Hardy-Littlewood Conjecture and Exceptional real Zero. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.
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1 Hardy-Littlwood Conjctur and Excptional ral Zro JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China Abstract. In this papr, w assum that Hardy-Littlwood Conjctur, w got a bttr uppr bound of th xcptional ral zro for a class of modul. Kyword. Hardy-Littlwood Conjctur, Excptional ral Zro MR000 Subjct Classification P3, M0 In this papr, w rval th rlationship btwn Hardy-Littlwood Conjctur and th xcptional ral zro. Assum that Hardy-Littlwood Conjctur, w obtaind a good rsults of th xcptional ral zro. In this papr, I gnraliz th rsults of my papr An Application of Hardy-Littlwood Conjctur. is th prim numbr and 3 mod 4 is improvd to is odd suar-fr and 3 mod 4. w know that th modul of th xcptional primitiv ral charactr ar suar-fr whn is odd positiv intgr. First, w giv Hardy-Littlwood Conjctur. Hardy-Littlwood Conjctur. hav 3 p,p N p +p N Whn N is vn intgr and N 6, w N N ϕn p log N p N
2 whr p, p, p ar th prim numbrs, ϕn is Eulr function. Undr th abov conjctur, w hav th following thorm Thorm. Lt is odd suar-fr and 3 mod 4, it has xcptional ral charactr χ, and its Dirichlt Ls, χ function has an xcptional ral zro β. If Hardy-Littlwood Conjctur is corrct, thn thr is a positiv constant c, w hav β c log Now, w do som prparation work. Lmma. Assuming that th Hardy-Littlwood Conjctur. Lt n is any positiv intgr, is odd intgr and larg sufficintly, thn 3 p,p n p +p n n d ϕ log n whr d 3 p p and p, p, p ar th prim numbrs. Proof. By Hardy-Littlwood Conjctur, whn N is larg sufficintly, w hav N N ϕn p log N p N 3 p,p N p +p N
3 bcaus p N p 3 p p and n ϕn p n p p p p p ϕ W choos N n, This complts th proof of Lmma. Lmma. Lt m is positiv intgr and n is intgr, thn m { kn m if n 0 mod m m 0 othrwis k whr x πix. Th lmma is obvious Lmma 3. Lt c b th positiv constant. if a,, thn πx;, a Lix ϕ χa x u β ϕ log u du + O x xp c log x whn thr is an xcptional charactr χ modulo and β is th concomitant zro. Whr Lix x du and xpx x log u Th lmma 3 follows from th Rfrncs [], Corollary.0 of th pag 38 3
4 It is asy to s that Lix x du log u x x log x + O log x and x u β log u du xβ x β β log x + O log x Lmma 4. if χ is a primitiv charactr modulo m, thn m nk χk χnτχ m k whr τχ m k χk k m. Th lmma 4 follows from th Rfrncs [], th pag 47. Lmma 5. if m is odd suar-fr and χ is a primitiv ral charactr modulo m, thn { m if m mod 4 τχ i m if m 3 mod 4 Th lmma 5 follows from th Rfrncs [], th thorm 3.3 of th pag 49. Lmma 6. W giv th valu of two sums, thy ar usd in th proof of th Thorm. k a a, ak k 4 a a, ak bk b b,
5 k a a, b b, a + bk a a, b b, a + bk k a a, b b,, a+b ϕ χk k a a, ak a a, χk k ak τχ a a, χa 0 whr χ is th primitiv charactr modulo, This complts th proof of Lmma 6. PROOF OF THEOREM. Th first part. By Lmma, whn x 4, w hav k kp k 3 p x 3 p x kp + p 5
6 3 p x 3 p x k kp + p 3 p,p x p +p 0 [ x ] n 3 p,p x p +p n by Lmma, th abov formula ϕ [ x ] n nd log n ϕ [ x ] n nd log x d 3 ϕ log x [ x ] n n d 3 ϕ log x [ x ][ x ] + d x 4ϕ log x + O x ϕ log x Th scond part. Whn k, w hav pk p, pk + Olog a a, ak + Olog p a by Lmma 3 and Lmma 4, th abov formula a a, ak Lix ϕ χa x u β ϕ log u du + O x xp c log x +Olog 6
7 Lix ϕ a a, ak τχχk x u β ϕ log u du + O x xp c log x whr χ is th xcptional primitiv ral charactr modulo. thrfor pk Lix ϕ a a, ak τχχklix ϕ a a, ak x u β χkτχ x log u du + u β ϕ log u du +O x xp c log x By Lmma 5 and Lmma 6, w hav k pk ϕ Lix x u β ϕ log u du 7
8 +O 3 x xp c log x x ϕ log x x β ϕβ log x + O x ϕ log 3 x + 3 x xp c log x W synthsiz th first part and scond part, w hav d x 4ϕ log x x ϕ log x x β ϕβ log x x +O log 3 x + x ϕ log x + 3 x xp c log x d x 4 log x x log x xβ x β log x + O log 3 x + x log x + 3 x xp c log x d 4 xβ + O β log x + x + 3 xp c 3 log x w tak log x 4 c 3 log, thn w tak log 8c 4 d, thn x β d 4 + c 4 log 8
9 x β d 8 β log d 8 log x log 8 8 d log x thrfor β c log This complts th proof of Thorm. Bcaus τχ 4 i and τχ 8, whn is odd suar-fr, by sam mthod, for modul 4, mod4 and modul 8, 3mod4, w hav th sam conclusion. REFERENCES [] 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,
An Application of Hardy-Littlewood Conjecture. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.China
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
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