Improvements on Waring s Problem

Transcription

1 Improvement on Warng Problem L An-Png Bejng, PR Chna apl@nacom Abtract By a new recurve algorthm for the auxlary equaton, n th paper, we wll gve ome mprovement for Warng problem Keyword: Warng Problem, Hardy-Lttlewood method, recurve algorthm, auxlary equaton

2 Introducton Warng problem now to fnd G ( ), the leat nteger, uch that each uffcent large nteger may be repreented a um of at mot th power of natural number The Hardy-Lttlewood method, that, o called crcle method man analy method, whch propoed by Hardy Ramanujan and Lttlewood n about, whch have been appled uccefully n olvng ome problem of number theory, eg Warng problem and Goldbach problem The nown bet reult for Warng problem up to now are a followng For uffcently large (Wooley [] ), G ( ) log( log ) O () And for maller, G(), G(), G(), G(), () For the detal referred to ee the Vaughan and Wooley urvey paper [] In th paper, by a new recurve algorthm, we wll gve ome mprovement for G ( ) Theorem For uffcently large, G ( ) o ( ) () Theorem For, let F ( ) be a n the Lt, then G ( ) F ( ) () F ( ) F ( ) F ( ) F ( ) Lt In ecton, there further progre

3 The Proof of Theorem Suppoe that P a uffcent large nteger, C ( P) a ubet of [, P ], a gven nteger, conder the equaton x x y y, x, y C ( P), () Denoted by S ( C ( P)) the number of oluton of (), or mply S ( P ), when the electon C ( P) clear n context The equaton above called auxlary equaton of Warng problem In the followng, we wll tae ue of teratve method to contruct C ( P) Suppoe that a real number, Let P P, P a et of prme number p n nterval [ P /, P ], wrte P Z, defne C( P ) xp xc ( P), pp () Wth repect to the contructon, we wll alo conder followng a relatve equaton p x x y yq x x y y, x, y C ( P), where p, qp, p q Denote by T, ( p, q) the number of oluton of (), and T, ( q) T, ( p, q) p () Lemma For nteger,, t ha S P Z T q () ( ), ( ) Proof A uual, wrte ex ( ) e x, let f( ) e x, f(, p) e p x, f( ) e y xc ( P) xc ( P) yc ( P) Then clearly, f ( ) e p x f(, p) pp xc ( P) pp Applyng Hölder nequalty, t ha S ( P ) f( ) d f( ) f( ) d pp qp f(, p) f(, q) d

4 d pq pq pq, P, pq ( ) ZS( P) Z f(, pf ) (, q) d pq, P, pq Z f( ) d ( f(, p) f(, q) ) d It clear that Z S ( P ) mnor for S ( P ) Moreover, for a non-negatve nteger, let (,, ) (, ) p q f p f(, q), then by Cauchy nequalty, t ha f (, p) f(, q) d (, p, q) (, q, p) d (, pq, ) d (,, ) q p d / / And / / f(, p) f(, q) d (,, ) (,, ) p q d q p d pq, P, pq pq, P, pq pq, P, pq (, pq, ) d Clearly, the nner ntegral the number of oluton of equaton () Denote by P [ ab, ] the et of prme number n the nterval[ ab, ] Smlar to contruct (), let,, be real number, /,, whch wll be determned later, and let Z P, P P [ Z /, Z], P P / Z, H P/ Z, Recurvely defne C ( P) x p xc ( P ), pp,( p, x),,, Smply wrte C C ( P),,,, In the followng, t wll be ued the notaton of dfference of a functon: A uual, for an nteger coeffcent of polynomal ( x), recurvely defne the forward dfference ( ( x ), t ) ( xt ) ( x ), ( ( x), h, h,, h, h ) ( ( x), h, h,, h), h,, Suppoe that t h m, m a contant, then we now that m ( ( x t) ( x)), n th cae we

5 defne modfed dfference ( ( ), ; ) ( ) ( ) x hm m x hm x ( ( x), h,, h, h ; m,, m, m ) ( ( x), h,, h; m,, m), h ; m Smply wrte, ( x, h,, h; p,, p ) ( x, h,, h; p,, p ),,, And defne f ( ) ex ( ), f(, p) epx ( ), pp, xc xc,( x, p) g ( ; h,, h; p,, p ) e( ( x) ), x And () () hh hh pp pp F(, q) g ( q ) F (, q) g ( q ) hh hh pp pp F (, q) g ( q ) Let p hh hh pp pp ( ) f(, p) f(, q) d ( ) () f ( ) F (, q) d, ( ) f ( ) F(, q) d, () () () Lemma ( ) () pc f (, p) F (, q) d ( Z ) P Z H S ( P ) P( Z ) P () Proof A uual, for a number x, denote by x wth that x x mod p Smply wrte ( x) ( x), and For a p P, let, p p p P p P h h h D(, p) x xc, ( x) mod p, ( p) C \ D (, p) Then the um of g ( q ) can be dvded two part, that, normal and ngular part, e one

6 wth ( x, y) ( p), and the other one not It not dffcult to demontrate that the ntegral n the ngular part econdary, for the mplcty, we ave the nvetgaton, and n the followng acquece n the normal part It clear that for equaton ( x) ( y) n mod p there are at mot O( p ) oluton wth that ( x, y) ( p), hence we can dvde ( p) nto ( ) O p, ayl clae, F p ( ) l, uch that n each cla F ( p) the equaton ha at mot two oluton mod p And denote by ( x) card y ( x, y) F ( p) It clear that j j j ( x) P Hence, t ha jl ( ) () ( ) (, ) (, ) (, ) (, ) pc p h pc ( ) f (, p) e(( ( x) ( y)) q ) d p h pc jl ( x, y) Fj( p) ( ) l f (, p) e(( ( x) ( y)) q ) d p h pc jl ( x, y) F j ( p) f p F q d f p g q d l f (, p) ( x) e( ( ( x), h; p) q ) l p h ( ) Fj j pc jl h x e( ( ( x), h; p) ( ( y), h ; p)) q ) d ( x, y) F j h, h ( ) l f (, p) Fj( p) d p h pc jl p l f (, p) e( ( ( x), h; p) ( ( y), h ; p)) q ) d p p l f (, p) ( ) f (, p) j( x) e( ( ( x), h; p)) q ) d h pc jl h x ( ) h pc jl ( x, y) Fj h, h ( ) lp Z f ( ) d h ( ) pc h x ( ) p h pc h x lp Z Z H S( P ) ( ) jl p h pc h x P e( ( ( x), h; p)) q ) d l f ( ) e( ( ( x), h; p) q ) d P f ( ) e( ( ( x), h; p)) q ) d lh p h ( ) f (, p) e( ( ( x), h; p) q ) d p h pc h x

7 lp Z ZHS( P ) Pl P From the proof above, we can now that, p ( ) T ( q) f(, p) f(, q) d Z P ZS( P) Z P ( P) P () Bede, / / ( ) ( ) ( ) f( ) F(, q) d f( ) d f( ) F(, q) d S P H Z / / / ( ) ( ) () where H H, Z Z j j j j In general, we have Lemma U V W () U S ( P) Z ( Z H ) ( lp Z Z H S ( P )), / ()/ / / V S ( P) / ()/ / / Z ZH Pl ( ) ( ), W S ( P) Z ( Z H ) ( P ) / ()/ / / Proof / / ( ) () ( ) ( ) () f ( ) (, ) ( ) ( ) ( (, )) F q d f d f F q d ( ) / () S ( P) f (, p) ( F (, q)) d pp / ( ) () S( P) Z f (, p) ( F (, q)) d pp / ( )/ ( ) () S( P) Z Z H f (, p) F (, q) d pp / ()/ / S ( P) Z ( Z H ) ( lp Z Z H S( P )) ( Pl ) ( P ) U V W / / / /

8 Let U V H P W H P, () / / / /,, wll be decded later Hence, t ha / ( Z) P ZZ HS( P) / / / / / / / / / PZ ( ) S ( P) ( HZ ) ( H) P ( P) ( H) P And t follow / / / / / ( Z) P ZZ H ( ) ( ) S P P H P / P ( H ) P That, / / / ( Z ) ZZ H S ( P) P( H) P () On the other hand, U V W U ( H ) P / / S ( P) Z ( Z H ) ( lp Z Z H S ( P )) ( H ) P / ()/ / / / / Combne the two equalte, ( Z ) Z Z H S ( P) P( H ) P S ( P) Z ( Z H ) ( lp Z Z H S ( P )) ( H ) P It follow / ()/ / / / / / ()/ / / ( ) ( ) ( ) ( ) / / ( Z ) ( H ) P S P Z H l H S ( P ) e () S ( P) Z ( H ) ( Z ) ( H ) () S P Z H P () ( ) ( ) ( ) Let ( ) S X X, () become P ( P/ Z ) ( P/ Z ) ( Z ) P Z P () Z ( ) e ( ) Z Z P Z Z And ( ) ( ) ( ) ( ) ( ) ( )

9 Denote by a () ( ), b, t ha ( ) j j a a j b a, and a ab a Bede, by (), t ha / / / / ( Z ) P Z Z H S ( P ) P ( H ) P It eay to now that Z HPS ( ) P, o t ha ( Z ) P Z Z H S ( P ) PZ H P S ( P )( H ) P And ( H ) P ( P/( Z) ) P ( ) Defne (), there a ab a a ( ) ( ) ( ) ( ) ( ) Epecally, a ( ) ( ) ( ) ( ) ( ) On the other hand, by Lemma wth, t ha S ( P ) Z ( P) Z U H P Z Z P ZS ( P) H P and / / / / ( )/ ( / ) ( ) ( ) Or, () ( ) () ( /) () ( ) ( ) ( ) When greater, t may ha ( ) () ( ) ( ) And let ( ) ( ) ( ), ( ) () Subttutng () and () n (), t follow

10 () ( ), ( ( ) ) ( ) where ( ) And then () ( ) ( ( ) ) ( ( ) ) So, ( d ) ( d) ( ( ) ) () d, or, or other It nown that,, e (), () Hence, ( ), or ( ( ) ) Let (, ) log log ( ) ( ( ) ) () From (), we can now that ( ) wll approach zero a tend to (, ) Hence t ha Lemma For uffcent large, and arbtrary mall, there ( (, ) ) uch that () Bede, for greater, t ha ( ( ) ) (, ) log a () Moreover, we now that (ee [] or []) ( u) G ( ) u ˆ where log( / ) log log ˆ, log log log O ( ) log Tae u ( (, ) ), and let, Theorem proved

11 The proof of Theorem For the maller, and () may be followed by recuron () and () from the ntal d, d,, or any nown better d by choong optmal value and n turn Wth nvolve earchng of two parameter and, and the retrcton of the ablty of PC, we have leen the earch range only n four dgt The reult n Lt are obtaned by PC () () Lt For, let ( ) f a power of, or /ele From the nown reult (ee [], []), we now that for two potve nteger tv,, f atfyng ) tv ( ), ) v ( t), then Wth Lt, we tae v a n the followng lt G ( ) t v () v( ) Lt And Theorem followed Further Improvement

12 Shortly after the paper appeared, we realze that the method of parameterzed recuron appled n the ecton, alo avalable for the recurve proce appled n paper [],and t unexpected that the reult are even better than the prevou one, the new reult are that Theorem For uffcent large, G ( ) o( ), f not a power of, ele () Theorem For, let F ( ) be a n the Lt, then G ( ) F ( ) () F ( ) F ( ) F ( ) F ( ) Lt The Proof of Theorem : Let, ( ) ( ) J T ( q) f(, p) f(, q) d, p J f ( ) F(, q) d, () Lemma of paper [] wll be ued n the followng proof, we retate here Lemma J ZPS ( P) J, J U V, U S ( P) Z P( H Z ) Z S ( P ) / ()/ V S ( P) Z ( H Z J ),, / ()/ / Where H H, Z Z j j j j / ()

13 Let U V H P,,, () / /, The parameter, wll be decded later Hence, t ha PH ( Z ) ZS ( P) ( H Z J) H P And J P( H Z ) Z S ( P) H P () On the other hand, / J U H P S ( P) Z P( H Z ) Z S ( P ) H P () / / / ()/ / / Combne the two equalte, t ha / PH ( Z ) ZS ( PH ) P S ( P) Z PHZ ( ) Z S ( P ) H P It follow, / ( )/ / / S ( P) Z ( H ) H, S P P H P / ()/ / / / / ( ) and S ( P) Z ( H ) H () S P P () ( ) Let ( ) S X X, t ha P ( P/ Z ) ( P/ Z ) P () Z P e ( Z ) Z P Z Z and ( ) ( ) Denote by a (() ) () ( ) ( ) (), b, t ha ( ) j j a a j b, and a Moreover, by () wth, t ha a ab a ()

14 / PH ( Z ) ZS ( P) ( H Z J) H P / / / And t eay to now that J PH Z S ( ) P, hence PH ( Z ) ZS ( P) ( H Z PHZS ( P)) H P H P P H ( P/ Z ) ( ) Z P ( ) Let () defned a before, there a ab a a ( ) ( ) ( ) ( ) ( ) Epecally, a ( ) ( ) ( ) ( ) ( ) On the other hand, by Lemma wth, t ha S ( P ) Z J ( P) Z U H P Z PZS ( P) H P And / / / / ( / ) ( / / ) ( ) ( ) ( ) Or, () ( ) ( /)( ) () () ( ) ( ) ( ) When greater, t may ha ( ) () ( ) ( ) And let ( ) ( ) ( ), ( ) () Subttutng () and () n (), t follow () ( ) () ( ( ) ) ( ( ) ) where ( ) And then

15 () ( ) ( ( ) ) ( ( ) ) For (), hence ( ), () ( ( ) ) Let (, ) log log ( ) ( ( ) ) () From (), we can now that ( ) wll approach zero a tend to (, ) Hence t ha Lemma For uffcent large, and arbtrary mall, there (, ) uch that () Clearly, for greater, there (, ) log () Denote by x /, and let, whch a root of the equaton x log x x x Hence, (, ) / log, a x We now that ( u) G ( ) u ˆ where e log( / ) log log ˆ, log log log O ( ) log G ( ) u( o()) ( u ) log Tae u (, ), and let x, Theorem proved

16 The Proof of Theorem : For maller, () may be followed by recuron () and () and the ntal () by choong optmally value and n turn Wth the retrcton of the ablty of PC, we have leened the earch range of parameter, only n fve dgt The reult n Lt are obtaned by PC A completed lt ncludng ntermedate reult poted behnd a appendx () () Lt A the proof of Theorem, wth () and Lt, we tae v ( ) a n the followng lt v( ) Lt And Theorem followed Further Improvement () In th ecton, we wll preent further mprovement when larger Theorem For uffcent large, G ( ) o( ), f not a power of,, otherwe ()

17 The Proof of Theorem : The notaton and ymbol ued here wll be ame a before Defne F (, q) e(( ( x) ( y)) q ), () () hh hh pp pp x, y F (, q) e(( ( x) ( y)) q ), () () hh hh pp pp xy Let ( ) J f(, p) f(, q) d, p ( ) J f ( ) F(, q) d, There () Lemma J ZPS ( P) J, J U V, U P HZS ( P), V S ( P) Z ( HZJ ),, / / ()/ / where H H, Z Z j j j j () Proof By Cauchy nequalty, ( ) ( ) ( ) ( ) (, ) ( ) ( ) (, ) / / ( ) ( ) () f ( ) d H Z f ( ) (, ) F q d / / J f F q d f d f F q d ( ) / ( ) ( ) () f( ) d ( HZ ) f( ) ( ) ( ) (, ) PHZ d f F q d S ( P) ( H Z ) / / / / ( ) / / () S ( P) ( H Z ) P( H Z ) S ( P) f (, p) F(, q) d pp ( ) () PHZ ( ) S ( P) Z f (, p) F(, q) d pp P H Z S ( P) S ( P) ( Z ) ( H Z ) f ( ) F (, q) d / / / / ( ) P H Z S P S P Z / / / / / ( ) ( ) ( ) ( HZ ) ( J ) / / / Let

18 ( ) / /, U V H P The parameter, wll be determned later So, t follow () Z () PH ( Z ) S ( P ) J ( H ) P () Bede, J U ( H ) P P H Z S ( P)( H ) P () / / / / / Combne the two dentte above, t follow S ( P ) P H ( H ) () S P H P e / / () Z / ( ) ( ) ( Z ) Z P ( P/ Z ) ( P/ Z ) Or, ( ) ( )/ / ( () ( )) ( ) / / P P P And, (() ( )) ( ) () ( ) ( ) Denote by a, b Then t ha j j a a a b, and a b a a j Moreover, by () wth, t ha PH ( Z ) S ( P ) S ( P ) Z ( H Z J)( H) P e ( ) PH Z S P Z J H P ( ) ( ) ( ) ( ) It eay to now that J PH Z S ( ) P, hence PH ( Z ) S ( P ) Z PHZS ( P)( H) P And S S e ( ) ( P ) ( ) Z H( H) P ( P ) ( ) ( ) ( ) Z P

19 It follow () ( ) Hence, ( ) ( ) ( ) a ( ) ( ( ) )( ( ) ) Epecally, ( ) ( ) ( ) a ( ) ( ( ) )( ( ) ) Furthermore, by Lemma wth, t ha S ( P ) Z J ( P) Z U H P Z PZS ( P) H P And / / / / ( / ) ( / / ) ( ) ( ) ( ) e () ( ) ( /)( ) () () ( ) ( ) ( ) Let ( ) ( ), ( ) () Then t follow ( ) ( ) () ( ), () ( ( ) ) ( ( ) ) where ( ), And then ( ) ( ) () ( ) ( ( ) ) ( ) ( ) ( ) ( ( ) ) e ( ) ( ) ( ) ( d ) ( d) () ( ) ( ( ) ) ( )

20 Let ( ) ( ) (, ) log log ( ( )) ( d) ( ) ( ( ) ) () From (), we can now that ( d) wll approach zero a tend to (, ) Hence t ha Lemma For uffcent large, and arbtrary mall, there (, ) d uch that () It clear that, when greater, ( ) ( ) ( d) (, ) log () ( ) ( ) We chooe ( d) uch that ( d) /, then we can tae, and Hence, t ha o( ) And then let, by (), t follow (, ) ( d) () On the other hand, from (there tae, ), we can now that d log () We have nown that (ee []) G ( ) u( o()) ( u ) log Tae u d (, ), Theorem proved In fact, Theorem can be alo proved along the way of ecton a followng The Second Proof of Theorem : By () and (), there ( ) ( ) () ( ) ( ( ) ) ( ( ) ) Where ( ) ( ),,,

21 And then, ( ) ( ) ( ) ( d ) ( d) () ( ) ( ( ) ) ( ) Let ( ) ( ) (, ) log log ( d)( ( ) ) ( ) ( ( ) ) () From (), we can now that ( d) wll approach zero a tend to (, ) Clearly, when greater, ( ) ( ) ( d) (, ) log () ( ) ( ) We chooe ( d) uch that ( d) /, then t can be taen that, and o( ), and then let, by (), t follow (, ) ( d) The ret ame a the frt proof Tae

22 Reference AP L, A note on Warng Problem, arxv: RC Vaughan, The Hardy-Lttlewood method, Cambrdge Unverty Pre, RC Vaughan, A new teratve method n Warng problem, Acta Math (), - RC Vaughan and TD Wooley, Warng problem: A urvey, Number Theory for the Mllennum III, A K Peter,, pp TD Wooley, Large mprovement n Warng problem, Ann of Math (), -

23 Appendx Intermedate Reult of Recuron for Theorem ()

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