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
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
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
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
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
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
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 / / / /
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 ( ) ( ) ( ) ( ) ( ) ( )
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
() ( ), ( ( ) ) ( ) 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
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
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 / ()
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 ()
/ 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
() ( ) ( ( ) ) ( ( ) ) 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
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 ()
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
( ) / /, 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
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) () ( ) ( ( ) ) ( )
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 ( ) ( ),,,
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
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 (), -
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