Improvements on Waring s Problem

Transcription

1 Imrovement on Warng Problem L An-Png Bejng 85, PR Chna al@nacom Abtract By a new recurve algorthm for the auxlary equaton, n th aer, we wll gve ome mrovement for Warng roblem Keyword: Warng Problem, Hardy-Lttlewood method, recurve algorthm, auxlary equaton

2 Introducton Warng roblem now to fnd G ( ), the leat nteger, uch that each uffcent large nteger may be rereented a um of at mot th ower of natural number The Hardy-Lttlewood method, that, o called crcle method man analy method, whch rooed by Hardy Ramanujan and Lttlewood n about 9, whch have been aled uccefully n olvng ome roblem of number theory, eg Warng roblem and Goldbach roblem The nown bet reult for Warng roblem u to now are a followng For uffcently large (Wooley [5] ), G ( ) log( log ) O () And for maller, G(5) 7, G(6) 4, G(7) 33, G(8) 4, () For the detal referred to ee the Vaughan and Wooley urvey aer [4] In th aer, by a new recurve algorthm, we wll gve ome mrovement for G ( ) Theorem For uffcently large, 3 G ( ) log(3/ ) o ( ) (3) 3 Theorem For 5, let F ( ) be a n the Lt, then G ( ) F ( ) (4) F ( ) F ( ) F ( ) F ( ) Lt

3 The Proof of Theorem Suoe 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 mly S ( P ), when the electon C ( P) clear n context The equaton above called auxlary equaton of Warng roblem In the followng, we wll tae ue of teratve method to contruct C ( P) Suoe that a real number, Let P P, P a et of rme number n nterval [ P /, P ], wrte P Z, defne C( P ) x xc ( P), P () Wth reect to the contructon, we wll alo conder followng a relatve equaton x x y yq x x y y, x, y C ( P), where, qp, q Denote by T, (, q) the number of oluton of (3), and T, ( q) T, (, q) (3) Lemma For nteger,, t ha S P Z T q (4) ( ), ( ) Proof A uual, wrte ex ( ) e x, let f( ) e x, f(, ) e x, f( ) e y xc ( P) xc ( P) yc ( P) Then clearly, f ( ) e x f(, ) P xc ( P) P Alyng Hölder nequalty, t ha S ( P ) f( ) d f( ) f( ) d P qp f(, ) f(, q) d

4 d q q q, P, q ( ) ZS( P) Z f(, f ) (, q) d q, P, q Z f( ) d ( f(, ) f(, q) ) d It clear that Z S ( P ) mnor for S ( P ) Moreover, for a non-negatve nteger, let (,, ) (, ) q f f(, q), then by Cauchy nequalty, t ha f (, ) f(, q) d (,, q) (, q, ) d (, q, ) d (,, ) q d / / And / / f(, ) f(, q) d (,, ) (,, ) q d q d q, P, q q, P, q q, P, q (, q, ) d Clearly, the nner ntegral the number of oluton of equaton (3) Denote by P [ ab, ] the et of rme 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 xc ( P ), P,(, x),,, Smly 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 olynomal ( x), recurvely defne the forward dfference ( ( x ), t ) ( xt ) ( x ), ( ( x), h, h,, h, h ) ( ( x), h, h,, h), h,, Suoe 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 Smly wrte, ( x, h,, h;,, ) ( x, h,, h;,, ),,, And defne f ( ) ex ( ), f(, ) ex ( ), P, xc xc,( x, ) g ( ; h,, h;,, ) e( ( x) ), x And () (4) hh hh P P F(, q) g ( q ) F (, q) g ( q ) hh hh P P F (, q) g ( q ) Let hh hh P P ( ) 4 f(, ) f(, q) d ( ) () f ( ) F (, q) d, ( ) f ( ) F(, q) d, 4 (5) (6) (7) Lemma ( ) (4) C f (, ) F (, q) d ( Z ) P Z H S ( P ) P( Z ) P (8) Proof A uual, for a number x, denote by x wth that x x mod Smly wrte ( x) ( x), and For a P, let, P P h h h D(, ) x xc, ( x) mod, ( ) C \ D (, ) Then the um of g ( q ) can be dvded two art, that, normal and ngular art, e one

6 wth ( x, y) ( ), and the other one not It not dffcult to demontrate that the ntegral n the ngular art econdary, for the mlcty, we ave the nvetgaton, and n the followng acquece n the normal art It clear that for equaton ( x) ( y) n mod there are at mot O( ) oluton wth that ( x, y) ( ), hence we can dvde ( ) nto ( ) O, ayl clae, F ( ) l, uch that n each cla F ( ) the equaton ha at mot two oluton mod And denote by ( x) card y ( x, y) F ( ) It clear that j j j ( x) P Hence, t ha jl ( ) (4) ( ) 4 (, ) (, ) (, ) (, ) C h C ( ) f (, ) e(( ( x) ( y)) q ) d h C jl ( x, y) Fj( ) ( ) l f (, ) e(( ( x) ( y)) q ) d h C jl ( x, y) F j ( ) f F q d f g q d l f (, ) ( x) e( ( ( x), h; ) q ) l h ( ) Fj j C jl h x e( ( ( x), h; ) ( ( y), h ; )) q ) d ( x, y) F j h, h ( ) l f (, ) Fj( ) d h C jl l f (, ) e( ( ( x), h; ) ( ( y), h ; )) q ) d l f (, ) ( ) f (, ) j( x) e( ( ( x), h; )) q ) d h C jl h x ( ) h C jl ( x, y) Fj h, h ( ) lp Z f ( ) d h ( ) C h x ( ) h C h x lp Z Z H S( P ) ( ) jl h C h x P e( ( ( x), h; )) q ) d l f ( ) e( ( ( x), h; ) q ) d P f ( ) e( ( ( x), h; )) q ) d lh h ( ) f (, ) e( ( ( x), h; ) q ) d h C h x

7 lp Z ZHS( P ) Pl P From the roof above, we can now that, ( ) 4 T ( q) f(, ) f(, q) d Z P ZS( P) Z P ( P) P (9) 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 3 U V W () U S ( P) Z ( Z H ) ( lp Z Z H S ( P )), / (5)/ / / V S ( P) / (5)/ / / Z ZH Pl ( ) ( ), W S ( P) Z ( Z H ) ( P ) / (5)/ / / Proof / / ( ) () ( ) ( ) () f ( ) (, ) ( ) ( ) ( (, )) F q d f d f F q d ( ) / () S ( P) f (, ) ( F (, q)) d P / 5 ( ) () S( P) Z f (, ) ( F (, q)) d P / ( 5)/ ( ) (4) S( P) Z Z H f (, ) F (, q) d P / (5)/ / S ( P) Z ( Z H ) ( lp Z Z H S( P )) ( Pl ) ( P ) U V W / / / /

8 Let U V H W H () /4 /, wll be decded later Hence, t ha ( Z ) P Z Z H S ( P) P( Z ) S ( P) ( H Z ) ( H ) ( P ) ( H ) / / / / / /4 / / And t follow / / / / ( Z) P ZZ H ( ) ( ) S P P H /4 P ( H ) That, / / ( Z ) ZZ H S ( P) P( H) (3) On the other hand, U V W U ( H ) / S ( P) Z ( Z H ) ( lp Z Z H S ( P )) ( H ) / (5)/ / / / Combne the two equalte, ( Z ) Z Z H S ( P) P( H ) S ( P) Z ( Z H ) ( lp Z Z H S ( P )) ( H ) It follow / (5)/ / / / / (4)/ / / ( ) ( ) ( ) ( ) ( ) ( H ) S P Z H l H / S ( P ) Z e S ( P) Z ( H ) ( Z ) ( H ) (5) S P Z H (4) ( ) ( ) ( ) (4) Let ( ) S X X, (5) become P ( P/ Z ) ( P/ Z ) ( Z ) P (4) Z ( Z) e ( ) Z Z P Z Z And ( 4) ( ) (4 ) (4) ( ) (4 ) Denote by a (4) ( ), b, t ha (4 )

9 j j a a j b a, and a ab a Bede, by (), t ha / / / ( Z ) P Z Z H S ( P ) P ( H ) It eay to now that Z HPS ( ) P, o t ha ( Z ) P Z Z H S ( P ) PZ H P S ( P )( H ) And ( ) / H H Defne (), there a ( ) ( ) ab a a ( ) ( ) ( ) ( ) Eecally, ( ) ( ) a ( ) ( ) ( ) ( ) On the other hand, by Lemma wth, t ha S ( P ) Z ( P) Z U H Z Z P ZS ( P) H and / / / ( / ) ( ) ( ) Or, (6) ( ) () ( /) (7) ( ) ( ) When greater, t may ha ( ) (8) ( ) ( ) And let ( ) ( ), (9) 3 Subttutng (8) and (9) n (7), t follow 3 () ( ) ( ( ) ) ( ( ) ) And then

10 3 3 () ( ) ( ( ) ) 3 ( ( ) ) o 3 ( d ) ( d) 3 ( ( ) ) 3 () d, or, or other It nown that,, e (), () Hence, 3 ( ),, or 3 ( ( ) ) 3 Let 3 ( ) log log 3 ( ) ( ( ) ) () From (), we can now that ( ) wll aroach zero a tend to ( ) Hence t ha Lemma 4 For uffcent large, and arbtrary mall, there ( ( ) ) uch that () We now (ee [3], or []), ( u) G ( ) 3u ˆ where log( / ) log log ˆ, log log log O 4( ) log Tae u ( ( ) ), and let /3, (/ 3) (8log(3/ ) / 3 o()), when great And note that3log(3/ ) / , o the cae that a ower of alo hold, and Theorem roved 3 The roof of Theorem For the maller, there are two way to obtan and (), the frt one that through the recuron (6) and (7) from the ntal d, d,, or any nown better d by choong otmal value n turn The econd one that, by (7) and (9), t follow

11 ( ) 3 ( ) ( ) ( d) ( d) (3) 3 ( ) ( ( ) ) 3 ( ) where wth tae d, or, or other nown better reult, and then choong otmally Wth the two way, by comuter we have obtaned followng reult a n Lt 3 I/II d / () 5 I 93 6 I I I I 96 8 I I I I II II I I I I I 9487 Lt 3 Where mar I or II ndcate calculaton by the way I or II For, let ( ) 4f a ower of, or3 /ele From the nown reult (ee [], [3]), we now that for two otve nteger tv,, f atfyng ) tv ( ), ) v ( t), then Wth Lt 3, we tae v a n the followng lt G ( ) t v (3)

12 v( ) 4 4 Lt 3 And Theorem roved The reult n Lt 3 wll be better f tae the harer ntal value 3 or 4

13 Reference AP L, A note on Warng Problem, arxv: 8377 RC Vaughan, The Hardy-Lttlewood method, Cambrdge Unverty Pre, 98 3 RC Vaughan, A new teratve method n Warng roblem, Acta Math 6(989), -7 4 RC Vaughan and TD Wooley, Warng roblem: A urvey, Number Theory for the Mllennum III, A K Peter,, TD Wooley, Large mrovement n Warng roblem, Ann of Math 35(99), 3-64