An Application of Hardy-Littlwood Conjctur JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China E-mail: fijinhuayoujian@msn.com Abstract. In this papr, w assum that wakr Hardy-Littlwood Conjctur, w got a bttr uppr bound of th xcptional ral zro for a class of prim numbr modul. Kyword. Hardy-Littlwood Conjctur, Excptional ral Zro MR000 Subjct Classification 11P3, 11M0 Goldbach s conjctur is on of th oldst and bst-known unsolvd problms in numbr thory and in all of mathmatics. It stats: Evry vn intgr gratr than can b xprssd as th sum of two prims. In 193, Hardy and Littlwood conjcturd 3 p 1,p N p 1 +p =N 1 N 1 N 1 ϕn p 1 log N p N whr N is vn intgr and N 6, p 1, p ar th prim numbrs, ϕn is Eulr function. Undr a wakr assumption, w got a bttr uppr bound of th xcptional ral zro for a class of th prim numbr modul. Wakr Hardy-Littlwood Conjctur. Lt N is vn intgr and N 6, p 1, p ar th prim numbrs. Thr is an absolut constant δ > 0, w hav 1
3 p 1,p N p 1 +p =N 1 δn log N Undr th abov conjctur, w hav th following thorm Thorm. Lt is a prim numbr and 3 mod 4, it has xcptional ral charactr χ, and its Dirichlt Ls, χ function has an xcptional ral zro β. If Wakr Hardy-Littlwood Conjctur is corrct, thn thr is a positiv constant c, w hav β 1 c log Now, w do som prparation work. Lmma 1. m { kn m if n 0 mod m = m 0 othrwis whr x = πix Th lmma 1 is obvious Lmma. Thr is a constant c 1 > 0 such that πx = Lix + O x xp c 1 log x uniformly for x. Whr Lix = du, and xpx = x log u Th lmma follows from th Rfrncs [], Thorm 6.9 of th pag 179. It is asy to s that Lix = du log u = x x log x + O log x
Lmma 3. Lt c b th positiv constant. if a, = 1, thn πx;, a = Lix ϕ χa u β 1 ϕ log u du + O x xp c log x whn thr is an xcptional charactr χ modulo and β is th concomitant zro. Th lmma 3 follows from th Rfrncs [], Corollary 11.0 of th pag 381 It is asy to s that u β 1 log u du = xβ x β β log x + O log x Lmma 4. if n, m = 1, thn m k,m=1 nk = µm m whr µm is Möbius function. Th lmma 4 follows from th Rfrncs [1], th pag 45. Lmma 5. if χ is a primitiv charactr modulo m, thn m nk χk = χnτχ m whr τχ = m χk k m. Th lmma 5 follows from th Rfrncs [1], th pag 47. Lmma 6. if m is odd suar-fr and χ is a primitiv ral charactr modulo m, thn 3
{ m if m 1 mod 4 τχ = i m if m 3 mod 4 Th lmma 6 follows from th Rfrncs [1], th thorm 3.3 of th pag 49. PROOF OF THEOREM. Th first part. By Lmma 1, whn x 4, w hav kp = 3 p 1 x 3 p x kp1 + p = 3 p 1 x 3 p x kp1 + p = 3 p 1,p x p 1 +p [ x ] 1 3 p 1,p x p 1 +p =n 1 by Wakr Hardy-Littlwood Conjctur, th abov formula [ x ] [ x δn ] log n δn log x δ log x [ x ] n = δ log x [ x ][ x ] + 1 δx x 4 log x + O log x Th scond part. 4
Whn 1 k 1, w hav pk = p,=1 pk + 1 = 1 + a=1 a,=1 ak 1 p a by Lmma 3, Lmma 4, Lmma 5 and Lmma 6, th abov formula = 1 + a=1 a,=1 ak Lix ϕ χa u β 1 ϕ log u du + O x xp c log x = µlix 1 τχχk 1 u β 1 log u du + O x xp c log x = i χk 1 u β 1 x log u du + O log x + x xp c log x thrfor pk = i χk x u β 1 1 log u du x +O 3 log x + x xp c log x 5
= u β 1 x 1 log u du + O 3 log x + x xp c log x x β x = β 1 log x + O log 3 x + x 3 log x + x xp c log x thrfor pk 1 = 1 + pk = x by Lmma, th abov formula log x x β β 1 log x +O x log 3 x + x log x + 3 x xp c 3 log x W intgratd th first part and scond part x β log x 1 δ 4 x log x + O x log 3 x + x log x + 3 x xp c 3 log x x β 1 δ 1 4 + O log x + 1 + 3 log x xp c 3 log x 6
w tak log x = 4 c 3 log, thn w tak log 8c 4 δ, thn x β 1 δ 4 + c 4 log x β 1 δ 8 β 1 log1 δ 8 log x = log 8 8 δ log x thrfor β 1 c log This complts th proof of Thorm. REFERENCES [1] Hnryk Iwanic, Emmanul Kowalski, Analytic Numbr Thory, Amrican mathmatical Socity, 004. [] Hugh L. Montgomry, Robrt C. Vaughan, Multiplicativ Numbr Thory I. Classical Thory, Cambridg Univrsity Prss, 006. 7