SMALL ZEROS OF QUADRATIC CONGRUENCES TO A PRIME POWER MODULUS

Transcription

1 SMALL ZEROS OF QUADRATIC CONGRUENCES TO A PRIME POWER MODULUS b ALI HAFIZ MAWDAH HAKAMI S, Kg Abdulazz Uverst, Saud Araba, 996 MS, Kasas State Uverst, USA, 004 AN ASTRACT OF A DISSERTATION subtted artal fulfllet of the requreets for the degree DOCTOR OF PHILOSOPHY Deartet of Matheatcs College of Arts ad Sceces KANSAS STATE UNIVERSITY Mahatta, Kasas 009

2 ASTRACT Let be a ostve teger, be a odd re, ad Z = Z /( ) be the rg of tegers odulo Let Q( ) = Q(,,, ) = a, be a quadratc for wth teger coeffcets Suose that s eve / ad detaq / 0 (od ) Set = (( ) det AQ / ), where ( / ) s the Legedre sbol ad cogruece cotaed = a Let V be the set of solutos the Q( ) 0 (od ), (#) Z ad let be a bo of ots = { Z a < a+ } where a, Z,,, Z of the te I ths dssertato we use the ethod of eoetal sus to vestgate how large the cardalt of the bo ust be order to guaratee that there ests a soluto of (#) I artcular we wll focus o cubes (all equal) cetered at the org order to obta rtve solutos wth sall For = ad 4 we obta a rtve soluto wth a{ 5, 8 } For / (/) + (/ ) (+ 4)/( ) 6, ad =+, we get a{, } =, Fall for a,, ad a osgular quadratc for we obta / [(/) + (/ )] ( + )/( ) /( ) a{6, } Others results are obtaed for boes wth sdes of arbtrar legths

3 SMALL ZEROS OF QUADRATIC CONGRUENCES TO A PRIME POWER MODULUS b ALI HAFIZ MAWDAH HAKAMI S, Kg Abdulazz Uverst, Saud Araba, 996 MS, Kasas State Uverst, USA, 004 A DISSERTATION subtted artal fulfllet of the requreets for the degree DOCTOR OF PHILOSOPHY Deartet of Matheatcs College of Arts ad Sceces KANSAS STATE UNIVERSITY Mahatta, Kasas 009 Aroved b: Maor Professor Todd Cochrae

4 Corght ALI HAFIZ MAWDAH HAKAMI 009

5 ASTRACT Let be a ostve teger, be a odd re, ad Z = Z /( ) be the rg of tegers odulo Let Q( ) = Q(,,, ) = a, be a quadratc for wth teger coeffcets Suose that s eve / ad detaq / 0 (od ) Set = (( ) det AQ / ), where ( / ) s the Legedre sbol ad cogruece cotaed = a Let V be the set of solutos the Q( ) 0 (od ), (#) Z ad let be a bo of ots = { Z a < a+ } where a, Z,,, Z of the te I ths dssertato we use the ethod of eoetal sus to vestgate how large the cardalt of the bo ust be order to guaratee that there ests a soluto of (#) I artcular we wll focus o cubes (all equal) cetered at the org order to obta rtve solutos wth sall For = ad 4 we obta a rtve soluto wth a{ 5, 8 } For / (/) + (/ ) (+ 4)/( ) 6, ad =+, we get a{, } =, Fall for a,, ad a osgular quadratc for we obta / [(/) + (/ )] ( + )/( ) /( ) a{6, } Others results are obtaed for boes wth sdes of arbtrar legths

6 TALE OF CONTENTS Lst of fgures Ackowledgeet Dedcato Itroducto ad Deftos 0 Defto of " sall rtve soluto" of a quadratc for odulo M 0 Hstorcal backgroud 0 Thess orgazato ad stateet of results Chater : Prelares 5 A bref stud of quadratc fors over the fte feld Z 5 Overvew 5 Method of roof 6 asc roertes of fte Fourer seres 8 A Suar of Cochrae s techque 9 Deterato of φ( V, ) odulo 9 Fudaetal dett 0 Sall solutos of the quadratc cogruece Q( ) 0 (od ) 0 A Sall roveet of Cochrae s estate 4 Eoetal sus odulo 5 asc results o quadratc fors odulo 5 v

7 5 Notos ad deftos 5 5 Dagoolzatato of quadratc fors odulo 7 Chater : Sall Zero of Quadratc Fors Modulo Itroducto Deterato of φ( V, ) odulo 4 Calculatg the su v S( f, ) 4 Evaluatg φ( V, ) for the case of a dagoal quadratc for 6 The su S λ 7 4 Forula for φ( V, ) for dagoal fors 8 5 Deterato of φ( V, ) for a geeral quadratc for 9 Sall solutos of the quadratc cogruece Q( ) 0 (od ) The fudaetal dett Uer bouds o ouds for the error ters the V fudaetal dett (od ) whe =+ 9 4 ouds for the error ters the fudaetal dett (od ) whe = 47 Chater : Sall Zeros of Quadratc Fors Modulo 55 Itroducto 55 Estatg V 56 ouds for the error ters the fudaetal dett (od ) The case of =+ 66

8 4 Secal cases that s a cube, = ouds for the error ters the fudaetal dett (od ) The case of = 75 6 Secal cases that s a cube, = 85 7 The a result 86 Chater 4: Sall Zeros of Quadratc Fors Modulo 87 4 Itroducto 87 4 Deterato of φ( V, ) odulo 88 4 Calculatg the su S( f, ) 88 4 Evaluatg φ( V, ) for the case of a dagoal quadratc for 90 4 The su S λ 9 44 Forula for φ( V, ) 9 4 Sall solutos of the quadratc cogruece Q( ) 0 (od ) 9 4 The fudaetal dett 9 4 Aular Lea o estatg the su = a( ) 95 4 ouds for the error ters of the fudaetal dett (od ) for the case of cube The a results 97 blograh 00 v

9 LIST OF FIGURES Fgure 4

10 ACKNOWLEDGMENTS All rase to Alght Allah the ost gracous ad ercful b whose grace ths research has bee coleted Frst, I a deel debted to advsor, Professor Todd Cochrae, for hs advce, ecourageet ad gudace He re-read a versos of ths thess, ad ever other research aer I have wrtte to date, ad rovded e wth coutless hours of hs te I cosder self ver fortuate to have had h as advsor He has had a studets, ad all that I have soke to are agreeet that he s a heoeal advsor It s hublg to kow that we ca ever ossbl equal h He otvates hs studets to work hard ad to go farther tha the ever thought the were caable of We ca ol tr to eulate h I would lke to thak the ebers of cottee To Professor Chrstoher Per for hs suort, for the a valuable dscussos ad for hs coets o the wrtg of ths dssertato I would also lke to thak Professor Adrew eett who was struetal gettg e to ths ot Ma thaks to Professor Davd Alle for hs terest work I would also lke to thak Professor Wlla Du for servg as char for fal ea I wat to eress secal thaks to the Deartet of Matheatcs at Kasas State Uverst, ad esecall to Professor Lous Pgo, the Deartet Head who gave e the rght leads ad advce ad for helg e throughout sta here I caot hel but eto how thakful I a to the reset Drector of Graduate Studes, Professor Davd Yetter, the Graduate Studes Secretar Haah Daveort, ad others for ther fredsh, ecourageet ad strog suort

11 I wat to thak fal for belevg e ad suortg e throughout work artcular wfe, Ea, for beg better half Also I wat to eress thaks ad arecato to brothers for all the have doe for e cludg hel, suort, ad atece durg log oure of stud I have to aologze to sos Halah ad Ibrahee for havg derved the of te whch orall would have bee devoted to the ad to fal lfe I would lke to thak Kg Khald Uverst Saud Araba for rovdg facal suort throughout graduate studes The thaks eted to all the eole who have heled e durg stud the Uted States, esecall the eole the Saud Cultural Msso to the USA A secal thaks go to brother Dr Abdullah Alwald (Drector of e-learg Ceter - Kg Khald Uverst) for the real fredsh ad suort I receved fro h durg all the te I set the USA Last, but ot least, I wsh gve a secal thak ou to the two eole who have stood behd e as a source of uwaverg suort, Mo ad Dad

12 DEDICATION To lovg arets, wfe, chldre, ad brothers I could ot have coleted stud wthout ther love, suort ad ecourageet

13 Itroducto ad Deftos The a urose of ths thess s to fd sall rtve solutos of a quadratc cogruece odulo, The hstorcal backgroud of ths subect s gve 0 below, but frst we gve the defto of a sall rtve soluto of quadratc for odulo a ostve teger M 0 Defto of "sall rtve soluto" of Let a quadratc for odulo M Q( ) Q(,,, ) a, = = be a quadratc for over Z ad M be a ostve teger Cosder the cogruece the or of a ot we ea Q( ) 0 (od M ) (0) = a ad b "sall soluto" we ea a ozero tegral soluto of (0) such that < M δ for soe ostve costat δ < The costat δ a deed o, but ot o M Our terest s fdg a rtve soluto of (0), a soluto wth gcd(,,, M ) =, that s, for a re dvsor of M, for soe Eale: Let Q( )= + + The t s clear that a ozero soluto of (0) ust satsf, a / M Thus / δ= s

14 the best ossble eoet for a sall soluto Our terest s rtve solutos to rule out trval sall solutos such as (,,, ) wth M = 0 Hstorcal backgroud Let Q( ) = Q(,, ) be a quadratc for over Z ad M be a ostve teger As we etoed we are lookg for otrval solutos of (0) wth δ < M for soe δ / Schzel, Schlckew ad Schdt (97, [7]) roved that (0) has a ozero soluto wth < (/ ) / ( ) M + for a ε > 0 we get a ozero soluto of (0) wth Thus for /+ M ε rovded s suffcetl large We ote that the soluto obtaed b ther ethod s ot ecessarl a rtve soluto Ideed, whe M= the use the trval soluto (,0,0,,0) Now let M=, wth a odd re, ad cosder fdg sall solutos to the quadratc cogruece wth Q( ) 0 (od ) (0) Heath-row (985, [5]) obtaed a ozero soluto of (0) / log for 4 (That s < < C / log for soe costat C ) Hs result was a roveet o the result of [7] ths case Wag Yua (988, [8]) geeralzed Heath-row s work to all fte felds 9 / 6 Cochrae (990, [8]) roved ths to < a{, 0} That s I a secal cases, t s kow that there ests a ozero soluto of (0) wth < /, for stace, whe ( Q ) = 0 or (987, [4]) Here ( Q ) s defed as followg / ( ) ( ) det Q / f det Q, ( ) = Q 0 f det Q,

15 where ( / ) deotes the Legedre sbol ad detq s the deterat of the atr reresetg Q ; see secto 6 We also get / < whe Q s of the for Q(, ) + Q(, 4) (989, Cochrae [5]), ad whe Q s a quadratc for wth > 4log + (989, Cochrae [5]) Wag Yua (989, [9]) oce ore has geeralzed Cochrae s work, to arbtrar fte felds If M= q, a roduct of two dstct res, we are seekg a soluto of the cogruece, We fd: Heath-row (99, [6]): Cochrae (995, [0]): Q( ) 0 (od q ) shareed the result of Heath-row /+ M ε, for > 4 / M, for > 4 ad ε > 0 Aga ths result 0 Thess orgazato ad stateet of results The outle of ths thess s as follows I Chater we stud brefl the dstrbuto of solutos to quadratc fors over Z ad gve the basc tools that we eed for our work I addto, we cocetrate o the ke deas of the techque of Cochrae for fdg a sall soluto of (0), whch aouts to fdg tegral solutos cotaed a sall bo cetered about the org We gve a sall roveet of the costat hs estate Theore 0: [Theore, ] For a quadratc for Q( ) wth 4 ad a re, there ests a rtve soluto of (0) wth < { /, 9 / } At the ed of ths chater we gve a quck look at eoetal sus ad the dagoalzato of a quadratc for over Z

16 I Chater we geeralze (od ) ethods for obtag a sall rtve soluto of the cogruece We show: Q( ) 0 (od ) (0) Theore 0 [Theore, 4] For a odd re ad osgular quadratc for Q( ) wth 4, there ests a rtve soluto of (0) wth 5 8 {, } a for = +, a for = 5 9 {,} Note that ths boud s best ossble (order I Chater, we stud the cogruece We rove: Q ( ) 0 (od ) / ( ) ) u to a costat (04) Theore 0 [Theore 5, 86] For a quadratc for Q( ) wth 4 ad a odd re, there ests a rtve soluto of (04) wth 7 / 8 {, } a for = +, / (/) (/ ) ( 4)/( ) a { +, + } for = Whe =, we have a best ossble te boud Fall, Chater 4, we address the questo of fdg sall solutos of for arbtrar re owers Q( ) 0 (od ), (05) We establsh: Theore 04 [Theore 4, 88] For a quadratc for Q( ) wth 4 ad a odd re, there ests a rtve soluto of (05) wth / [(/) + (/ )] ( + )/( ) /( ) a{6, } 4

17 Chater Prelares I ths research we shall follow closel the ethod of Cochrae [8] for fdg sall zeros of quadratc fors odulo Thus we shall gve a suar of the ke deas of that ethod I the followg sectos we shall establsh aalogous results but for odulo, Also we wll gve a sall roveet of hs result for od (see Theore ths chater) A bref stud of quadratc fors over the fte feld Z The a of ths secto s to revew the ost ortat cocets that wll be eeded our work, o the dstrbuto of zeros of a quadratc for over Z the fte feld eleets, where s a re For ore detals the reader s referred to [], [6], [7] Overvew Let Q( ) Q(,,, ) a, = = be a quadratc for wth teger coeffcets -varables, V= V ( Q) the algebrac subset of Z defed b the equato Q ( ) = 0 Our terest s the roble of fdg ots V wth the varables restrcted to 5

18 a bo of the te { Z a < a + } = where a, Z, ad 0< < for,, detfed Z wth the set of tegers {0,,, }) (Here we have If V s the set of zeros of a " osgular" Quadratc for Q ( ) (see 5), the oe ca show that / V = + O^ ( log) h, () for a bo where the brackets are used to deote the cardalt of the set sde the brackets (see []) It s aaret fro () that V s oet rovded For a, ( /) + (log ) Z, we let deote the ordar dot roduct, π = = For a Z, let = ( ) / e e We use the abbrev- ato = Z for colete sus The ke gredet obtag the dett () s a ufor uer boud o the fucto e φ = ( ) for 0, (, ) V V V for = 0 Method of roof I order to show that Let α ( ) be a real valued fucto o () V s oet we ca roceed as follows Z such that α ( ) 0 for all ot If we ca show that α ( ) > 0 the t wll follow that V s oet Now α ( ) has a fte Fourer easo where for all Z Thus α ( ) = a( ) e ( ), a( ) = α( ) e ( ), 6

19 α( ) = a( ) e( ) = a( ) e ( ) Sce a( 0) = α ( ), we obta 7 = a( 0) V + a( ) e ( ) 0 α( ) = V α( ) + a( ) φ( V, ), () 0 where φ ( V, ) s defed b () A varato of () that s soetes ore useful s α ( ) = α( ) + a( ) φ( V, ), (4) whch s obtaed fro () b otcg that whece ( ) ( 0)[ (, 0) ] ( ) (, ) α = a φv + + a φv 0 = + a( 0) a( ) φ( V, ) V = φ( V, 0 ) +, Equato () ad (4) eress the " colete" su α V ( ) as a fracto of the "colete" su α ( ) lus a error ter I geeral V so that the fractos the two equatos are about the sae I fact, f V s defed b a " osgular" quadratc for Q ( ) the V = + O / ( ) (That s / φ( V, 0) ) To show that α ( ) s ostve, t suffces to show that the V error ter s saller absolute value tha the (ostve) a ter o the rght-had sde of () or (4) Oe tres to ake a otal choce of α( ) order to ze the error ter Secal cases of () ad (4) have aeared a uber of tes the lterature for dfferet tes of algebrac sets V ; Chalk [], Tetavae [4], ad Merso [] The frst case treated was to let α ( ) be the characterstc fucto χ S( ) of a subset S of Z, whece (4) gves rse to forulas of the te

20 = + Error (5) V S S Equato () s obtaed ths aer Partcular atteto has bee gve to the case where S=, a bo of ots Z Aother oular choce for α s let t be a covoluto of two characterstc fucto, α= χs χt for, cole valued fuctos defed o α( ), β ( ), wrtte αβ ( ), s defed b for S T Z We recall that f α( ), β ( ) are Z the the covoluto of α β( ) = α( u) β( u) = α( u) β( v ), u u+v= Z If we take α( ) = χ χ ( ) the t s clear fro the S defto that α( ) s the uber of was of eressg as a su s+ t wth s S ad t T Moreover ( S+ T) V s oet f ad ol f α V ( ) > 0 T asc roertes of fte Fourer seres We ake use of a uber of basc roertes of fte Fourer seres, whch are lsted below The are based o the orthogoalt relatosh, f = 0, e( ) = Z 0 f, 0 ad ca be routel checked Meawhle b vewg odule, the Gauss su 8 Z ( ) S ( Q, ) = e Q( ) +, s well defed whether we take Z or Z as a Z- Z Let α( ), β ( ) be cole valued fucto o Z wth Fourer easos α( ) = a( ) e ( ), β( ) = b( ) e ( ) The α β( ) = a( ) b( ) e ( ) (6) αβ( ) = α( ) β( ) = ( ab)( ) e ( ) (7)

21 ( )( ) ( α β)( ) = α( ) β( ) (8) ( )( ) ( αβ)( ) α( ) β( ) (9) The last dett s Parseval s equalt a( ) = α( ) (0) A suar of Cochrae s techque I ths secto we gve a suar of the strateg that Cochrae follows to fd a sall soluto of a quadratc for odulo (for ore detals see [5], [6], [8]) Let Q( ) = Q(,, ) be a quadratc for wth teger coeffcets ad be a odd re Set = a Let V= V ( Q) be the set of zeros of Q cotaed Z Whe s eve we let / (( ) det Q / ) f det Q, ( Q) = 0 f det Q, where ( / ) s the Legedre sbol We outle the ke deas of Cochrae s techque for obtag sall solutos of the case whe s eve Q( ) 0 (od ), () Deterato of φ ( V,) odulo Usg dettes for the Gauss su 9 S= e a + b, oe obtas = ( ) Lea [see eg[5], Lea ]Whe eve ad =±, / ( ) f Q ( ) = 0, φ( V, ) = / f Q ( ) 0, where Q s the quadratc for assocated wth the verse of the atr for Q od

22 Fudaetal dett Recall, (4) we saw the dett α α φ ( ) = ( ) + a( ) ( V, ) 0 Isertg the value φ( V, ) Lea elds (see eg [5]), Lea [The fudaetal dett] Suose s eve For a α( ) o Z, ad a quadratc for ( ) 0 Q wth ( ) =±, Q / / ( ) ( ) ( ) α = α α 0 + a( ) V Q ( ) = 0 a ter error ters Sall solutos of the quadratc cogruece Q( ) 0 (od ) Let our set be a bo of ots of the te ad vew ths bo as a subset of = { Z : a < a +, }, Z ad let χ be ts characterstc fucto wth Fourer easo χ ( ) =a ( ) e ( ) The for a Z, s( π / ) a e a ` ( ) = ` +, s( π / ) = where the ter the roduct s take to be f = 0 We al the fudaetal dett wth α( ) = χ χ the covoluto of χ ad χ where, are boes such that two cases: + Now we have ) = I ths case we let be cetered at org ad take = = The the coeffcets a( ) are ostve reals, so the fudaetal dett gves us α( ) > α( ) α( ) 0 = ( /) ( /)

23 We see that α( ) > 0, rovded > Sce α s suorted o we have V φ /, that s > / ) = I ths case we eed to estate ( ) 0 a( ), but we do t sst o beg cetered at the org A ke tool for estatg the error ter ( ) 0 a( ) s a good uer boud o V Frst [8] establshes, Q Q =, the uber of solutos of () wth Lea [[8], Lea ] Let S be a closed star-shaed rego about the org R wth / < for all S [A rego of ots S R s sad to be star- shaed about the org f for a ot P S the le seget og Pad the org s cotaed S ] For 0< γ< let γs= { γ S} Let of a for varables over ZThe γ γs V + S V γ V Z be the set of zeros od The usg the fudaetal dett ad lea oe obtas Lea 4 [[8], Lea ] Suose that 4 s eve, ( ) = ad V= V( Q) Let be a bo of ots of the te = Z,, { } for soe oegatve tegers < /, Let t be a gve os- tve teger If < t, for, or > + ( + t) + / V + t = Q / t the A secod aeal to the fudaetal dett elds Theore [[8], Theore ] Suose that 4 s eve, ad that ( Q ) = If for, ad / > 0, 4+ 6

24 the cotas a ozero soluto of () The et theore follows fro Theore uo settg all but 4 varables equal to zero ad lettg be a cube cetered about the org Theore [[8], Theore ] For a quadratc for Q( ) wth 4 ad a re, there ests a ozero soluto of () wth { 9 /, 0 6 } < a, () A sall roveet of Cochrae s estate a lttle work, the boud () last secto ca be roved to the followg Theore For a quadratc for Q( ) wth 4 ad a re, there ests a ozero soluto of () wth { /, 9 / } < () Proof settg varables equal to zero, we a assue = 4 The boud [7] gves for, a ozero soluto of () wth Whe = 4 (4) gves /+ / f odd, (4) /+ /( ) f eve / (5) We cobe ths uer boud wth the boud of Theore The two uer bouds are grahed fgure below Observe that 9 / = 0 6 / = 0 6 = 6 0 =64 0 (6) O the other had, coarg (5) ad (), we have that ad / = 9 / /6 = 9 = 9 6 = 0 4, (7) / = 0 6 = / 0 6 / = 0 9 = (8)

25 So collectg (6), (7), (8), oe deduces that f use / < ad f Thus () follows 4 > we use 4 <, we 9 / < (see fgure ) 4 Eoetal sus odulo I order to roceed fro cogrueces (od ) to cogrueces (od ), we eed to geeralze results for eoetal sus Let Z = Z /( ) Aga we have the basc orthogoalt relatosh that for a Z, f = 0, e ( ) = (9) Z 0 f 0 We also wll use the followg lea a tes Let G( ) deote the ultlcatve grou of uts odulo Lea 5 Let λ, a Z For a odd re ad a ostve teger, λ G ( ) e, f a ( λ ) = a f a, 0 f a Proof The su λ G ( ) e( λa) be wrtte as follows:

26 Now, t s well kow that ( ) ( ) e λa = e λa e ( λa) λ G( ) λ= λ 0 f a, λ = e ( a) λ= (0) f a So for the secod su f we set λ= λ, we have (usg (0)) (0) 0 f a, e ( ) λa e ( λ a) e ( λ a) = = = f a λ Therefore, λ = λ = e 0 f a ( λ ) = a 0 f a f a λ G ( ) - Ths coletes the roof of Lea 5 where Let f be a oloal wth teger coeffcets ad let we defe = ( ), S( f, ) e f( ) 4 = s a re ower wth For a oloal f over Z ( ) t= t( f) : = ord f ( X), where f = f ( X) deotes the dervatve of f( X ) Also we defe the set of crtcal ots assocated wth the su S( f, ) to be the set of zeros of the cogruece A= A (, ) : = {,, }, f α α D t f ( ) 0 (od ), () where t= ord ( f ) For a α A let ν= να deote the ultlct of α as a zero of the cogruece () Theore 4 [[], Theore ]: Let be a odd re ad f be a o-costat oloal defed over Z If t+ the for a teger α we have:

27 () If α A the Sα ( f, ) = 0 () If α s a crtcal ot of ultlct ν the t /( ν+ ) ( ( /( ν+ )) S ( f, ) ν () α () If α s a crtcal ot of ultlct oe the ( + t)/ e ( f( α )) f t s eve, Sα( f, ) = ( + t )/ χ( A) e α ( f( α )) G f t s odd, where α s the uque lftg of α to a soluto of the cogruece t [( + t )/] t f ( ) 0 (od ), ad A f ( α )(od ) I artc- ular, we have equalt () Here G s the classcal Gauss su, f (od4), G = = : e( ) = f (od4), ad χ s the quadratc character od The roof of Theore 4 s gve [] α 5 asc results o quadratc fors odulo I ths secto we shall dscuss as backgroud for our work soe of the geeral roertes of quadratc fors over the rg Z = Z /( ), wth a odd re ad a ostve teger 5 Notatos ad Deftos Recall that a quadratc for Q( ) over Z s a oloal of the te Q( ) = Q(,,, ) = a, wth a Z, We assocate wth Q( ) a setrc atr A= AQ gve b a a a a a a a a A Q= a a a a a a a a 5

28 That s Observe that where Here A= [ a ], = a 6 a for <, = a for >, a for = t Q( ) = A t, = [ ] t deotes the trasose of the atr O the other had ote that f the atr A s dagoal (A atr A s dagoal f a = 0 wheever the dagoal reresetato ), the the corresodg quadratc for Q has t Q( ) = A= a + + a, e, the quadratc for wll cota o "cross roduct" ters I the sae wa we call Q a dagoal quadratc for (od ) for a re ower f Q cota o "cross roduct" ters whe read (od ) The deterat of Q, abbrevated detq, s defed to be the deterat of the atr A Q We sa that Q( ) s osgular over Z f detq 0 Slarl for a odd re ower Q( ) s osgular od Aga let f detq we sa be a odd re ower Let Q( ) ad Q ( ) be two quadratc fors over Z wth assocated atrces A, A resectvel We ow vew the etres of these atrces as eleets of Z /( ), ad regard / as the ultlcatve verse of (od ) (Alteratvel we ca relace ½ wth ( + ) ad regard A Q as havg teger etres) We sa that Q( ) s equvalet to Q ( ) (od ), wrtte Q( ) Q ( ) (od ), f there s a vertble atr T over Z /( ), such that Q ( ) Q( T) (od ), that s Q Q

29 t A T A T (od ) Q Q It s clear that " " s a equvalece relato Note that Eale: Let The where That s dett det det ( ) Q= Q (od ) be a odd re ower ad Q( ) = t A, A= Q( ) = + + = [ ] akg the sle observato that we ca wrte wth eve t A Q Q( ) = + + Q( ) = ( + ) + (od ), t Q( ) A (od ), + A = + M ( Z ) Note that sce s odd, the etres of A are all tegers Thus we a assue that A M ( ) Z whe workg wth cogrueces odulo odd res Q 5 Dagoalzato of quadratc fors odulo I ths subsecto we rove Theore 5 For a odd re ower 7, ad quadratc for Q( ) over Z, Q( ) s equvalet to a dagoal quadratc for (odulo )

30 5 Dagoalzato of quadratc fors odulo I ths subsecto we rove Theore 5 For a odd re ower 8, ad quadratc for Q( ) over Z, Q( ) s equvalet to a dagoal quadratc for (odulo ) Proof The theore s well kow ad a roof ca be foud for eale [5] We shall ol deal wth the case of osgular Q(od ), the te of for we deal wth ths thess We roceed b ducto o Whe =, t s well kow (see [9]) that Q ca be dagoalzed over the fte feld F Sa t T A T D (od ), Q for soe T, D M ( Z ) wth T osgular (od ) ad D a dag- oal atr Lets lft ths to a soluto U= T+ X, (od ) Let where X= [ ] s a atr of varables We wsh to solve t U AU D (od ) Ths s equvalet to Q t ( T+ X) A( T+ X) D (od ) t t t Q Q T AT+ T A X+ X A T D Q (od ) t T AQ T D t t + T A Q X+ X A Q T 0 (od ) t t D T AQ T Y+ Y (od ), t T AQ X Y t Q where Y= ad = ( D T A T)/ Note that s a setrc atr wth teger etres Let b b b Y b b b b b b b 0 (od )

31 The t Y+ Y = Thus we are left wth solvg the cogruece t T A X Y (od ) Q Sce T ad A Q are osgular (od ), ths equato has a uque soluto X A ( T ) Y(od ) t Q I the sae aer oe ca lft a soluto (od ), to (od ) + for a Ideed, roceedg as above, suose that t T AT D (od ), for soe T, D M ( Z ) wth T osgular (od ) ad D a dagoal atr Let U= T+ X, where X s a atr of varables ad solve Ths s equvalet to where t U AU D (od ) Q t ( T+ X) A( T+ X) D (od ) T AT T A X X AT D + t + t + t (od ) t T AT D t t 0 (od ) + T AX+ X AT Y t t D T AT Y+ Y (od ), Y t t = T AX ad = ( D T AT)/ s a setrc atr wth teger etres Let 9 0 β β β Y β β β (od ) β β β β t The Y+ Y = (We ote that the choce of Y s ot uque) Hece we are left wth solvg the cogruece t T AX Y (od ) As T ad A are osgular (od ), ths equato has a uque soluto

32 t X A T Y ( ) (od ) Ths coletes the ducto ste Eales: Let A Q(, ) = [ ] = + + Note that Q(, ) s alread a dagoal for whe read (od ) We roceed to dagoalze Q(, ) (od ) t D T AT = = Solve AX Y (od ), Check : T= I = 0 0, = 0 0 Y= =, 0 (od ) (od ) (od ) U= T+ X= t 0 0 U AU= Thus Q( ) + (od ) Let A Q(, ) = [ ] 0 = + (od ) What haes f A sgular? (od )

33 Here A s ot vertble, so we caot drectl follow the ethod gve our roof Let us tr to solve t T AX Y (od ) Frst, we see that T= I sce A s alread dagoal (od) Let ( + ) The + A= ad the latter atr has teger etres (od ), D A = = 0 0 = + 0 If we roceed as the roof we would let Now solve Ths s equvalet to Y= X (od ) + (od ) (od ) (od ), whch gve us a cotradcto(0 = ) ad hece there s o soluto of ths sste The Net, let us tr the choce Y Y t + = Y= α α 0 0 α 0 α 0 + α (od ) α 0 = (od )

34 Solve + 0 α X α 0 or, equvaletl α 0 0 α 0 Let α= The obvousl so that (od ), (od ), (od ) 0 X= 0 0 Hece, t follows oe ca ake the chage of varable to dagolze the quadratc for +, Q (, )(od ) Ideed, ( + ) ( + ) ( ) (od ) (od ) Our roof of Theore 5 actuall elds the stroger result Corollar If s a odd re, Q( ) s a quadratc for over Z, osgular (od ) ad equvalet to dagoal for a (od ), the Q( ) s equvalet to the sae dagoal for = a (od ) for a = Note: Ths fals for osgular fors Ideed but + / (od ) + (od ),

35 Chater Sall Zeros of Quadratc Fors Modulo Itroducto Let Q( ) = Q(,,, ) = a be a quadratc for wth teger coeffcets ad be a odd re Set = a Let V = V ( Q) be the set of zeros of Q cotaed Z Whe s eve we let / ( Q ) ( ) det A / f det AQ, ( Q) = 0 f det A, where ( / ) deotes the Legedre-Jacob sbol ad A Q s the defg atr for Q () For Z set where e ( ) for, 0 φ( V, ) = ( ) V for = 0, / e ( ) = e π Our goal ths chater s to geeralze (od ) ethods for obtag a sall rtve soluto of Q Recall b rtve we ea () ( ) 0 (od ) for soe sall we ea δ C( ), for soe δ>0 ad costat C Ideall, we would lke < Note, oe has trval ozero sall solutos of () such as Q

36 (,0,0,,0), but these solutos are ot rtve Our strateg s to frst calculate the Gauss su 4 ( λ ), () S= S( f, ) = e a + = ad the use ths su to calculate the fucto φ( V, ) After that we use ths calculato to fd a fudaetal dett aalogous to Lea Fall we roceed to boud the error ters the fudaetal dett We ca state our a result ths chater; see Corollar ad Corollar 5 Theore For a odd re ad osgular quadratc for Q( ) wth 4, eve, there ests a rtve soluto of () wth 5 8 a {, } for = +, () 5 9 a {,} for = (4) Deterato of φ ( V, ) odulo Calculatg the su S( f, ) We eed to use Theore 4 of Chater wth = The followg fudaetal lea holds Lea Let be a odd re, λ, a, Z wth a ad S be as () The e ( 4 aλ ) f λ, S= χ( λ a) G ( 4 ), e λ a f λ ad 0 f λ ad, where χ s Legedre Sbol, λ = λ, =, ad λ, λ, a are verses od Proof Assue that a We cosder two cases:

37 Case (): or, equvaletl λ The crtcal ot cogruece s where t ord ( f ) The clearl = t f ( ) 0 (od ), t ( λ a+ ) 0 (od ), (5) t ( aλ, ) t= 0, because aλ Thus b alg (Theore 4, art ()) we have t = 0= (eve) So turg back to (5), we ow have ad hece Thus where od λ a (od ), α= aλ (od ) ( )/ ( ( )) + t ( ) S = S = e f = e a + α α λα α, * α s the uque lftg of α to a soluto of (5) [/] verses = od We take od The α aλ (od ) where,, λ f( α ) = λα a + α λa( aλ) + ( ) aλ (od ) a are ad so Sα= e ( 4 aλ ) λa aλ aλ (od ) λa4 aλ (od ) 4 aλ (od ), Case (): λ, the aga the crtcal ot cogruece s as (5) Now f ) t =, t sce ( aλ, ), ( λ, ), ad a Thus b settg =, we have = + + ( λa ) = ( λ a ) S e e = = 5 = λ λ, ad

38 ( λ ) = χλ ( a) Ge` λ a = 4λ = e a + So S= χ( ) G e ( a ) λ a ) t= 0, 4 t because ( aλ, ), ad ( a, ) Returg, oce ore, to (5), we ow have or, equvaletl λ a (od ), 0 (od ), a cotradcto Thus there s o crtcal ot, so S = 0 Lea s coletel roved Evaluatg φv (, ) for the case of a dagoal quadratc for Assue that Q( ) == a, wth a 6, The t follows fro the orthogoalt roert of eoetal sus that Now, f = e = ae ( λq ) ke Z λ= 0 ( ) ( ) ( ) = e ( λq( ) + ) λ = e ( ) + e ( λq( ) + ) λ 0 S 0, the we have ( ) ( ) = + S S = = V 0 V V φ(, ) If 0 The, b (9) of Chater, sce soe 0, Also, = = = = = S e ( ) e ( ) 0 S e λq = ( ( ) + ) λ 0 S

39 ( λ ) = e ( a+ a+ + a) λ 0 Thus we ow have S = φ ( V, ) for all Heceforth we shall use φ ( V, ) to ea S ad vce versa ut frst we eed to treat the sde su S λ searatel The su S λ We rewrte S λ as follows: 7 S λ ([ ] [ ]) S e a a = λ λ λ = e ( λa + ) e ( λa + ) = e ( λa + ) = = Gauss su we shall ea that (6) for all ad vce versa As a cosequece of Lea we have the followg Lea Lea Let be eve ad let S λ be as (6) Assue a, a,, a The f λ, 0 f λ, ad for soe, * S e ( 4 Q ( )) f, λ= λ λ / * e( 4 λ Q ( )) f λ, ad for all, / where = χ( ( ) ) χ ( a a ) ad χ= χ Proof We wll dvde the roof to two cases accordg to whether dvdes λ or ot Case (): λ The, b Lea, for a, S = e ( 4 aλ ) e ( 4 aλ ) e ( 4 aλ ) λ = e ( 4aλ + 4aλ aλ) = e * ( λq ) 4 ( ), * * λ 4 λq (,, ) = 4 Q ( ),

40 where Q ( ) s the quadratc for assocated wth verse of the atr for Q od Case (): λ The we have the followg two subcases: ) for soe The certal, vew of Lea, S λ = 0 ) for all The aga b Lea, Sλ= χ ( λ a) G ( 4 λ ) χ ( λ ) ( 4 λ e a a G e a ) = G χλ ( a λ a ) e 4 λ Q ( ) / / * = χ ( ) χ( ) 4 λ ( a a e Q ) / * 8 s eve = e 4 λ Q ( ), ad ths coletes the roof of Lea 4 Forula for φ ( V, ) for dagoal fors Lea Whe s eve ad 0, The φ * f for soe ad Q * ( ), f for soe ad Q ( ), * ( V, ) = 0 f for soe ad Q ( ), ( /) + ( ) f for all ad Q ( ), ( /) ( ) + ( ) f for all ad Q ( ), where = Proof We have two cases: Case (): for soe The we frst use Lea, so obvousl f λ, we have S λ = 0 ad φ(, ) = V = Sλ+ Sλ Sλ λ λ λ λ 0 = e ( 4 λq ( )) = * λ e Q ( λ ( ) ) * λ,

41 sce 4λ rus through G( ) as λ does Net, we al Lea 5 of Chater to the last su, to get f Q ( ), f Q ( ), φ = * ( V, ) f Q ( ), = * f Q ( ), * * 0 f Q ( ) 0 f Q ( ) Case (): for all, (that s, 9 les that aga b Lea, f λ, S λ= / * e ( 4 λ ( Q )) f λ, where λ= λ Thus we ow have φ(, ) V = Sλ+ Sλ λ λ λ 0 / * ( 4 λ Q ( )) λ = λ = e + Q ( ) ) The f Q ( ) / 0 (od ) / = + ( ) f Q ( ) 0 (od ) It follows that / + ( ) f Q ( ), φ( V, ) = / ( ) + ( ) f Q ( ) Cobg cases () ad (), Lea follows 5 Deterato of φ ( V, ) for a geeral quadratc for I the last secto we calculated φ ( V,) for the case of dagoal quadratc fors Suose ow that Q () s a quadratc for Let V be the set of soluto of the quadratc cogruece Q( ) 0 (od ) Let =T( u ) where T s a trasforato that dagoalzes Q, so that Q( T( u)) = Q( u ), a dagoal quadratc for Let V be the set of soluto of the quadratc cogruece Q ( u ) 0(od ) Set

42 t T ( ) = v We frst show that φ (, ) = V φ (, v ) Note that, sce T s a osgular trasforato od, 0 (od ) s equvalet to v 0 (od ) If 0 (od ), the φ ( ) ( ) ( V, ) = = = φ( V, v ) V For / 0 (od ), we have φ( V, ) = e ( ) t = φ( V, T ( )) Q( ) 0(od ) Q( T( u)) 0(od ) Q ( u) 0(od ) u V V V = e ( ) = e ( T( u) ) t = e ( u T ( )) t = e ( u T ( )) = φ( V, v) t Sa Q( ) = A Q, where A Q s the assocated atr for Q The Ad ( ) ( ) ( ) Q( u) = Q T( u) = T( u) AQ T( u) = u T AQ T u t t ( T ( )) ( T ) t t t t t t Q ( ) Q T AQ ( T ) t v = = T ( ) = AQ = Q ( ) Thus b our result for dagoal fors we have for the orgal quadratc for that * ( V, ) = 0 f for soe ad Q ( ), ( /) + ( ) f for all ad Q ( ), φ * f for soe ad Q ( ), * f for soe ad Q ( ), ( /) ( ) ( ) f + for all ad Q ( ) (7) A Q 0

43 Sall solutos of the quadratc cogruece Q( ) 0 (od ) We start ths secto b fdg The fudaetal dett Let α( ) be a cole valued fucto defed o Z easo α ( ) = The Sce wth Fourer a( ) e ( ), where a( ) = α( ) e ( ) α( ) = a( ) e ( ) ( ) 0 ( ) = a( ) e ( ) = a( 0) V + a( ) e ( ) 0 a = α, we obta α( ) = V α( ) + a( ) φ( V, ), 0 where φ( V, ) as we defed Also b otcg that we obta that because V = φv + ( ) (, 0 ), α ( ) = α( ) + a( ) φ( V, ), (8) ( ) ( ) = ( 0)[ (, 0) + ] + ( ) (, ) α a φv a φv 0 = a 0 + a φv Now we ca rove: ( ) ( ) (, ) Lea 4 [The fudaetal dett] For a cole valued α( ) o Z α( ) = α( ) + a( ) a( ) Q ( ) Q ( ) ( /) ( /) a( ) + a( ), (od ) Q ( ) (od ) (9)

44 where = Ths lea s secal case of the geeral Lea 44 of Chater 4 Proof We shall use the abbrevato to ea for soe ad to ea for all α = α + φ Isertg (7) (8), ad slfg, we get ( ) ( ) a( ) ( V, ) (od ) Net, deote ( ) = α( ) + a( ) a( ), Q ( ), Q ( ) *, Q ( ) ( /) ( ( ) ) T + + a( ), Q ( ) T ( /) ( ( ) ( ) ) + + T= T+ T + T + T 4 The after soe aulato, T reduces to 4 T a( ) T= a a ( /) ( ) ( ) ( ) Q ( ) ( ) + ( ) Now, we ote that (0) the su ( /) a a *, Q ( ), Q ( ) T a, Q, Q, Q ( ) ( ) ( ) k (0) a( ) = a( ) a( ) () O the other had, we observe that () the su ad the su a = a a, Q ( ) Q ( ) a ( ) ( ) ( ) a a a, Q ( ) Q ( ) a k k, () ( ) = ( ) ( ) ()

45 So, b usg () ad (), the su () becoes a( ) = a( ) a( ) (4), Q ( ) Q ( ) Q ( ) Cosequetl, b usg (4), (0) becoes ( /) ( ) ( ) ( ) T= a a a Q ( ) Q ( ) = ( /) + a( ) a( ) + a( ) Q ( ) Q ( ) = Q ( ) ( /) = a( ) a( ) Q ( ) = Hece, t follows that ( /) a( ) + a( ) Q ( ) Q ( ) α = α + a a ( ) ( ) ( ) ( ) Q ( ) Q ( ) ( /) ( /) a( ) + a( ), = whch s the asserto of Lea 4 Uer bouds o Let be the bo of ots V Z gve b, Q ( ) { Z a < a+, } =, (5) where = q + r, 0 r< ad q, r Z Thus the uber of ots (cardalt of ) s = = Our terst ths secto s deterg the uber of solutos of Q( ) 0 (od ), (6) wth Frst we treat the case where all I ths case we ca vew the bo (5) as a subset of Z ad let χ be ts charact- erstc fucto wth Fourer easo χ ( ) = a ( ) e ( ) The for a Z, s π / ( ) =, s π / = a

46 ad so (b work of []), s( π / ) 4 4 a ( ) log log( ) ` + ` s( π / ) = π π (7) Lea 5 Let be a bo of te (5) cetered at the org wth all, ad V = V ( Q) deote to the set of solutos of (6) Z If =+, the Q / 4 a) V + ` log( ) π (8) b) V / ` + (9) Proof Sce Q=, the fudaetal dett (odulo) s / / α( ) = α( ) α( 0) + a( ), (0) V Q ( ) = 0 b Lea of Chater Lettg α( ) = χ ( ) = a ( ) e ( ) (0), we have ad so (usg 7), χ ( ) + 4 / a, ( ), / 4 V + ` log( ), π coletg the roof of art (a) For art (b) set α= χ χ, the covoluto of χ wth tself, e; α( ) = χ ( u) χ ( u) = χ ( u) χ ( v) u u+ v= = a ( ) e ( u ) a ( z) e ( z ( u)) u z = a ( ) a ( z) e ( z ) e ( u ( z)) z u a e = ( ) ( ), so that the Fourer coeffcets a( ) of α( ) are a ( ) Sce s cetered at the org the Fourer coeffcets a ( ) are all real Thus the coeffcets a( ) of χ χ are all ostve usg Parseval s dett,

47 ((0), Chater ), 5 a( ) = a ( ) = χ ( ) = () Also Net b (0), we observe that α / ( ) α( ) + a( ) / = ( χ χ) ( ) + a( ) The, usg the dett ((9), Chater ) ad (), the above s ( ) ( ) / ( ) ( ) ( ) α χ χ u v + u v = + / = + O the other had, for a, we cla that /, () α( ) = χ χ ( ) () To see ths, we shall argue as follows Let I= [ M, M ] be a terval setrc about 0 We eed frst to rove that for I, χiχi ( ) I = (M+ ) (4) To ths ed we have to cout the uber of ots ( u, v) I I such that u+ v= We have two cases If M 0, the the uber of ots s M+ +, secfcall = u+ ( u), M u + M Thus lal the total uber of the ots s greater tha or equal to M M+ = M+ I If 0< M, the we have M + ots, secfcall = u + ( u ), M u M, ad thus oce aga the total uber of the ots s greater tha or equal to M M+ = M+ I The two cases l (4) Thus t follows edatel that for I I=,

48 α( ) = χ χ ( ) I =, I I = = b (4) whch s () Now we retur to colete rovg the lea Fro () t follows that α( ) = V (5) Thus, uttg () ad (5) together ad slfg we coclude that V / ` + The lea s thereb roved Lea 5 s stated for boes cetered at the org I the et Lea we wll dro ths hothess ad rove the lea for arbtrar boes We wll get the sae result Lea 6 Let be a bo of te (5) wth all 6, ad V= V( Q) deote to the set of solutos of (6) Z If Q=+, the V / ` + (6) Proof Aga as Q=+, the fudaetal dett (odulo) s / / α( ) = α( ) α( 0) + a( ) (7) V Q ( ) = 0 Let α( ) = χ χ where = c The value c s chose such that s "earl" cetered at the org: c= a+ The α ( ) = =, (8) α( 0) =, (9) u v u+ v= 0

49 a( ) = a ( ) a ( ) 7 Thus, usg the Cauch-Schwartz equalt (see eg [8]) ad Parseval s dett, (0) of Chater, we obta a( ) = a ( ) a ( ) / / a ` a ( ) ` ( ) / / ( ) χ ( ) χ ` ` / / = = (0) Thus b the fudaetal dett (7) ad (8), (9), ad (0), f =+, Now we cla that α α + a / ( ) ( ) ( ) Q ( ) = 0 / α( ) + a( ) + / () α( ) = V () To see (), we are gog to argue as follows Let The f s odd, c Thus for a I= { a, a +,, a + } = a+, ad hece {,, } I = I c= I, u I v I u+ v= + If s eve, so that c = a+, the I = I c= { +,, }

50 Hece for a So I, u I v I u+ v= α( ) = χ χ ( ), ad the cla follows Now we cobe () ad (), we get V / ` +, whch coletes the roof of Lea 6 defe Net we cosder larger boes where the = a eceed We N = b + l () Lea 7 Let V, Z= V, Z ( Q) be the set of teger solutos of the cogruece (6) ad let = The for a bo of te (5), V, + N / Z ` (4) Proof Partto to N = N saller boes, =, N where each has all of ts edge legths be aled to each We obta V, Z = N = (Lea 6) N ` + 8 = N = + N = V = ` + Thus Lea 6 ca / / N So the roof of Lea 7 s colete /

51 ouds o the error ters the fudaetal dett odulo whe = + Let be a bo of ots all Z as (5) cetered about the org wth, ad vew ths bo as a subset of 9 Z Let χ be ts chara- cterstc fucto wth Fourer easo χ ( ) = a ( ) e ( ) Let α( ) = χ χ = a( ) e ( ) The for a Z, s π / a ( ) =, (5) s π / = where the ter the roduct s take to be f we take / for all, ( ) a, = ` f = 0 I artcular, Suose s cetered about the org If =+, the we al the fudaetal dett (9) to α( ) = χ χ α * to get ( /) ( ) α( ) a( ) a( ) Q ( ) (od ) Ma Ter E E Ma Ter E E, (6) sce our bo s cetered about the org, ad so the Fourer coeff- cets a ( ) are all ostve The a ter (6) s α( ) = χ χ ( ) =, ad the others are error ters We roceed to boud these error ters We shall, ths secto ad et, refer to a error ter or to the value whch bouded that error ter b {,,,4} For the error ter E we frst observe E= a a Q ( ) 0 (od ) Q ( ) 0 (od ) E or E,, ( ) ( ) (7) The t s clear ow we ol eed to boud the su Let be a abbrevato for Q ( ) 0(od ), < / ( ) Q ( ) 0od Defe ρ b a

52 k for k, ρ= 0 for k= 0 Usg the fact that ( ) a, = elds `, (8) * Q ( ) 0(od ) / a( ), 4 k= 0 k= 0 = k ρ / / k k= 0 k = 0 = k / k / 40 4( / ) = k k= 0 k = 0 = For o-egatve tegers k, k,, k, let k = Z, Put k = +, so that (9) k = ` + = 4 (40) k+ k+ = = = = Now, fro the uer boud (4), we have where b utlzg (), N V, Z + N, (4) / = + = + b l b l (4) = = k /4 The last equalt (4) follows, sce k+ k < + < < 4

53 ut the rght -had sde of (4), s less tha or equal to It follows that k+ k = = k /4 k /4 ` ` = k /4 4 k N ` + (4) Al the uer boud (4) to the er su gves * Q ( ) 0(od ) / a( ) V, Z k k= 0 k = 0 = + * (9) Ths / N k k= 0 k = 0 = ` σ+ σ, (44) where b elog the equalt (40), we fd that σ = k k= 0 k = 0 = = 8 4 k k k 0 k 0 = = = = k= 0 k = 0 a a = k k 8 = 6, ad b the equalt(4), σ / = N k k= 0 k = 0 = a ka k + k / k k= 0 k 0 = = = k /4 k / k k = 0 ` k = k = k k < /4 /4 k ` (45)

54 4 k + k = / k 0 k = = k / k =, / k 0 = = k / /, + = Thus b equaltes (7), (44), (45) ad (46), we have 4 E Now assue that The for ad for, ` (46) 6 +, / + ` = =, + /+ + ` E =, >, 4 E 4 8 l l+, (47) < (48) ,8` + (49) ad (50), we ca wrte `, (49) 4 4 0,8 + ` + l ` (50) l l l l `, =, (5) ` l l = = =+ l = = ad cosequetl (usg (47) ad (5)) l 4 4, /+ l = 4 = = / + l l = 4 l ( /) = = =+ l 4 l ( /) = =+ l E (5)

55 Therefore b equaltes (47) ad (5), we arrve at 4 4 l ( /) E + E =+ l, E We et estate the error ter E, but to do that ad also for future referece, we frst rove Lea 8 Let be a bo of te (5) ad α( ) = χ χ ( ) Suose The we have l l+ 4, < (5) Z Proof We frst observe, l l ( ) =+ l a a e ( ) = α( ) ( ) = = = = α( ) e( ) = = = = 0 (od ) = = 0 (od ) u v u+ v 0(od ) α( ) α( ) + = b l (54) To obta the last equalt (54) we ust cout the uber of solutos of the cogruece wth u+ v 0 (od ), u, v For each choce of v, there are at ost = ([ / ] + ) choces for u So the total uber of solutos s less tha or equal to

56 + b l = Usg the hothess (5) the cotug fro (54) we have = a( ) + l The lea s establshed ` = =+ l l ` l =+ l =+ l Now vew of Lea 8, t s clear that the error ter E has the estate ( /) l l ( /) E= a (od ) =+ l ( ) The followg theore suarzes the fal outcoe of our vestgato for the error ters the case of =+ Theore Suose that 4 s eve, ad that ( ) =+ The for a bo cetered at the org, where Error 44 α( ) Error, Q 4 4 l ( /) l l ( /) E =+ l =+ l, E, E + + I Theore we have dcated below each ter, the error ter bouded b the gve value ter Net we coare each error ter Theore to the a / To ake the left-had sde greater tha /4 of the a ter, we ake each error ter less tha /4 of the a ter For the error ter E,, we eed ,

57 ad for the error ter E,, 4 l = l = Fall for the error ter E, 4 l = 45 4 l ( /) 4 Puttg the eces together, we deduce 4+ l ( /) + 4 ( /) l l ( /) l l ( /) l / =+ l =+ l =+ l Theore Suose that 4 s eve, ad that ( ) =+ If 4 + ad l= 0), the I artcular Q l ( / ) 4 ( /) = (where L H S= f α( ) = 4 4 V V ( + ) 4 Recall that a soluto of () s called rtve f soe coordate s ot dvsble b, e; rtve ots I fact for soe We wrte for - Corollar Uder the hotheses of Theore, + cotas a rtve soluto of () Proof We eed to show that α( ) > α( )

58 Frst b Lea 8, l l α( ) = α( ) = a( ) V, = = 4 l l l =+ l 46 =+ l The last equalt s guarateed b our hothess (Theore ) that More recsel, assue (55), the certal l = l 4 ( /) (55) = ( /) l 4+ l < + = + l 4 l l < 4 So we have ow o the oe had, l = =+ l = =+ l α( ) < 4 O the other had, b Theore, we have We therefore get α( ) 4 α( ) α( ) 0 4 The roof of the Corollar s colete Now we colete the cture b rovg the followg secal cases Corollar Let ( ) = ad let be a cube cetered at the org wth all =, Q 4 + (/ ) >, ad (8 4)/( ) + The +

59 cotas a rtve soluto of () Proof Suose that 4 + (/ ) > ad ad the hotheses of Corollar are satsfed 47 (8 4)/( ) + The l= 0 Corollar Let be a rectagular bo cetered at the org wth (4+ )/( l ) /( l ) = = l=, l+ = =, for soe l ( /) ad 4( )/( ) + l The + cotas rtve soluto () Proof Suose that the hotheses of Corollar hold The we have that l = ` Also, 4 ( /) So Corollar ales l 4 ( /) = (4 )( )/( ) ( )/( ) (4 ) + l l l l + 4( )/( ) + l 4 ouds o the error ters the fudaetal dett odulo whe = I ths subsecto we focus our work o the case = Aga we cosder the case of a bo setrc about the org We start b otcg that Lea 6 could be rewrtte ths case as follows: Lea 9 Let be a bo of te (5) wth all, ad ( ) V= V Q deote to the set of solutos of (6) Z If =, the / + V ` + Proof The roof s slar to the roof of Lea 6 The fudaetal dett odulo whe = s gve b / / α( ) = α( ) + α( 0) a( ) (56) V Q ( ) = 0

60 Let α be as gve the roof of Lea 6 (56),(8),(9) ad (0), α ( /) / ( ) + + ( ) / a / ut, the roof of Lea 6, we roved that Thus, t follows 48 ` α( ) = V / + V ` +, whch s the asserto of the lea A edate result fro the recedg Lea s Lea 0 Let V, Z= V, Z ( Q) be the set of teger solutos of the cogruece (6) ad let = The for a bo of te (5), /, + V + N Z ` (57) where N s gve () Proof We roceed ust as the roof of Lea 7 Partto to N = N saller boes Ths eas = N, where each has all of ts edge legths each, we thus obta V, Z = N = (Lea 9) N ` + = = V Al Lea 9 to + / + N = + N /

61 = ` + + / N, fshg the roof of the lea the fudaetal dett (9) aled to α( ) = χ χ wth =, ad usg the fact that a( ) 0 for all we have α ( /) ( ) α( ) a( ) a( ) Q ( ) Q ( ) Ma Ter (od ) E E V * 49 (58) Net we are seekg to boud the error ters (58) For the error ter E we have alread see the case =+ how ths error ter bouded The sae strateg wll work the case =, ecet we shall ake use of the uer boud (57) Lea 0 stead of the uer boud (4) Lea 7 Ideed we fd that * Q ( ) 0(od ) / a( ) = V Z k k= 0 k = 0 = / ` N k k= 0 k = 0 = /, + ` = Thus, t follows that E, + /+ + E =, Assue (as before) that The for ad for, >, ` (59) < l l ,8` + `, E,

62 4 4 0,8 + ` + 50 ` takg accout of these two equaltes, we have l l l l `, = (60) l a k l l = = =+ l = = Usg (59) ad (60), we fer that E 4 4 l 4+ =, / l = + / + l l = = 4+ l ( /) 4+ l ( /) = = = =+ l =+ l To estate the error ter E, we ust eed to al Lea 8 It s easl see that ( /) l l ( /) E = a (od ) =+ l ( ) (6) However let us derve a good estate for E, wthout usg Lea 8, hog to get a better boud tha the oe (6) Let rus * through the set { Z : Q ( ) 0(od )} Rewrte (5) to be for a Z, wth < /, where = = a ( ) a ( ), s π / a( ) =, s π / ad the ter the roduct s take to be f = 0 (as before) The lal 4 ( ) a,, = 4 4 Relace each b The, wth < /, we have Thus a( ), 4 ` ` (6)

63 a( ) a ( ) = a ( ) = = < /, = / 4 < = π (Usg the fact: = ad cotug), Suose that The obvousl Hece t follows ` = = 6 + / / 4 > 5 π + + = 4 6 ` + + = < l l+ a( ) = l = =+ l =+ l ` ` E ` (6) ( /) =+ l A secal case, whe l= 0 (6), we have ( /) [( /) + ] E = Coarg these two estates (6) ad (6), we coclude that the estate (6) stll s better Hece, we suarze Theore 4 Suose that 4 s eve, ad that ( ) = The for a bo cetered at the org, where Error α( ) Error, Q l ( /) l ( /) E =+ l =+ l, E, E + +

64 As before, order to obta a ostve su we seek codtos such that each error ter s less tha ¼ of the a ter E,: E, : 4 4 E : Thus we obta, l 4+ l ( /) 4 ( /) =+ l = l l l ( /) l ( /) =+ l = Theore 5 Suose that 4 s eve, ad that ( ) = If 4 + ad l= 0), the I artcular 5 Q l ( / ) 4 ( /) = (wth L H S= f α( ) = 4 4 V V ( + ) 4 As a cosequece of Theore 5, we have the followg aalogue of Corollar for rtve solutos Corollar 4 Uder the hotheses of Theore 5, + cotas a rtve soluto of () Proof Everthg alost works the sae as Corollar We ust rove that α( ) > α( ) Frst we have wth the hel of Lea 8, l l α( ) = α( ) = a( ) V, = = 4 l l l =+ l =+ l (64)

65 Here the last equalt (64) s true b our hothess (Theore 5) that l 4 ( /) (65) = Let us ause the roof for oet ad verf that ths hothess gves us the last equalt So we a assue (65) The l / l 4+ l ( /) 4+ = = 4 l ( /) 5 =+ l =+ l l 4 We resue our roof Sce we ow have ad Theore 5 elds we thus obta α( ) α( ) <, 4, 4 α( ) α( ) > 0 4, whch gve us the desred cocluso 4 We ow tur our atteto to the followg secal cases Corollar 5 Let be a cube cetered at the org wth all 4 + (/ ) > ad a rtve soluto of () (8 6)/( ) + ad = = The + cotas Proof Ths follows as the roof of the Corollar of the recedg secto Just b assug that 4 + (/ ) >, ad (8 4)/( ) + have the l= 0 ad the hotheses of Corollar 4 are satsfed, we

66 Corollar 6 Let be a rectagular bo cetered at the org wth (4+ )/( l ) /( l ) = = l=, l+ = =, for soe l ( /) ad 4(+ )/( l) The + cotas a rtve soluto () Proof Assue that the codtos of Corollar 6 hold The s led b l = ( / ) 4 ( /), 4(+ )/( l) l Ideed, b hothess ( / ) l =, ad f ad ol f We also have, =, l 4 / (4+ )/( l) = (4 + )( l )/( l ) ( l )/( l ) (4 + ) Hece Corollar 4 ales We coclude ths chater wth Proof of Theore Ths theore follows edatel fro Corollar (gves us ()) ad Corollar 5 (gves us (4)) uo settg = 4 54

67 Chater Sall Zeros of Quadratc Fors Modulo Itroducto Let Q( ) = Q(,,, ) = a be a quadratc for wth teger coeffcets ad be a odd re Set = a Let V = V ( Q) be the set of zeros of Q cotaed Z Whe s eve we let / ( Q ) ( ) det A / f det AQ, ( Q) = 0 f det A, where ( / ) deotes the Legedre-Jacob sbol ad A Q s the defg atr for Q( ) For Z set where to wth φ e ( ) e π ( V, ) / = 55 e = ( ) V = ( ) for 0, for 0 We shall devote ths chater to geeralze the ethod for Q (od ) (od ) Our goal s to fd a rtve soluto of the cogruece Q ( ) 0 (od ), (), where s a bo wth suffcetl large I artcular we wsh to obta the estece of a otrval soluto of

68 () wth as sall as ossble To ths ed we shall buld leas, theores ad corollares aalogous to those Chater I Chater 4, we shall rove the followg fudaetal dett (Theore 4): For a cole valued fucto α( ) defed o wth fte Fourer easo α ( ) = a( ) e ( ), where a( ) = ( ) α ( ) we have e α( ) = α( ) (od ) / / δc = 0 = = Q ( ) Q ( ) + a( ) a( ) where δ s defed: f s eve, δ= f s odd Thus whe =, we have α( ) ( ) / = α + c a( ) a( ) = = Q ( ) Q ( ) / c + a( ) a( ) = = Q ( ) Q ( ) + c a( ) a( ) () = = Q ( ) The frst ter o the rght-had sde s the a ter, ad the reag ters are the error ters I order to estate the error ters we frst eed to obta good uer bouds for V Z Estatg V I ths secto we tr to fd ad rove the aalogue of Lea 5, Lea 6 ad Lea 7 of Chater, for a arbtrar bo 56

69 Let where { Z a < a+, } = () a The, Z,, = =, the card- alt of Vew the bo () as a subset of Z ad let χ be t characterstc fucto wth Fourer easo χ = Cosder the cogruece ( ) ( ) ( ) Q 57 a e ( ) 0 (od ), (4) where Q ( ) s a quadratc for Later we eed to develo a good boud for the error ter estate V frst Q a( ) ad to do ths we eed to ( ) Lea Let be a odd re, V = V ( Q) be the set of zeros of (4) Z, ad be a bo as gve () cetered at the org wth all where If =±, the V ϑ` + (5) ( /) + ` +, =, ϑ= (6) ( /) + `+, =+ Proof The dea of the roof s slar to the deas used to rove Lea 5 of Chater We beg b wrtg the fudaetal dett (od ): ( ) = α( ) + ( ) ( ) α a a V = = Q ( ) Q ( ) ( /) ( /) a( ) + a( ) = = Q ( ) (7) Set α= χ χ =a( ) e ( ) The the Fourer coeffcets of α( )

70 are gve b a ( ) = a ( ) ad sce s cetered at the org, these are ostve real ubers Parseval s dett we have a( ) a ( ) χ ( ) = = = (8) Thus, t follows fro (8), a( ) a( ) (9) = Q ( ) Notce that the a ter (7) s 58 α( ) = χ χ ( ) = Lea 8 of Chater, we have ad ( /) l l ( /) a( ) = =+ l (0), () ( /) ( /) l l ( /) a( ) a( ),() = =+ l Q ( ) where as defed chater, l s defed b The case ( Q ) = : < l l+ Now gog back to (7), f =, we have ( /) ( ) α( ) + a( ) + a( ) α () V = = Q ( ) The b the equaltes (9), (0), ad (), we obta l l ( /) α( ) + + (4) =+ l We et detere whch of the ters doat ter We cosder two cases: Case (): Suose l The coare,, ad (4) s the

71 = = = l l ( /) l ( /) l =+ l l ( /) l l / =+ l = l ( /) l l ( /) / / ( /) + ` =, whch les that 59 ` Case (): Suose whch leads to ( /) + ( /) + l ( /) or =+ l l The coare = = l l ( /) =+ l l l ( /) =+ l / l ( /) ( l ) l / l = / / l ( /) or =+ l So for a l, alwas we have or, ( /) + / ` +, ( /) + / l l ( /) + =+ l Returg to (4), we ow ca wrte where α( ) + + ` ( /) + / ( /) + / = `+ + `+ ϑ +, ` (5) ( /) + ϑ = + ( / ) O the other had, we kow ( ) α V (6) To esure that the above equalt (6) s true see the roof of

72 Lea 5 of Chater Hece t follows b cobg (5) ad (6) that The case ( Q ) =+ : V ϑ ` + (7) 60 If =+, aga b (7), we have α α + + ( /) ( ) ( ) a( ) a( ) (od ) l l ( /) + (8) + =+ l (0),(9)&() 4 We do a slar vestgato (as before) to detere whch of the ters,, ad4of the equalt (8) s the doat ter I case () we fd 4/ ( /) +, whch eas that 4 ( /) + Ad / case () we fd 4/ / /, whch gves us that 4 / Hece for a l, we alwas have or, ` / ( /) + 4 +, ` / l l ( /) ( /) + + =+ l Now lookg at (8), oe easl deduces where α ϑ ` +, ( /) + ( ) ( ) ` ( /) ϑ + = + Thus b (6), Lastl lettg V / α( ) ϑ = ` + ϑ ϑ = f = ad ϑ= ϑ f =+ we get fro (5) ad (9) that for =±, oe alwas has V ϑ` + Ths acheves the roof of the lea