ 0 χ D =χ 0 χa β = D y=1 e φ 19 yβφ = D D yβ e = 0 D y=1

28 by If = 2, 3 we have that the characters are defined by χ 1 = e 2 c 0, 1 c 0 2, and χ5 = e 2 2c, 1 c 2 2. Thus we have kth ower characters for the k, 2 2 = D solutions to c c k od 2 2 when D c, along with the 2, k solutions to c 0 c 0k od 2 when 2, k c 0. If k, 2 = 1 then D = 1 and there is only the rincial character with χ D = χ 0, if k, 2 = 2 there are 2D characters with this roerty. Thus if b is a kth ower od 2 2D, if k, 2 > 1, χb = χ D =χ 0 1, if k, 2 = 1. If b is not a kth ower then b = 1 w 5 β where k, 2 w and D β, giving k,2 χb = χ 1 w χ5 β = e 2 xw χ D =χ 0 χ D =χ 0 k,2 D yβ = e 2 xw e = 0. D y=1 D y=1 e 2 2 yβ2 2 D by The Duality of the Gauss Su is another useful roerty given in the following lea. Lea If χ is a riitive character od j, j 1, then j y=1 χye jay = χagχ, j. Proof. For A this is lain fro y A 1 y. If A and j = 1 the su equals y=1 χy = 0. For j 2 as χ is riitive there exists a z 1 od j 1 with χz 1, there ust be 20

29 soe a b od j 1 with χa χb, and we can take z = ab 1 so, since Az A od j, j j j χye jay = χzye jazy = χz χye jay 2.21 y=1 y=1 y=1 and j y=1 χye jay = 0. An alternate way of showing this for j 2 and odd is writing y = a u+φj 1 v, for a a riitive root od, χa = e φ j c, u = 1,..., φ j 1, v = 1,..,, j y=1 χye jay = φ j 1 u=1 χa u e jaa u e cv = v=1 2.7 Reduction Method of Cochrane and Zheng In [4] Cochrane and Zheng establish a reduction ethod for evaluating exonential sus of the for Sχ, x, fx, = which was then generalized to sus of the for χxe fx 2.23 Sχ, gx, fx, = χgxe fx in [5] with g, f rational functions over Z. The ethod for evaluating 2.23 involves finding the set, A, of all nonzero residues od satisfying the congruence t 1 rxf x + c 0 od

30 with the integers r and c defined in 2.8 and 2.10, where t 1 rxf X + c. We write Sχ, x, f, = 1 χxe fx = S α, α=1 where for any integer α with α, S α = S α χ, x, f, := x α od χxe fx. Theore T. Cochrane, Z. Zheng [4]. Let be an odd rie, f be any olynoial over Z and t 1 be as above and t such that t f. Suose that t Then for any integer α with α we have 1. If α / A, S α χ, f, = If α is a critical oint of ultilicity ν 1 then t = t 1 and S α χ, x, f, ν t ν ν If α is a critical oint of ultilicity one then χα e S α χ, x, f, fα +t 2, If t is even, = χα e fα χ 2 A α G χ Q, +t 1 2, if t is odd, where α is the unique lifting of α to a solution of the congruence t Rcf x+c 0 od [ t+1/2], and A α 2α t f α + αf α od. In articular, we have equality in Here χ Q is the Legendre sybol 2.4 and so G χ Q, is the quadratic Gauss su discussed earlier, and R is the -adic integer R := 1 log1 + r. 22

31 When fx = nx k with k we have the twisted gauss su χxe nx k 2.26 and 2.24 takes the sile for rkx k + c 0 od, 2.27 we have that either 2.26 is zero or a su of 1, k S α sus, deending whether there is a solution to 2.27 or not. When the critical oints have ultilicity one the S α can be evaluated exlicitly. For exale if fx = x then as observed in Cochrane and Zheng [4, 9] the critical oint congruence is sily rx + c 0 od. For odd and 2, if χ is iriitive there is no critical oint and Sχ, x, = 0, while if χ is riitive there is one critical oint of ultilicity one and where x 2rc Sχ, x, = χα e α /2 ɛ, 2.28 denotes the Jacobi sybol, 1, if 1 od 4, ε := i, if 3 od 4, 2.29 and Rα c od [+1/2] A sall adjustent is needed in 2.30 in the case = = 3, see 5.15, and ore generally in [4, Theore 1.1iii] when = t = 3. The sae forula 2.28 occurs in Mauclaire [16] with α defined by χ1 + /2 = e /2 α when is even and χ1 + 23

32 1/ = e +1/2 α when is odd. Mauclaire also deals with the case = 2 in the second art of [16]. A variation of 2.28 was obtained by Odoni [19] see also Berndt, Evans and Willias [2, Theores ]. In Chater 3 we will evaluate the fx = x su using the reduction ethod but relacing -adic integer R with a slightly siler constant, as well as dealing with the case = 2. In the later chaters we will be evaluating sus with critical oints of ultilicity greater than one and obtaining exlicit evaluations. 24

33 Chater 3 Evaluation of Gauss Sus The od Gauss su is given as Gχ, = χxe x In this chater we will give an exlicit evaluation of the Gauss su for all, illustrating the reduction ethods of Cochrane and Zheng discussed in 2.7. By using the congruence relationshis in 2.4 we get an evaluation articularly useful for the exlicit evaluation of the general Jacobi su, in Chater 7. J nχ 1,..., χ k, = x 1 =1 χ 1 x 1 χ k x k, > n, x k =1 x 1 + +x k = n 3.1 Evaluation of the Gauss Su We shall need an exlicit evaluation of the od, 2, Gauss sus. The for we use coes fro alying the technique of Cochrane & Zheng [4] as forulated in [20]. For odd 25

34 this is essentially the sae as [5, 9] but for = 2 sees new. Variations can be found in Odoni [19] and Mauclaire [16] see also [2, Chater 1]. Theore Suose that χ is a od character with 2. If χ is iriitive, then Gχ, = 0. If χ is riitive, then Gχ, = 2 χ cr 1 j e cr 1 j 2 c 2rc ε, if 2, and 3 3, ω c, if = 2 and 5, 3.1 for any j when is odd and any j + 2 when = For the reaining cases Gχ, 27 = χ cr 1 j e3 3 10cR 1 j 2rc i, 3 and Gχ, 2 = 2 2 i, if = 2, ω 1 χ 1, if = 3, χ ce 16 c, if = Here x 1 denotes the inverse of x od, and r, R j and c are as in 2.8 and 2.10 or 2.12 and ω := e πi/4. It is iortant to note that although we are evaluating the Gauss su using an arbitrary R j during the course of the roof we get the evaluation 2rc ε, if 2, 3 3, Gχ, = 2 χ α e α 1, if = 2 and 5 is even, 1 i c, if = 2 and 5 is odd,

35 where α is a solution to c + R 2 x 0 od 2, 3.4 unless χ is iriitive in which case there is no solution to 3.4, relatively rie to and Gχ, = 0. The Gauss su evaluation of the theore becoes useful when evaluating 1.4 in Chater 7. Proof. For odd, let a be a riitive root of for all owers of. We can write Gχ, = χxe x = φ γ=1 χa γ e a γ. For an interval I 1 of length φ 2 it is clear that γ = φ 2 u + v with u I 2 = [ 1, 2 ], v I 1, runs through a colete set of residues od φ. Hence, Gχ, = v I 1 χa v u I 2 χa φ 2 u e a φ 2 u+v = v I 1 χa v u I 2 e 2 cue 1 + R 2 2 u a v. Observing 1 + R 2 2 u 1 + R 2 2 u od gives Gχ, = v I 1 χa v u I 2 e 2 cue 1 + R 2 2 ua v = v I 1 χa v e a v u I 2 e 2 cue R 2 2 ua v = χa v e a v e uc + R 2 2 a v. v I 1 u I 2 27

36 Noting that the u su is over a colete set of residues od 2 gives Gχ, = 0 unless c + R 2 a v 0 od 2, 3.5 has a solution α = a v 0. If c there is no solution and Gχ, = 0. When c, there exists a solutions when α cr 1 2 od 2. To silify our result we choose α to be a solution to the stronger congruence c + R J x 0 od J, 3.6 where J := 2, to satisfy 3.5. Given any two solutions, a v 0 and a v 1, to 3.5 we have a v 0 a v 1 od 2 or equivalently v 0 v 1 od φ 2. When is even = thus there can only be one solution in the range of v. Taking 2 2 I 1 to contain a v 0 gives the result for even. When is odd, given a solution a v 0, we have a v 0+yφ 2 for y = 1,..., are all the solutions in an interval of length φ 2. Taking I 1 to contain these solutions and letting L = 2 = 1 2 we get Gχ, = L χa v 0+yφ L e y=1 = L χa v 0 = L χa v 0 a v 0+yφ L χa yφl e a v R L L y y=1 e cy e 2 a v R L L y. y=1 28

37 As long as 3 we have 1 + R L L y = 1 + yr L L + y R 2 2 L 1 + y R 3 3 L R L L 2 1 R 2 L 1 y R 2 L 1 y 2 od Thus Gχ, = L χa v 0 e a v 0 = L χa v 0 e a v 0 e cy e 2 a v 0 R L L 2 1 RL 2 1 y RL 2 1 y 2 y=1 e yc + R 2 La v 0 e a v RL 2 1 y RL 2 1 y 2. y=1 3.7 We here note that R L R J R 2 L L 3 1 R 3 L 1 od J where the last ter is zero unless = 3, = 3. This can be seen fro 1 + R J J = a φj = a φl = 1 + R L L 1 + R L L + R 2 2 L 2L + R 3 3 L 3L 1 + J R L 2 1 R 2 L L R 3 L 1 od +1, ilying that R J R L 2 1 R 2 L L R 3 L 1 od +1 J. Using this congruence as well as the fact that a v 0 is a solution to the stronger characteristic 29

38 equation 3.6 we have e 2 yc + R La v 0 = e 2 yc + a v 0 R J R 2 L L 3 1 R 3 L 1 = e 2 ya v RL 2 L 3 1 RL 3 1 = e ya v RL RL Thus the y su becoes, when not in the secial case = 3, = 3 y=1 e ya v RL 2 1 e a v RL 2 1 y RL 2 1 y 2 = e a v RLy 2 2 y=1 a v RL 2 = e y 2 y=1 2cr = e y 2 2cr = G χ Q,. y=1 Here G χ Q,. is the quadratic gauss su which faously sus to or i as 1 or 3 od 4. Note that for a solution, a v 0, to the equation c + R J x 0 od J, by 2.9 we ay take a v 0 = cr 1 J cr 1 j od J for any j J and cr 1 j will be a solution as well. This together with 3.7 gives the result for odd excet when = = 3. 30

39 When = 3, = 3 we get the y su 3 e 27 ya v RL9 2 RL9 3 e 27 a v RL9y RL9y 2 2 y=1 = = 3 e 3 a v RLy 2 2 RLa 3 v 0 y y=1 3 e 3 a v y 2 + R L y = e 3 a v e 3 a v y R l 2 y=1 = e 3 a v 0 3 w=1 coleting the odd case. e 3 a v 0 w 2 = 3 1/2 2rc 3 y=1 e 3 cr 1 2 i For = 2, 6 siilarly write the su in ters of the generators 1 and 5 giving, Gχ, 2 = 2 χxe 2 x = A { 1,1} 2 2 γ=1 χa5 γ e 2 A5 γ. We let γ = u v where v I 1 and u I 2 where I 1 is an interval of length and I 2 = [ 1, 2 2 ]. Thus after silification siilar to the odd case we have Gχ, 2 = A { 1,1} = A { 1,1} = A { 1,1} = A { 1,1} χa5 v e 2 2 cu e 2 v I 1 u I 2 χa5 v e 2 2 cu e 2 v I 1 u I 2 χa5 v e 2 A5 v e 2 2 cu e 2 v I 1 u I 2 χa5 v e 2 A5 v e 2 2 v I 1 u I 2 A5 v 1 + R u A5 v + A5 v ur A5 v ur u c + A5 v R 2. So Gχ, 2 = 0 unless we have a solution to either the A = 1 or 1 characteristic equation c + A5 v R 2 0 od

40 Notice that both c ± 5 v R 2 0 od 2 2 cannot be siultaneously satisfied as 5 v1 5 v 2 od 2 2 so cr 1 2 is either 5v or 5 v for soe v. Clearly we can have no solution to equation 3.8 if 2 c, thus χ ust be a riitive character. For the sake of silification we take our solutions to be solutions of the stronger characteristic equation c + A5 v R 2 0 od For two solutions A 0 5 v 0 and A 0 5 v 1, we have 5 v 0 5 v 1 od 2 2. Thus v 0 v 1 od 2 2 2, which is recisely the length of I 1 when is even. Taking I 1 to contain this solution we get Gχ, 2 = 2 /2 χ A 0 5 v 0 e 2 A 0 5 v For odd we take I 2 to contain the two solutions, A 0 5 v 0 and A 0 5 v giving Gχ, 2 = 2 2 χa5 v 0 e 2 A 0 5 v 0 + χ A 0 5 v e 2 A 0 5 v = 2 2 χa 0 5 v 0 e 2 A 0 5 v 0 + χ A 0 5 v e 2 A 0 5 v = 2 2 χa 0 5 v 0 e 2 A 0 5 v χ e 2 A 0 5 v A 0 5 v 0 = 2 2 χa 0 5 v 0 e 2 A 0 5 v χ e 2 = 2 2 χa 0 5 v 0 e 2 2 A 05 v 0 A 0 5 v 0 R e 2 2 c + A 0 5 v 0 R 2. We know fro 2.13 that R 2 R R 2 2 od Couled with the 32

41 solution to the stronger characteristic equation fro 3.9 we get c + A 0 5 v 0 R 2 = c + A 0 5 v 0 R R 2 2 A 0 5 v od 2 2 since R 2 j 1 od 4 for j 2. Using this gives e 2 c + A v 0 R 2 = e 2 2 A 0 5 v 0 = e 2 2 cr 1 2 = i c, as R j 1 od 4 for any j 3. Thus for = 2 we have 1, if is even, Gχ, = 2 /2 χαe 2 α 1 i c 2, if is odd, 3.11 where α is a solution to c + R J x 0 od 2 J, 3.12 and zero if there is no solution or χ is iriitive. If 2 c and j J + 2 then using 2.14 and R j 1 od 4, j 3 we can take α cr 1 J cr j + 2 J 1 1 cr 1 j 2 J 1 od 2 J+1, and χαe 2 α = χ cr 1 j e 2 cr 1 j χ1 R j 2 J 1 e 2 c2 J 1, where, checking the four ossible c od 8, 1 i c 2 = ω c. 2 c 33

42 Now e 2 c2 J 1 = e 2 2c2 J 3 = χ 5 2J 3 = χ 1 + R J 1 2 J 1, where, since R j R J 1 2 J 2 od 2 J+1, 1 Rj 2 J R J 1 2 J 1 = 1 + R J 1 R j 2 J 1 R j R J 1 2 2J J 3 + R J 1 2 2J R 2J 3 2 2J 3 od 2. Hence χ1 R j 2 J 1 e 2 c2 J 1 = χ 5 22J 5 = e 2 2c2 2J 5 = ω c, ω 2c, if is even, if is odd. One can check nuerically that the forula still holds for the 2 2 riitive od 2 characters when = 5. For = 2, 3, 4 one has 3.2 instead of 2iω, ω 2, 2 2 χce 2 4cω c so our forula 3.1 requires an extra factor ω 1, ω 1 χ 1 or χ 1ω 2c resectively. 34

43 Chater 4 Rewriting Sus in Ters of Gauss Sus to Obtain Weil and Weil Tye Bounds For a general ixed exonential su of the for Sχ, gx, fx, = χgxe fx with f, g rational with the oles of g oitted, a rather well known result of Weil [24] is the uer bound on such sus. If f is a olynoial and the su is non-degenerate then Sχ, gx, fx, degf + l 1 1/2, 4.1 where l denotes the nuber of zeros and oles of g see Castro & Moreno [3] or Cochrane & Pinner [8] for a treatent of the general case. Here we are dealing with secial sus that can be written in ters of Gauss sus which can be used to give the Weil bound in the od case and Weil tye bounds for general rie owers, which in certain cases are 35

44 shar. 4.1 Gauss Sus and Weil tye bounds For a character χ od j, j 1, we let Gχ, j denote the Gauss su j Gχ, j = χxe jx. Recall see for exale Section 1.6 of Berndt, Evans & Willias [2] that Gχ, j = j/2, if χ is riitive od j, 1, if χ = χ 0 and j = 1, 4.2 0, otherwise. For the classical od Gauss su, letting d = k, 1 we can write 1 1 e Ax k = 1 + e Ax d x=0 = 1 + = 1 + χ d =χ 0 u=1 1 χue Au χagχ, 1 χ d =χ 0 χ χ 0 = χagχ, χ d =χ 0 χ χ 0 by Lea 2.6.2, and Lea Fro 4.2 we get exactly the Weil bound of 1 e Ax k d x=0 36

45 In this chater we show the sus we are considering can all be written in ters of rie ower Gauss sus. 4.2 Twisted onoial sus as Gauss Sus In this section we write the twisted onoial Gauss S 1 = S 1 χ, x, nx k, = χxe nx k in ters of Gauss sus. Here the Weil bound coes fro writing S 1 as a su of k, φ when is odd and 2, kk, 2 2 when = 2 sus of absolute value, giving S1 χ, x, nx k, D /2, 4.3 when is odd, and S1 χ, x, nx k, 2, kd2 2, when = 2, 3, where k, φ, for odd, D = k, 2 2, for = 2, 3. By Lea S 1 is zero unless χ = χ k 1 for soe character χ 1 od, thus we can write S 1 = = χxe nx k = χ 1 x k e nx k, χ k 1xe nx k 37

46 and by Lea and Lea 2.6.1, S 1 = χ D 2 =χ 0 u=1 χ 1 χ 2 ue nu = = χ 1 χ 2 ngχ 1 χ 2,. χ D 2 =χ 0 χ D 2 =χ 0 u=1 χ 1 χ 2 nχ 1 χ 2 ue u When is odd, there are D = k, φ characters χ 2 where χ D 2 = χ 0, and so one iediately obtains a Weil tye bound Sχ, x, nx k, D / When = 2, 3 an additional factor of 2, k is needed by the fact that there are 2, kd characters with χ D 2 = χ Binoial Character Sus as Gauss Sus In this section we write the binoial characters su S 2 = S 2 χ 1, χ 2, Ax k + B = χ 1 xχ 2 Ax k + B in ters of Gauss sus. Fro Lea we know that this su is zero unless χ 1 = χ k 3 for soe character χ 3 od, in which case the su can be written as a su of k, φ od Jacobi like sus χ 5xχ 2 Ax + B and again be exressed in ters of Gauss sus. Theore Let be an odd rie. If χ 1, χ 2 are characters od with χ 2 riitive and χ 1 = χ k 3 for soe character χ 3 od, A, B Z with B and n and A are given by A = n A, 0 n <, A,

47 then χ 1 xχ 2 Ax k + B = n χ 4 X χ 3 χ 4 A χ 2 χ 3 χ 4 B Gχ 3χ 4, n Gχ 2 χ 3 χ 4,, 4.6 Gχ 2, where X denotes the od characters χ 4 with χ D 4 = χ 0, D = k, φ, such that χ 3 χ 4 is a od n character. We iediately obtain the Weil tye bound Sχ1, χ 2, Ax k + B, k, φ +n/ Before roving the Theore we note a nuber of secial cases. For = 1 and A this gives us the bound where 1 χ x l Ax k + B w d 1 2, d = k, For l = 0 we can slightly irove this for the colete su, 1 χax k + B d 1 1 2, 4.9 x=0 since, taking χ 1 = χ 3 = χ 0, χ 2 = χ, the χ 4 = χ 0 ter in Theore equals χb, the issing x = 0 ter in 4.9. These corresond to the classical Weil bound 4.1 after an aroriate change of variables to relace k by d. For t + 1 the bound 4.7 is d +n 2 +t, as we shall see in 6.12 we have equality in 4.7 for n + 2t + 2, but not for t + n + 1 < < 2t + n + 2. Notice that if k, φ = 1 and χ 1 = χ k 3 and, in case A for soe od n character, 39

48 χ 3, then we have the single χ 4 = χ 0 ter and Thus χ 1 xχ 2 Ax k + B = n χ 3 A χ 2 χ 3 B Gχ 3, n Gχ 2 χ 3,. Gχ 2, χ 1xχ 2 Ax k + B = +n/2 if χ 2, χ 2 χ 3 and χ 3 are riitive od and n. Noting that Gχ, = χ 1Gχ, this can be written Gχ 3, n G 2 χ 2, /Gχ 2 χ 3, and we lainly recover the for Jχ 1, χ 2, = Gχ 1, Gχ 2, Gχ 1 χ 2, 4.10 in that case. For the ultilicative analogue of the classical Kloosterann su, χ assued riitive and A, Theore gives a su of two ters of size /2, χax + x 1 = χ 3A Gχ3, 2 + χ Gχ, Q AG χ 3 χ Q, 2 when χ = χ 2 3 otherwise the su is zero, where χ Q here denotes the od extension of the Legendre sybol taking χ 2 = χ, χ 1 = χ, k = 2 we have D = 2 and χ 4 = χ 0 or χ Q. For = 1 this is Han Di s [10, Lea 1]. Cases where we can write the exonential su exlicitly in ters of Gauss sus are rare. Best known after the quadratic Gauss sus are erhas the Salié sus, evaluated by Salié [25] for = 1 see Willias [27],[28] or Mordell [17] for a short roof and Cochrane & Zheng [7, 5] for 2; for AB 1 χ Q xe Ax + Bx e 2γ + e 2γG χ Q,, = χ Q B 1 2 χ Q γe 2γ + χ Q γe 2γ, odd, even, if AB = γ 2 od, and zero if χ Q AB = 1. Cochrane & Zheng s 2 ethod works 40

49 with a general χ as long as their critical oint quadratic congruence does not have a reeated root, but forulae see lacking when = 1 and χ χ Q. For the Jacobsthal sus we get essentially Theores & of [2] 1 =1 k + B = 1 =0 k + B = B k 1 B j=0 k 1 j=1 χb 2j+1 Gχ2j+1, Gχ 2j+1 χ,, Gχ, χb 2j Gχ2j, Gχ 2j χ,, Gχ, when 1 od 2k and B, where χ denotes a od character of order 2k see also [13]. Proof of Theore Observe that if χ is a riitive character od j, j 1, then by the duality Lea 2.6.2, j y=1 χye jay = χagχ, j Hence if χ 2 is a riitive character od we have Gχ 2, χ 2 Ax k + B = y=1 χ 2 ye Ax k + By, and, since χ 1 = χ k 3 and D = k, φ, Gχ 2, χ 1 xχ 2 Ax k + B = = χ 3 x k y=1 χ 3 x D y=1 χ 2 ye Ax k + By χ 2 ye Ax D + By. 41

50 By Lea we have Gχ 2, χ 1 xχ 2 Ax k + B = χ 3 uχ 4 u χ D 4 =χ 0 u=1 y=1 = χ 2 ye By χ D 4 =χ 0 y=1 u=1 = χ 2 ye Au + By χ 2 χ 3 χ 4 ye By χ D 4 =χ 0 y=1 u=1 χ 3 χ 4 ue Auy χ 3 χ 4 ue Au. Since B we have y=1 χ 2 χ 3 χ 4 ye By = χ 2 χ 3 χ 4 BGχ 2 χ 3 χ 4,. If χ 3 χ 4 is a od n character then u=1 n χ 3 χ 4 ue Au = n u=1 χ 3 χ 4 ue na u = n χ 3 χ 4 A Gχ 3 χ 4, n. If χ 3 χ 4 is a riitive character od j for soe n < j then by Lea j χ 3 χ 4 ue Au = j χ 3 χ 4 ue j j n A u = 0, u=1 u=1 and the result follows. Notice that if n+2 then by 4.2 the set X can be further restricted to those χ 4 with χ 3 χ 4 riitive od n. Hence if t k, with n+t+2 and we write χ 3 a = e φ c 3, χ 4 a = e φ c 4 we have 1 t c 4, n c 3 + c 4, giving n c 3. Letting χ 1 = χ l = χ k 3, for soe od character, χ, this yields n+t c 3 k = c 1 = cl and n+t l. If n > 0, letting χ 2 = χ w, we deduce that t l + wk. Moreover when n = 0 reversing the roles of A and B 42

51 gives t l + wk. Hence when n + t + 2 we have Sχ 1, χ 2, Ax k + B, = 0 unless n+t l, t l + wk, 4.12 holds. For = n + t + 1 we siilarly still have n+t l. 4.4 The Generalized Jacobi Su as Gauss Sus Finally we show that the generalized Jacobi su J nχ 1,..., χ k, = x 1 =1 χ 1 x 1 χ k x k, > n 4.13 x k =1 x 1 + +x k = n can be exressed in ters of Gauss sus, a fact that will be central in our roof of Theore It is well known that the classical od Jacobi sus, Jχ 1, χ 2, = χ 1 xχ 2 1 x, 4.14 and their generalization to finite fields can be written in ters of Gauss sus see for exale Theore of [2] or Theore 5.21 of [14]. This extends to the od sus. For exale when χ 1, χ 2 and χ 1 χ 2 are riitive od Jχ 1, χ 2, = Gχ 1, Gχ 2,, 4.15 Gχ 1 χ 2, and Jχ 1, χ 2, = /2 see Lea 1 of [31] or [32]; the relationshi for Jacobi sus over ore general residue rings odulo rie owers can be found in [33]. Writing 4.15 in ters of Gauss sus is well known for the od sus and the corresonding result for 4.13 when n = 0 can be found, along with any other roerties of Jacobi sus, in Berndt, 43

52 R. J. Evans and K. S. Willias [2, Theore & Theore ] or Lidl-Niederreiter [14, Theore 5.21]. There the results are stated for sus over finite fields, F, so it is not surrising that such exressions exist in the less studied od case. When χ 1,..., χ k and χ 1 χ k are riitive, Zhang & Yao [30, Lea 3] for k = 2, and Zhang and Xu [32, Lea 1] for general k, showed that Jχ 1,..., χ k, = k i=1 Gχ i, Gχ 1... χ k, In Theore we obtain a siilar exansion for J nχ 1,..., χ k,, with > n. As we showed in Theore the od Gauss sus can be evaluated exlicitly using the ethod of Cochrane and Zheng [4] when 2. We shall need the counterart of 4.16 for the J nχ 1,..., χ k, along with the evaluation of the Gauss su fro Chater 3 in order to evaluate J nχ 1,..., χ k,. We state a less syetrical version to allow weaker assutions on the χ i : Theore Suose that χ 1,..., χ k are characters od with > n and χ k riitive od. If χ 1 χ k is a od n character, then J nχ 1,..., χ k, = n Gχ 1 χ k, n Gχ k, If χ 1 χ k is not a od n character, then J nχ 1,..., χ k, = 0. k 1 Gχ i, Fro the well known roerty of Gauss sus see for exale Section 1.6 of [2], i=1 j/2, if χ is riitive od j, Gχ, j = 1, if χ = χ 0 and j = 1, , otherwise, when χ 1 χ k is a riitive od n character and at least one of the χ i is a riitive 44

53 od character, we iediately obtain the syetric for J nχ 1,..., χ k, = k i=1 Gχ i, Gχ 1... χ k, n In articular we recover 4.16 under the sole assution that χ 1 χ k is a riitive od character. Proof. We first note that if χ is a riitive character od j, j 1, then by Lea j y=1 χye jay = χagχ, j. Hence if χ k is a riitive character od we have χ k 1Gχ k, = χ k 1 = = y=1 y y=1 y x 1 =1 x 1 =1 x k 1 =1 χ k ye n y x k 1 =1 χ 1 x 1... χ k 1 x k 1 χ k n x 1 x k 1 χ 1 x 1... χ k 1 x k 1 x 1 =1 χ 1... χ k ye n y = χ 1... χ k 1 y=1 y y=1 χ 1 x 1 e x 1 y x 1 =1 χ 1 x 1 e x 1 χ k ye n x 1 x k 1 y x k 1 =1 k 1 χ 1... χ k ye n y Gχ i,. i=1 χ k 1 x k 1 e x k 1 y x k 1 =1 χ k 1 x k 1 e x k 1 If > n and χ 1... χ k is a od n character, then y=1 y n χ 1... χ k ye n y = n y=1 y χ 1... χ k ye ny = n Gχ 1... χ k, n. 45

54 If χ 1... χ k is a riitive character od j with n < j, then by the sae reasoning as in Lea y=1 y j χ 1... χ k ye n y = j χ 1... χ k ye j j n y = 0, y=1 and the result follows on observing that Gχ, = χ 1Gχ,. 46

55 Chater 5 Evaluating the Twisted Monoial Sus Modulo Prie Powers We use the Cochrane and Zheng reduction ethod to show that the su S 1 χ, nx γt, = χxe nx γt has an exlict evaluation for sufficiently large. For a ultilicative character χ od q and fx Z[x] we define the twisted Gauss su Sχ, fx, q := q χxe q fx where e q x = e 2πix/q. We are concerned here with evaluating these sus when fx = nx k is a onoial and the odulus is a rie ower q = with 2. Obtaining satisfactory bounds, other than the Weil bound [24], reains a difficult roble when = 1 see for exale Heath-Brown and Konyagin [12]. For higher owers though, ethods of Cochrane and Zheng [4] can often be used to reduce the odulus of an exonential su and soeties evaluate the su exactly. 47

56 When the odulus q is squarefull, i.e. q 2 q, and 2nk, q = 1, Zhang and Liu [37] consider the fourth ower ean value of Sχ, nx k, q, averaged over the characters χ od q, and obtained χ od q Sχ, nx k, q 4 = qφ 2 q k, 1 2, 5.1 q their forula contains an additional factor when there are ries q with k, 1 = 1 due to an aarent iscount in their Lea 5. In the quadratic case, Sχ, nx 2, q, He and Zhang [29] obtain siilar exact exressions for the sixth and eighth ower eans when q is squarefull and corie to 2n, aking the conjecture, subsequently roved by Liu and Yang [34], that χ od q Sχ, nx 2, q 2l = 4 l 1ωq q l 1 φ 2 q, ωq := q 1, 5.2 for any integer l 2. Siilarly Guo Xiaoyan and Wang Tingting [26] consider ower eans averaged over the araeter n for quadratic and cubic sus, again q squarefull with 2n, q = 1 and 6n, q = 1 resectively, showing that for any real l 0, q n=1 n,q=1 Sχ, nx 2, q 2l = 2 2l 1ωq q l φq, 5.3 when χ is the square of a riitive character od q and zero otherwise, and q n=1 n,q=1 Sχ, nx 3, q 4 = 27 ω 1q q 2 φq, ω 1 q := q 3 1 1, 5.4 when χ is the cube of a riitive character od q and zero otherwise. These average results all generalize to arbitrary onoials nx k and arbitrary real ower eans as we show in Corollary below. 48

57 Actually, the ethods in Cochrane and Zheng [4] can be used to evaluate the individual sus Sχ, nx k, directly when 2,, 2nk = 1, with no need to average. Moreover due to the straightforward relationshi between the α satisfying 2.27 for general fx = nx k, 2nk, = 1, 2, the S α arising in Cochrane and Zheng s ethod will, with a little work, silify down to a single ter of odulus k, 1 /2. A fact that is just a secial case of our ain Theore of this chater, Theore When k, though, the critical oints are ultile roots so one has to do ore work. However we show here that Cochrane and Zheng s ethod can be adjusted to deal with the case k. Additionally our aroach reduces to finding a single solution of a certain characteristic equation 5.13 or 5.14, avoiding the need to su as with the original S α. Working od we write fx = nx γt, γn, 5.5 and define d = γ, Analogous to the squarefull condition in [26], [29], [34] and [37] we shall assue that t Theore Let be an odd rie, χ be a character od and suose that 5.5 and 5.7 hold. If χ is the d t -th ower of a riitive character od and an aroriate characteristic equation 5.13 or 5.14 has a solution then S 1 χ, nx γt, = d τ 49

58 where 1, if t + 1 < 2t + 2, τ = t, if 2t Otherwise S 1 χ, nx γt, = 0. Theore is an iediate consequence of our ain Theore 5.1.1, where we state an exlicit forula for S 1 χ, nx γt,. The corresonding result for = 2 is given in Averaging over the n or χ we iediately obtain: Corollary Under the sae hyotheses of Theore 5.0.2, for any real b > 0, χ od S 1 χ, nx γt and when χ is a d t -th ower of a riitive character od,, b = d τ b φ2 d 2 ax{ 2t 2,0}, 5.9 n=1 n,=1 S 1 χ, nx γt, b = d τ b φ d ax{ t 1, t+1} The corresonding results for coosite oduli including then follow iediately fro the ultilicativity discussed in Section 2.2. Since Theore shows that the S 1 χ, nx γt, can assue only one nonzero value, ower eans are soewhat artificial here, with 5.9 and 5.10 aounting only to a count on the nuber of non-zero cases we include the for coarison with results in the literature and to ehasize that the restriction to certain integer ower eans is unnecessary. The condition 5.7 is aroriate here. For t the exonent reduces by Euler s Theore and as shown in the roof of Theore 2.1 see 5.22 when = t + 1 the su is zero unless χ is a od character, in which case it reduces to a Heilbronn tye od su 1 Sχ, nx γ 1, = 1 χxe nx γ 1. 50