ACTA ARITHMETICA LXXVI (6 Almos all shor inervals conaining prime nmbers by Chaoha Jia (Beijing Inrocion In 37, Cramér [] conjecred ha every inerval (n, n f(n log 2 n conains a prime for some f(n as n In 43, assming he Riemann Hypohesis, Selberg [] showed ha, for almos all n, he inerval (n, n f(n log 2 n conains a prime providing f(n as n In he same paper, he also showed ha, for almos all n, he inerval (n, n n 77 ε conains a prime In 7, Mongomery [6] improved he exponen 77 o 5 The zero densiy esimae of Hxley [7] gives he exponen 6 In 82, Harman [3] sed he sieve mehod o prove ha, for almos all n, he inerval (n, nn 0 ε conains a prime Heah-Brown [5] and Harman [4] menioned ha he exponen 2 can be achieved In [], Jia Chaoha invesigaed he problem of he excepional se of Goldbach nmbers in he shor inerval As a by-proc, he proved ha, for almos all n, he inerval (n, n n 3 ε conains prime nmbers Li Hongze [4] improved he exponen 3 o 2 27 Recenly, Jia Chaoha [2] showed ha, for almos all n, he inerval (n, n n 4 ε conains prime nmbers Wa [20] also obained he same resl Their mehods are differen In [2] only classical mehods are sed and in [20] a new mean vale esimae of Wa [2] is sed in addiion Li Hongze [5] combined hese mehods o improve he exponen 4 o 5 In his paper, we prove he following: Theorem Sppose ha B is a sfficienly large posiive consan, ε is a sfficienly small posiive consan and X is sfficienly large Then for posiive inegers n (X, 2X, excep for O(X log B X vales, he inerval (n, n n 20 ε conains a leas 0005n 20 ε log n prime nmbers Projec sppored by he Tian Yan Iem in he Naional Naral Science Fondaion of China [2]
22 C Jia We apply a mean vale esimae of Deshoillers and Iwaniec [2] (see Lemma 2 Using he classical mean vale esimae insead of ha of Deshoillers and Iwaniec, we can ge he exponen 8 We refer o [3] and he explanaion in [] Throgho his paper, we always sppose ha B is a sfficienly large posiive consan, ε is a sfficienly small posiive consan and ε = ε 2, δ = ε 3 We also sppose ha X is sfficienly large and ha x (X, 2X, η = 2X 20 ε Le c, c and c 2 denoe posiive consans which have differen vales a differen places m M means ha here are posiive consans c and c 2 sch ha c M < m c 2 M We ofen se M(s (M may be anoher capial leer o denoe a Dirichle polynomial of he form M(s = a(m m s, m M where a(m is a complex nmber wih a(m = O( The ahor wold like o hank Profs Wang Yan and Pan Chengbiao for heir encoragemen 2 Mean vale esimae (I Lemma Sppose ha X δ H X, MH = X, M(s is a Dirichle polynomial and H(s = Λ(h h s h H Le b = / log X, T 0 = log B ε X Then for T0 T X, we have ( min 2 η, 2T M(b ih(b i 2 η 2 log 0B x T c H<h c 2 H T P r o o f Le s = b i By he zero-free region of he ζ fncion, for 2X we have Λ(h ( h s = (c 2H s (c H s O(log 2B ε x s So, for T 0 2X, (2 H(s log B ε x According o he discssion in [6], here are O(log 2 X ses S(V, W, where S(V, W is he se of k (k =,, K wih he propery r s (r s Moreover, V M 2 M(b ik < 2V, W H 2 H(b ik < 2W,
Shor inervals conaining prime nmbers 23 where X M 2 V, X H 2 W and V M 2, W H 2 log B ε x Then (3 2T T M(b ih(b i 2 V 2 W 2 x log 2 x S(V, W O(x 2ε, where S(V, W is one of ses wih he above properies Assme X k H < X k, where k is a posiive ineger, k and kδ Applying he mean vale esimae (see Secion 3 of [] or Lemma 7 of [] o M(s and H k (s, we have S(V, W V 2 (M T log d x, S(V, W W 2k (H k T log d x, where d = c/δ 2 Applying he Halász mehod (see Secion 3 of [] or Lemma 7 of [] o M(s and H k (s, we have Ths, where S(V, W (V 2 M V 6 MT log d x, S(V, W (W 2k H k W 6k H k T log d x V 2 W 2 S(V, W V 2 W 2 F log d x, F = min{v 2 (M T, V 2 M V 6 MT, W 2k (H k T, I will be proved ha ( (4 min 2 η, V 2 W 2 F η 2 x log B ε x T and so We consider for cases (a F 2V 2 M, 2W 2k H k Then V 2 W 2 F V 2 W 2 min{v 2 M, W 2k H k } V 2 W 2 (V 2 M 2k (W 2k H k 2k = V k W M 2k H 2 x log B ε x ( min 2 η, V 2 W 2 F η 2 x log B ε x T (b F > 2V 2 M, 2W 2k H k Then W 2k H k W 6k H k T } V 2 W 2 F V 2 W 2 min{v 2 T, V 6 MT, W 2k T, W 6k H k T } V 2 W 2 (V 2 3 2k (V 6 M 2k (W 2k k T = M 2k T
24 C Jia Since k, we have H X k X k 0, M 2k X min 2 ( η, T (c F 2V 2 M, F > 2W 2k H k Then V 2 W 2 F η T x 20 T η 2 x ε 20, and so V 2 W 2 F V 2 W 2 min{v 2 M, W 2k T, W 6k H k T } V 2 W 2 (V 2 M 3k (W 6k H k T 3k MH 3 T 3k, since V M 2 As H X k X 20 min 2 ( η, T 2k ε, we have V 2 W 2 F η 2 3k T 3k x 20 3k T 3k x ε η 2 x ε (d F > 2V 2 M, F 2W 2k H k Then V 2 W 2 F V 2 W 2 min{v 2 T, V 6 MT, W 2k H k } If k 0, hen H X k ( min 2 η, T V 2 W 2 (V 2 T 3 2k (V 6 MT 2k (W 2k H k k = M 2k HT k If k =, hen X 0 H X, M X 76 min 2 ( η, T (k (k X 0(2k, M X 0(2k, and so V 2 W 2 F η k x 20 ( k η 2 x ε, and so V 2 W 2 F η 2 (x 20 ε 8 x 38 45 η 2 x ε Combining he above, we obain (4 Hence, Lemma follows Lemma 2 Sppose ha N(s is a Dirichle polynomial and L(s = l s T c L<l c 2 L Le T Then 2T ( I = 4 ( 2 L 2 i N 2 i ( ( ( T N 2 T 5 2 N 4 T min L, T 2 NL 2 T 2 T ε L
Shor inervals conaining prime nmbers 25 P r o o f Firs we assme c L T 2 If N T, hen by he discssion in Secion 2 of [2], we have I (T N 2 T 2 N 5 4 (T L 2 T ε If N > T, he mean vale esimae yields I (L 2 N T (LN ε (T N 2 T 2 T ε Now we assme T 2 < c L 2T By he discssion in Secion 2 of [2], we ge ( ( I T N 2 T 5 2 N 4 T T 2 T ε L Lasly we assme 2T < c L I follows from Theorem on page 442 of [8] ha Hence, and c L<l c 2 L l = (c 2 2L i (c L 2 i ( 2 i 2 i O L 2 ( L 2 i L 2 L 2 L 2 2T ( I L2 2 T 4 N 2 i L2 NL2 (N T T 4 T 2 T Combining he above, we ge Lemma 2 Lemma 3 Sppose ha MNL = X, M(s, N(s are Dirichle polynomials, and L(s = l s l L Le b = / log X, T = L Assme frher ha M and N lie in one of he following regions: (5 (i M X 32, 2 N M 3 X 0 ; (ii X 32 53 M X 60, 23 N X 80 ; (iii X 53 60 3 M X 32, N M 2 6 3 X (iv X 3 32 32 M X 680, N M 2 X 2 20 Then for T T X, we have ( min 2 η, 2T M(b in(b il(b i 2 η 2 x 2ε T T 20 ;
26 C Jia P r o o f I is sfficien o show ha ( I = min 2 η, 2T ( ( ( 2 T N M 2 i 2 i L 2 i η 2 x 2ε T We shall show ha he above ineqaliy holds, providing M and N saisfy he following condiions: M 2 N X 2 20, M 2 N 3 X 6 40, M 6 N 7 X 4 0, N X 23 80, N 3 M 2 X 3 0 Using he mean vale esimae and Lemma 2, we have 2T ( ( ( 2 N M 2 i 2 i L 2 i T ( 2T ( 4 ( 2 M 2 i N 2 i 2 T ( 2T ( 4 ( 2 L 2 i N 2 i 2 T (M 2 N T 2 (T N 2 T 2 N 5 4 (T L 2 NL 2 T 2 2 T ε Hence, ( I min 2 η, (M 2 N T 2 (T N 2 T 5 2 N 4 (T L 2 2 T ε η 2 T 2 x ε T η 2 (M 2 N η 2 (η N 2 η 2 N 5 4 (η L 2 2 x ε η 2 x 2ε η 2 x 2ε In every region of (5, or condiions are saisfied So Lemma 3 follows Lemma 4 Under he assmpions of Lemma 3, (5 being replaced by he region (6 M X 2 40, N X 60, for T T X, we have ( min 2 η, 2T M(b in(b il(b i 2 η 2 x 2ε T T P r o o f I is sfficien o show ha ( I = min 2 η, 2T ( ( ( 2 T N M 2 i 2 i L 2 i η 2 x 2ε T
Shor inervals conaining prime nmbers 27 Using he mean vale esimae and Lemma 2, we have ( I min 2 η, ( 2T ( 4 T M 2 i 2 ( 2T T min 2 ( η, T N 5 2 T ( 4 ( 4 L 2 i N 2 i ( (M 2 T 2 T N 4 T 2 2 ( ( T min L, T 2 N 2 L 2 T 2 2 T ε L η 2 (M 2 η 2 (η N 4 η 2 2 x ε η 2 MN 5 4 (η L 4 x ε N 5 4 η 2 7 8 x ε η 2 T 2 x ε η 2 x 2ε, since min(l, T/L T 2 Ths Lemma 4 follows 3 Mean vale esimae (II Lemma 5 Sppose ha MHK = X and M(s, H(s and K(s are Dirichle polynomials, and G(s = M(sH(sK(s Le b = / log X, T 0 = log B ε X Assme frher ha for T0 2X, M(b i log B ε x and H(b i log B ε x Moreover, sppose ha M and H saisfy one of he following hree condiions: MH X 20, X 0 H, M 2 /H X 0, X 3 0 M, H 2 /M X 3 5, X 0 M 2 H ; 2 MH X 26 45, M 2 H X 5 4, X 38 45 M 2 H, M 2 H X 2 0, X 2 M 58 H 4 ; 3 MH X 25 44, X 00 H, M 6 H X 5, X 70 M, MH 8 X 23 0, X 20 M 6 H 5 Then for T 0 T X, we have ( (7 min 2 η, 2T G(b i 2 η 2 log 0B x T T P r o o f Using he mehod of Lemma, we only show ha for T = /η = 2X 20 ε, (8 I = 2T T G(b i 2 log 0B x
28 C Jia I Firs, we assme condiion On applying he mean vale esimae and Halász mehod o M 3 (s, H 5 (s and K 2 (s, we ge where I U 2 V 2 W 2 x F log c x, F = min{v 6 (M 3 T, V 6 M 3 V 8 M 3 T, W 0 (H 5 T, W 0 H 5 W 30 H 5 T, U 4 (K 2 T, U 4 K 2 U 2 K 2 T } We discss he following cases: (a F 2V 6 M 3, 2W 0 H 5, 2U 4 K 2 Then U 2 V 2 W 2 min{v 6 M 3, W 0 H 5, U 4 K 2 } U 2 V 2 W 2 (V 6 M 3 3 0 (W 0 H 5 5 (U 4 K 2 2 = V 5 M 0 HK x log B x (b F 2V 6 M 3, 2W 0 H 5, F > 2U 4 K 2 Then U 2 V 2 W 2 min{v 6 M 3, W 0 H 5, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 6 M 3 3 (W 0 H 5 5 (U 4 T 20 (U 2 K 2 T 60 = T 7 5 MHK 30 x ε (c F 2V 6 M 3, F > 2W 0 H 5, F 2U 4 K 2 Then U 2 V 2 W 2 min{v 6 M 3, W 0 T, W 30 H 5 T, U 4 K 2 } U 2 V 2 W 2 (V 6 M 3 3 (W 0 T 3 20 (W 30 H 5 T 60 (U 4 K 2 2 = T 6 MKH 2 x ε (d F 2V 6 M 3, F > 2W 0 H 5, 2U 4 K 2 Then U 2 V 2 W 2 min{v 6 M 3, W 0 T, W 30 H 5 T, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 6 M 3 3 (W 0 T 5 (U 4 T 20 (U 2 K 2 T 60 = T 2 3 MK 30 x ε (e F > 2V 6 M 3, F 2W 0 H 5, 2U 4 K 2 Then U 2 V 2 W 2 min{v 6 T, V 8 M 3 T, W 0 H 5, U 4 K 2 } U 2 V 2 W 2 (V 6 T 7 60 (V 8 M 3 T 60 (W 0 H 5 5 (U 4 K 2 2 = T 3 0 M 20 HK x ε
Shor inervals conaining prime nmbers 2 (f F > 2V 6 M 3, F 2W 0 H 5, F > 2U 4 K 2 Then U 2 V 2 W 2 min{v 6 T, V 8 M 3 T, W 0 H 5, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 6 T 3 (W 0 H 5 5 (U 4 T 20 (U 2 K 2 T 60 = T 4 5 HK 30 x ε (g F > 2V 6 M 3, 2W 0 H 5, F 2U 4 K 2 Then U 2 V 2 W 2 min{v 6 T, V 8 M 3 T, W 0 T, W 30 H 5 T, U 4 K 2 } U 2 V 2 W 2 (V 6 T 3 (W 0 T 3 20 (W 30 H 5 T 60 (U 4 K 2 2 = T 2 H 2 K x ε (h F > 2V 6 M 3, 2W 0 H 5, 2U 4 K 2 Then U 2 V 2 W 2 min{v 6, V 8 M 3, W 0, W 30 H 5, U 4, U 2 K 2 }T U 2 V 2 W 2 (V 6 7 60 (V 8 M 3 60 (W 0 5 (U 4 2 T = T M 20 x ε, since M X II Nex, we assme condiion 2 On applying he mean vale esimae and Halász mehod o M 2 (sh(s, H 5 (s and K 2 (s, we ge where I U 2 V 2 W 2 x F log c x, F = min{v 4 W 2 (M 2 H T, V 4 W 2 M 2 H V 2 W 6 M 2 HT, W 0 (H 5 T, W 0 H 5 W 30 H 5 T, U 4 (K 2 T, U 4 K 2 U 2 K 2 T } We consider several cases: (a F 2V 4 W 2 M 2 H, 2W 0 H 5, 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 M 2 H, W 0 H 5, U 4 K 2 } U 2 V 2 W 2 (V 4 W 2 M 2 H 3 8 (W 0 H 5 8 (U 4 K 2 2 = V 2 M 3 4 HK x log B x
30 C Jia (b F 2V 4 W 2 M 2 H, 2W 0 H 5, F > 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 M 2 H, W 0 H 5, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 4 W 2 M 2 H 2 (W 0 H 5 0 (U 4 T 7 20 (U 2 K 2 T 20 = T 2 5 MHK 0 x ε (c F 2V 4 W 2 M 2 H, F > 2W 0 H 5, F 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 M 2 H, W 0 T, W 30 H 5 T, U 4 K 2 } U 2 V 2 W 2 (V 4 W 2 M 2 H 2 (U 4 K 2 2 = W H 2 MK x log B x (d F 2V 4 W 2 M 2 H, F > 2W 0 H 5, 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 M 2 H, W 0 T, W 30 H 5 T, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 4 W 2 M 2 H 2 (W 30 H 5 T 30 (U 4 T 20 (U 2 K 2 T 60 = T 2 MH 2 3 K 30 x ε (e F > 2V 4 W 2 M 2 H, F 2W 0 H 5, 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 T, V 2 W 6 M 2 HT, W 0 H 5, U 4 K 2 } U 2 V 2 W 2 (V 4 W 2 T 7 20 (V 2 W 6 M 2 HT 20 (W 0 H 5 0 (U 4 K 2 2 = T 2 5 M 0 H 20 K x ε (f F > 2V 4 W 2 M 2 H, F 2W 0 H 5, F > 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 T, V 2 W 6 M 2 HT, W 0 H 5, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 4 W 2 T 7 20 (V 2 W 6 M 2 HT 20 (W 0 H 5 0 (U 4 T 2 = T 0 M 0 H 20 x ε
Shor inervals conaining prime nmbers 3 (g F > 2V 4 W 2 M 2 H, 2W 0 H 5, F 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 T, V 2 W 6 M 2 HT, W 0 T, W 30 H 5 T, U 4 K 2 } U 2 V 2 W 2 (V 4 W 2 T 20 (V 2 W 6 M 2 HT 60 (W 30 H 5 T 30 (U 4 K 2 2 = T 2 M 30 H 60 K x ε (h F > 2V 4 W 2 M 2 H, 2W 0 H 5, 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2, V 2 W 6 M 2 H, W 0, W 30 H 5, U 4, U 2 K 2 }T U 2 V 2 W 2 (V 4 W 2 20 (V 2 W 6 M 2 H 60 (W 30 H 5 30 (U 4 2 T = T M 30 H 60 x ε III Lasly, we assme condiion 3 On applying he mean vale esimae and Halász mehod o M 3 (s, H 4 (s and K 2 (s, we ge where I U 2 V 2 W 2 x F log c x, F = min{v 6 (M 3 T, V 6 M 3 V 8 M 3 T, W 8 (H 4 T, W 8 H 4 W 24 H 4 T, U 4 (K 2 T, U 4 K 2 U 2 K 2 T } Consider he following cases: (a F 2V 6 M 3, 2W 8 H 4, 2U 4 K 2 Then U 2 V 2 W 2 min{v 6 M 3, W 8 H 4, U 4 K 2 } U 2 V 2 W 2 (V 6 M 3 4 (W 8 H 4 4 (U 4 K 2 2 = V 2 M 3 4 HK x log B x (b F 2V 6 M 3, 2W 8 H 4, F > 2U 4 K 2 Then U 2 V 2 W 2 min{v 6 M 3, W 8 H 4, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 6 M 3 3 (W 8 H 4 4 (U 4 T 3 8 (U 2 K 2 T 24 = T 5 2 MHK 2 x ε
32 C Jia (c F 2V 6 M 3, F > 2W 8 H 4, F 2U 4 K 2 Then U 2 V 2 W 2 min{v 6 M 3, W 8 T, W 24 H 4 T, U 4 K 2 } U 2 V 2 W 2 (V 6 M 3 3 (W 8 T 8 (W 24 H 4 T 24 (U 4 K 2 2 = T 6 MKH 6 x ε (d F 2V 6 M 3, F > 2W 8 H 4, 2U 4 K 2 Then U 2 V 2 W 2 min{v 6 M 3, W 8 T, W 24 H 4 T, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 6 M 3 3 (W 8 T 8 (W 24 H 4 T 24 (U 4 T 2 = T 2 3 MH 6 x ε (e F > 2V 6 M 3, F 2W 8 H 4, 2U 4 K 2 Then U 2 V 2 W 2 min{v 6 T, V 8 M 3 T, W 8 H 4, U 4 K 2 } U 2 V 2 W 2 (V 6 T 5 24 (V 8 M 3 T 24 (W 8 H 4 4 (U 4 K 2 2 = T 4 M 8 HK x ε (f F > 2V 6 M 3, F 2W 8 H 4, F > 2U 4 K 2 Then U 2 V 2 W 2 min{v 6 T, V 8 M 3 T, W 8 H 4, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 6 T 5 24 (V 8 M 3 T 24 (W 8 H 4 4 (U 4 T 2 = T 3 4 M 8 H x ε (g F > 2V 6 M 3, 2W 8 H 4, F 2U 4 K 2 Then U 2 V 2 W 2 min{v 6 T, V 8 M 3 T, W 8 T, W 24 H 4 T, U 4 K 2 } U 2 V 2 W 2 (V 6 T 3 (W 8 T 8 (W 24 H 4 T 24 (U 4 K 2 2 = T 2 H 6 K x ε (h F > 2V 6 M 3, 2W 8 H 4, 2U 4 K 2 Then U 2 V 2 W 2 min{v 6, V 8 M 3, W 8, W 24 H 4, U 4, U 2 K 2 }T U 2 V 2 W 2 (V 6 5 24 (V 8 M 3 24 (W 8 4 (U 4 2 T = T M 8 x ε, since M X 2 5 (he laer follows from MH X 25 44 and X 00 H Combining he above, we obain (8 Hence, Lemma 5 follows
Shor inervals conaining prime nmbers 33 Lemma 6 Under he assmpion of Lemma 5, M and H lie in one of he following regions: (i X 70 3 M X 0, M 6 5 X 00 H M 23 8 X 80 ; (ii X 3 0 M X 60, M 2 X 0 H M 23 8 X 80 ; (iii X 60 247 M X 770, X 0 H M 23 8 X 80 ; (iv X 247 770 35 M X 00, X 0 H M X 25 44 ; (v X 35 00 48 M X 450, X 0 H M 6 X 5 ; (vi X 48 450 443 M X 305, X 0 H M X 20 ; (vii X 443 305 M X 55, X 0 H M 2 2 X 0 ; (viii X 55 243 M X 5620, M 4 X 8 H M 2 2 X 0 ; (ix X 243 5620 363 M X 860, M 4 X 8 H M 2 5 X 76 ; (x X 363 860 433 M X 860, X H M 2 5 X 76 Then (7 holds for T 0 T X P r o o f In he regions: X 3 0 M X 60, M 2 X 0 H M 2 X 3 45 ; X 60 M X 55, X 0 H M X 20, we apply Lemma 5 wih condiion In he regions: X 443 305 M X 55, M X 20 H M 2 X 2 0 ; X 55 M X 243 5620, M 4 X 8 H M 2 X 2 0 ; X 243 5620 M X 363 860, M 4 X 8 H M 2 X 5 76 ; X 363 860 M X 433 860, X H M 2 X 5 76, we apply Lemma 5 wih condiion 2 In he regions: X 70 M X 3 0, M 6 5 X 00 H M 8 X 23 80 ; X 3 0 M X 60, M 2 X 3 45 H M 8 X 23 80 ; X 60 M X 247 770, M X 20 H M 8 X 23 80 ; X 247 770 M X 35 00, M X 20 H M X 25 44 ; X 35 00 M X 48 450, M X 20 H M 6 X 5, we apply Lemma 5 wih condiion 3 Ping ogeher he above regions, we ge Lemma 6
34 C Jia Lemma 7 Under he assmpion of Lemma 5, sppose ha M and H also saisfy one of he following hree condiions: MH X 7 0, X 0 H, M 2 H X 63 0, X 33 20 M, H /M X 5, X 38 5 M 2 H ; 2 M 2 H X 52 45, M 58 H X 5 2, X 45 M, MH 5 X 2 20, X 4 M 2 H 0 ; 3 MH X 5 70, X 00 H, M 6 H X 63 20, X 44 M, H 7 /M X 3 0, X 5 M 6 H 5 Then (7 holds for T 0 T X P r o o f We only show ha (8 holds for T = /η = 2X 20 ε I Firs, we assme condiion We apply he mean vale esimae and Halász mehod o M 2 (s, H 5 (s and K 3 (s o ge where I U 2 V 2 W 2 x F log c x, F = min{v 4 (M 2 T, V 4 M 2 V 2 M 2 T, W 0 (H 5 T, W 0 H 5 W 30 H 5 T, U 6 (K 3 T, U 6 K 3 U 8 K 3 T } We consider several cases: (a F 2V 4 M 2, 2W 0 H 5, 2U 6 K 3 Then U 2 V 2 W 2 min{v 4 M 2, W 0 H 5, U 6 K 3 } U 2 V 2 W 2 (V 4 M 2 2 (W 0 H 5 6 (U 6 K 3 3 = W 3 H 5 6 MK x log B x (b F 2V 4 M 2, 2W 0 H 5, F > 2U 6 K 3 Then U 2 V 2 W 2 min{v 4 M 2, W 0 H 5, U 6 T, U 8 K 3 T } U 2 V 2 W 2 (V 4 M 2 2 (W 0 H 5 5 (U 6 T 7 60 (U 8 K 3 T 60 = T 3 0 MHK 20 x ε (c F 2V 4 M 2, F > 2W 0 H 5, F 2U 6 K 3 Then U 2 V 2 W 2 min{v 4 M 2, W 0 T, W 30 H 5 T, U 6 K 3 } U 2 V 2 W 2 (V 4 M 2 2 (W 0 T 3 20 (W 30 H 5 T 60 (U 6 K 3 3 = T 6 MKH 2 x ε
Shor inervals conaining prime nmbers 35 (d F 2V 4 M 2, F > 2W 0 H 5, 2U 6 K 3 Then U 2 V 2 W 2 min{v 4 M 2, W 0 T, W 30 H 5 T, U 6 T, U 8 K 3 T } U 2 V 2 W 2 (V 4 M 2 2 (W 0 T 3 20 (W 30 H 5 T 60 (U 6 T 3 = T 2 MH 2 x ε (e F > 2V 4 M 2, F 2W 0 H 5, 2U 6 K 3 Then U 2 V 2 W 2 min{v 4 T, V 2 M 2 T, W 0 H 5, U 6 K 3 } U 2 V 2 W 2 (V 4 T 20 (V 2 M 2 T 60 (W 0 H 5 5 (U 6 K 3 3 = T 7 5 M 30 HK x ε (f F > 2V 4 M 2, F 2W 0 H 5, F > 2U 6 K 3 Then U 2 V 2 W 2 min{v 4 T, V 2 M 2 T, W 0 H 5, U 6 T, U 8 K 3 T } U 2 V 2 W 2 (V 4 T 2 (W 0 H 5 5 (U 6 T 7 60 (U 8 K 3 T 60 = T 4 5 HK 20 x ε (g F > 2V 4 M 2, 2W 0 H 5, F 2U 6 K 3 Then U 2 V 2 W 2 min{v 4 T, V 2 M 2 T, W 0 T, W 30 H 5 T, U 6 K 3 } U 2 V 2 W 2 (V 4 T 2 (W 0 T 3 20 (W 30 H 5 T 60 (U 6 K 3 3 = T 2 3 H 2 K x ε (h F > 2V 4 M 2, 2W 0 H 5, 2U 6 K 3 Then U 2 V 2 W 2 min{v 4, V 2 M 2, W 0, W 30 H 5, U 6, U 8 K 3 }T U 2 V 2 W 2 (V 4 2 (W 0 5 (U 6 7 60 (U 8 K 3 60 T = T K 20 x ε, since K X II Nex, assme condiion 2 We apply he mean vale esimae and Halász mehod o M 2 (s, H 5 (s and K 2 (sh(s o ge I U 2 V 2 W 2 x F log c x,
36 C Jia where F = min{v 4 (M 2 T, V 4 M 2 V 2 M 2 T, W 0 (H 5 T, W 0 H 5 W 30 H 5 T, U 4 W 2 (K 2 H T, U 4 W 2 K 2 H U 2 W 6 K 2 HT } Consider he following cases: (a F 2V 4 M 2, 2W 0 H 5, 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4 M 2, W 0 H 5, U 4 W 2 K 2 H} U 2 V 2 W 2 (V 4 M 2 2 (U 4 W 2 K 2 H 2 = W H 2 MK x log B x (b F 2V 4 M 2, 2W 0 H 5, F > 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4 M 2, W 0 H 5, U 4 W 2 T, U 2 W 6 K 2 HT } U 2 V 2 W 2 (V 4 M 2 2 (W 0 H 5 0 (U 4 W 2 T 7 20 (U 2 W 6 K 2 HT 20 = T 2 5 MH 20 K 0 x ε (c F 2V 4 M 2, F > 2W 0 H 5, F 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4 M 2, W 0 T, W 30 H 5 T, U 4 W 2 K 2 H} U 2 V 2 W 2 (V 4 M 2 2 (U 4 W 2 K 2 H 2 = W H 2 MK x log B x (d F 2V 4 M 2, F > 2W 0 H 5, 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4 M 2, W 0 T, W 30 H 5 T, U 4 W 2 T, U 2 W 6 K 2 HT } U 2 V 2 W 2 (V 4 M 2 2 (W 30 H 5 T 30 (U 4 W 2 T 20 (U 2 W 6 K 2 HT 60 = T 2 MH 60 K 30 x ε (e F > 2V 4 M 2, F 2W 0 H 5, 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4 T, V 2 M 2 T, W 0 H 5, U 4 W 2 K 2 H} U 2 V 2 W 2 (V 4 T 7 20 (V 2 M 2 T 20 (W 0 H 5 0 (U 4 W 2 K 2 H 2 = T 2 5 M 0 HK x ε
Shor inervals conaining prime nmbers 37 (f F > 2V 4 M 2, F 2W 0 H 5, F > 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4 T, V 2 M 2 T, W 0 H 5, U 4 W 2 T, U 2 W 6 K 2 HT } U 2 V 2 W 2 (V 4 T 7 20 (V 2 M 2 T 20 (W 0 H 5 0 (U 4 W 2 T 2 = T 0 M 0 H 2 x ε (g F > 2V 4 M 2, 2W 0 H 5, F 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4 T, V 2 M 2 T, W 0 T, W 30 H 5 T, U 4 W 2 K 2 H} U 2 V 2 W 2 (V 4 T 20 (V 2 M 2 T 60 (W 30 H 5 T 30 (U 4 W 2 K 2 H 2 = T 2 M 30 H 2 3 K x ε (h F > 2V 4 M 2, 2W 0 H 5, 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4, V 2 M 2, W 0, W 30 H 5, U 4 W 2, U 2 W 6 K 2 H}T U 2 V 2 W 2 (V 4 20 (V 2 M 2 60 (W 30 H 5 30 (U 4 W 2 2 T = T M 30 H 6 x ε III Lasly, we assme condiion 3 We apply he mean vale esimae and Halász mehod o M 2 (s, H 4 (s and K 3 (s o ge where I U 2 V 2 W 2 x F log c x, F = min{v 4 (M 2 T, V 4 M 2 V 2 M 2 T, W 8 (H 4 T, W 8 H 4 W 24 H 4 T, U 6 (K 3 T, U 6 K 3 U 8 K 3 T } Consider he following cases: (a F 2V 4 M 2, 2W 8 H 4, 2U 6 K 3 Then U 2 V 2 W 2 min{v 4 M 2, W 8 H 4, U 6 K 3 } U 2 V 2 W 2 (V 4 M 2 2 (W 8 H 4 6 (U 6 K 3 3 = W 2 3 H 2 3 MK x log B x
38 C Jia (b F 2V 4 M 2, 2W 8 H 4, F > 2U 6 K 3 Then U 2 V 2 W 2 min{v 4 M 2, W 8 H 4, U 6 T, U 8 K 3 T } U 2 V 2 W 2 (V 4 M 2 2 (W 8 H 4 4 (U 6 T 5 24 (U 8 K 3 T 24 = T 4 MHK 8 x ε (c F 2V 4 M 2, F > 2W 8 H 4, F 2U 6 K 3 Then U 2 V 2 W 2 min{v 4 M 2, W 8 T, W 24 H 4 T, U 6 K 3 } U 2 V 2 W 2 (V 4 M 2 2 (W 8 T 8 (W 24 H 4 T 24 (U 6 K 3 3 = T 6 MKH 6 x ε (d F 2V 4 M 2, F > 2W 8 H 4, 2U 6 K 3 Then U 2 V 2 W 2 min{v 4 M 2, W 8 T, W 24 H 4 T, U 6 T, U 8 K 3 T } U 2 V 2 W 2 (V 4 M 2 2 (W 8 T 8 (W 24 H 4 T 24 (U 6 T 3 = T 2 MH 6 x ε (e F > 2V 4 M 2, F 2W 8 H 4, 2U 6 K 3 Then U 2 V 2 W 2 min{v 4 T, V 2 M 2 T, W 8 H 4, U 6 K 3 } U 2 V 2 W 2 (V 4 T 3 8 (V 2 M 2 T 24 (W 8 H 4 4 (U 6 K 3 3 = T 5 2 M 2 HK x ε (f F > 2V 4 M 2, F 2W 8 H 4, F > 2U 6 K 3 Then U 2 V 2 W 2 min{v 4 T, V 2 M 2 T, W 8 H 4, U 6 T, U 8 K 3 T } U 2 V 2 W 2 (V 4 T 2 (W 8 H 4 4 (U 6 T 5 24 (U 8 K 3 T 24 = T 3 4 HK 8 x ε (g F > 2V 4 M 2, 2W 8 H 4, F 2U 6 K 3 Then U 2 V 2 W 2 min{v 4 T, V 2 M 2 T, W 8 T, W 24 H 4 T, U 6 K 3 } U 2 V 2 W 2 (V 4 T 2 (W 8 T 8 (W 24 H 4 T 24 (U 6 K 3 3 = T 2 3 H 6 K x ε
Shor inervals conaining prime nmbers 3 (h F > 2V 4 M 2, 2W 8 H 4, 2U 6 K 3 Then U 2 V 2 W 2 min{v 4, V 2 M 2, W 8, W 24 H 4, U 6, U 8 K 3 }T U 2 V 2 W 2 (V 4 2 (W 8 4 (U 6 5 24 (U 8 K 3 24 T = T K 8 x ε, since X 3 5 MH (he laer follows from X 44 M and X 00 H Combining he above, we obain (8 Hence, Lemma 7 follows Lemma 8 Under he assmpion of Lemma 5, sppose ha M and H lie in one of he following regions: (i X 45 M X 44, M 2 0 X 40 H M 2 5 X (ii X 44 87 M X 72, M 2 0 X 40 H M 2 5 X (iii X 44 87 M X 72, M 6 5 X 25 3 H M 7 X 70 ; (iv X 87 72 33 M X 20, X H M 2 5 X 00 ; (v X 87 72 33 M X 20, M 6 5 X 25 3 H M 7 X 70 ; (vi X 33 20 M X 40, X 3 H M 7 X 70 ; (vii X 40 53 M X 0, X H M X 5 70 ; (viii X 53 0 33 M X 700, X H M 5 X 8 ; (ix X 53 0 33 M X 700, X 0 H M X 5 70 ; (x X 33 700 4 M X 00, X H M 5 X 8 ; (xi X 33 700 4 M X 00, X 0 H M 6 X 63 20 ; (xii X 4 00 M X 2, X 0 H M X 7 0 Then (7 holds for T 0 T X P r o o f In he regions: X 33 20 M X 40, M 2 X 38 55 H M X 5 ; X 40 M X 2, X 0 H M X 7 0, we apply Lemma 7 wih condiion In he regions: 00 ; 00 ; X 45 M X 87 72, M 2 0 X 40 H M 5 X 2 00 ; X 87 72 M X 33 20, X H M 5 X 2 00 ; X 33 20 M X 40, X H M 2 X 38 55 ;
40 C Jia X 40 M X 53 0, X H X 0 ; X 53 0 M X 4 00, X H M X 5 8, we apply Lemma 7 wih condiion 2 In he regions: X 44 M X 33 20, M 6 5 X 25 H M 7 X 3 70 ; X 33 20 M X 40, M X 5 H M 7 X 3 70 ; X 40 M X 33 700, M X 7 0 H M X 5 70 ; X 33 700 M X 4 00, M X 7 0 H M 6 X 63 20, we apply Lemma 7 wih condiion 3 Ping ogeher he above regions, we ge Lemma 8 Lemma Under he assmpion of Lemma 5, sppose ha M and H lie in one of he following regions: 88 5280 M X 275, M 70 7 5 X 25 H M (i 4 X X 88 363 275 M X 860, X H M (ii 4 X 8 ; X 063 433 2640 M X 860, M 2 5 X 76 H M 35 7 (iii 23 X X 843 Then (7 holds for T 0 T X P r o o f Firs we show ha (8 holds for T = /η = 2X 20 ε, providing M and H saisfy he following condiions: MH X 25 44, M 35 H 23 X 7 0, X 22 M 2 H, M 2 H 3 X 3 7 0, X 5 M 70 H 5 We apply he mean vale esimae and Halász mehod o M 2 (sh(s, H 6 (s and K 2 (s o ge I U 2 V 2 W 2 x F log c x, where F = min{v 4 W 2 (M 2 H T, V 4 W 2 M 2 H V 2 W 6 M 2 HT, 8 ; 230 W 2 (H 6 T, W 2 H 6 W 36 H 6 T, U 4 (K 2 T, U 4 K 2 U 2 K 2 T } Consider he following cases: (a F 2V 4 W 2 M 2 H, 2W 2 H 6, 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 M 2 H, W 2 H 6, U 4 K 2 } U 2 V 2 W 2 (V 4 W 2 M 2 H 2 (U 4 K 2 2 = W H 2 MK x log B x
Shor inervals conaining prime nmbers 4 (b F 2V 4 W 2 M 2 H, 2W 2 H 6, F > 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 M 2 H, W 2 H 6, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 4 W 2 M 2 H 2 (W 2 H 6 2 (U 4 T 3 8 (U 2 K 2 T 24 = T 5 2 MHK 2 x ε (c F 2V 4 W 2 M 2 H, F > 2W 2 H 6, F 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 M 2 H, W 2 T, W 36 H 6 T, U 4 K 2 } U 2 V 2 W 2 (V 4 W 2 M 2 H 2 (U 4 K 2 2 = W H 2 MK x log B x (d F 2V 4 W 2 M 2 H, F > 2W 2 H 6, 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 M 2 H, W 2 T, W 36 H 6 T, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 4 W 2 M 2 H 2 (W 36 H 6 T 36 (U 4 T 24 (U 2 K 2 T 72 = T 2 MH 2 3 K 36 x ε (e F > 2V 4 W 2 M 2 H, F 2W 2 H 6, 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 T, V 2 W 6 M 2 HT, W 2 H 6, U 4 K 2 } U 2 V 2 W 2 (V 4 W 2 T 3 8 (V 2 W 6 M 2 HT 24 (W 2 H 6 2 (U 4 K 2 2 = T 5 2 M 2 H 3 24 K x ε (f F > 2V 4 W 2 M 2 H, F 2W 2 H 6, F > 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 T, V 2 W 6 M 2 HT, W 2 H 6, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 4 W 2 T 3 8 (V 2 W 6 M 2 HT 24 (W 2 H 6 2 (U 4 T 2 = T 2 M 2 H 3 24 x ε
42 C Jia (g F > 2V 4 W 2 M 2 H, 2W 2 H 6, F 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 T, V 2 W 6 M 2 HT, W 2 T, W 36 H 6 T, U 4 K 2 } U 2 V 2 W 2 (V 4 W 2 T 24 (V 2 W 6 M 2 HT 72 (W 36 H 6 T 36 (U 4 K 2 2 = T 2 M 36 H 3 72 K x ε (h F > 2V 4 W 2 M 2 H, 2W 2 H 6, 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2, V 2 W 6 M 2 H, W 2, W 36 H 6, U 4, U 2 K 2 }T U 2 V 2 W 2 (V 4 W 2 2 (W 2 2 (U 4 3 8 (U 2 K 2 24 T = T K 2 x ε, since X 2 5 MH (he laer follows from X 7 5 M 70 H 5 In every region, or condiions are saisfied So he proof of Lemma is complee Lemma 0 Under he assmpion of Lemma 5, sppose ha M and H lie in one of he following regions: X 37 554 00 M X 340, M 82 33 6 X 230 H M 70 7 (i 5 X X 554 88 340 M X 275, X H M 70 7 (ii 5 X 25 ; X 433 45 M X 860, M 35 7 23 X 230 H M 4 42 (iii 27 X Then (7 holds for T 0 T X P r o o f Firs we show ha (8 holds for T = /η = 2X 20 ε, providing M and H saisfy he following condiions: MH X 73 30, M 4 H 27 X 42 20, X 65 M 2 H, M 2 H 5 X 33 3 0, X 0 M 82 H 6 We apply he mean vale esimae and Halász mehod o M 2 (sh(s, H 7 (s and K 2 (s o ge I U 2 V 2 W 2 x F log c x, where F = min{v 4 W 2 (M 2 H T, V 4 W 2 M 2 H V 2 W 6 M 2 HT, 25 ; 540 W 4 (H 7 T, W 4 H 7 W 42 H 7 T, U 4 (K 2 T, U 4 K 2 U 2 K 2 T } Consider he following cases:
Shor inervals conaining prime nmbers 43 (a F 2V 4 W 2 M 2 H, 2W 4 H 7, 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 M 2 H, W 4 H 7, U 4 K 2 } U 2 V 2 W 2 (V 4 W 2 M 2 H 2 (U 4 K 2 2 = W H 2 MK x log B x (b F 2V 4 W 2 M 2 H, 2W 4 H 7, F > 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 M 2 H, W 4 H 7, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 4 W 2 M 2 H 2 (W 4 H 7 4 (U 4 T 28 (U 2 K 2 T 28 = T 3 7 MHK 4 x ε (c F 2V 4 W 2 M 2 H, F > 2W 4 H 7, F 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 M 2 H, W 4 T, W 42 H 7 T, U 4 K 2 } U 2 V 2 W 2 (V 4 W 2 M 2 H 2 (U 4 K 2 2 = W H 2 MK x log B x (d F 2V 4 W 2 M 2 H, F > 2W 4 H 7, 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 M 2 H, W 4 T, W 42 H 7 T, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 4 W 2 M 2 H 2 (W 42 H 7 T 42 (U 4 T 3 28 (U 2 K 2 T 84 = T 2 MH 2 3 K 42 x ε (e F > 2V 4 W 2 M 2 H, F 2W 4 H 7, 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 T, V 2 W 6 M 2 HT, W 4 H 7, U 4 K 2 } U 2 V 2 W 2 (V 4 W 2 T 28 (V 2 W 6 M 2 HT 28 (W 4 H 7 4 (U 4 K 2 2 = T 3 7 M 4 H 5 28 K x ε (f F > 2V 4 W 2 M 2 H, F 2W 4 H 7, F > 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 T, V 2 W 6 M 2 HT, W 4 H 7, U 4 T, U 2 K 2 T } U 2 V 2 W 2 (V 4 W 2 T 28 (V 2 W 6 M 2 HT 28 (W 4 H 7 4 (U 4 T 2 = T 3 4 M 4 H 5 28 x ε
44 C Jia (g F > 2V 4 W 2 M 2 H, 2W 4 H 7, F 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2 T, V 2 W 6 M 2 HT, W 4 T, W 42 H 7 T, U 4 K 2 } U 2 V 2 W 2 (V 4 W 2 T 3 28 (V 2 W 6 M 2 HT 84 (W 42 H 7 T 42 (U 4 K 2 2 = T 2 M 42 H 5 28 K x ε (h F > 2V 4 W 2 M 2 H, 2W 4 H 7, 2U 4 K 2 Then U 2 V 2 W 2 min{v 4 W 2, V 2 W 6 M 2 H, W 4, W 42 H 7, U 4, U 2 K 2 }T U 2 V 2 W 2 (V 4 W 2 2 (W 4 4 (U 4 28 (U 2 K 2 28 T = T K 4 x ε, since X 3 0 MH (he laer follows from X 65 M 2 H In every region, or condiions are saisfied, so he proof of Lemma 0 is complee Lemma Under he assmpion of Lemma 5, sppose ha M and H lie in one of he following regions: (i X 843 5280 37 M X 00, M 70 7 5 X 25 H M 4 X 8 ; (ii X 37 00 554 M X 340, M 82 33 6 X 230 H M 4 X 8 ; (iii X 554 340 363 M X 860, X H M 4 X 8 ; (iv X 063 2640 M X 45, M 2 5 X 76 H M 35 7 23 X 230 ; (v X 45 433 M X 860, M 2 5 X 76 H M 4 42 27 X 540 Then (7 holds for T 0 T X P r o o f Ping ogeher regions in Lemmas and 0, we can ge Lemma Lemma 2 Under he assmpion of Lemma 5, sppose ha M and H lie in one of he following regions: X 433 26 860 M X 550, M 35 2 X 40 H M 2 (i 0 X 40 ; X 26 87 550 M X 72, X H M 2 (ii 0 X 40 ; X 233 4 480 M X 00, M 5 X 8 H M 70 7 (iii X X 4 244 00 M X 550, X H M 70 7 (iv X 55 Then (7 holds for T 0 T X 55 ;
Shor inervals conaining prime nmbers 45 P r o o f Firs we show ha (8 holds for T = /η = 2X 20 ε, providing ha M and H saisfy he following condiions: M 2 H X 25 22, M 70 H X 7 5, X 44 M, MH 6 X 3 7 20, X 0 M 35 H 2 We apply he mean vale esimae and Halász mehod o M 2 (s, H 6 (s and K 2 (sh(s o ge I U 2 V 2 W 2 x F log c x, where F = min{v 4 (M 2 T, V 4 M 2 V 2 M 2 T, W 2 (H 6 T, W 2 H 6 W 36 H 6 T, U 4 W 2 (K 2 H T, U 4 W 2 K 2 H U 2 W 6 K 2 HT } Consider he following cases: (a F 2V 4 M 2, 2W 2 H 6, 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4 M 2, W 2 H 6, U 4 W 2 K 2 H} U 2 V 2 W 2 (V 4 M 2 2 (U 4 W 2 K 2 H 2 = W H 2 MK x log B x (b F 2V 4 M 2, 2W 2 H 6, F > 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4 M 2, W 2 H 6, U 4 W 2 T, U 2 W 6 K 2 HT } U 2 V 2 W 2 (V 4 M 2 2 (W 2 H 6 2 (U 4 W 2 T 3 8 (U 2 W 6 K 2 HT 24 = T 5 2 MH 3 24 K 2 x ε (c F 2V 4 M 2, F > 2W 2 H 6, F 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4 M 2, W 2 T, W 36 H 6 T, U 4 W 2 K 2 H} U 2 V 2 W 2 (V 4 M 2 2 (U 4 W 2 K 2 H 2 = W H 2 MK x log B x (d F 2V 4 M 2, F > 2W 2 H 6, 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4 M 2, W 2 T, W 36 H 6 T, U 4 W 2 T, U 2 W 6 K 2 HT } U 2 V 2 W 2 (V 4 M 2 2 (W 36 H 6 T 36 (U 4 W 2 T 24 (U 2 W 6 K 2 HT 72 = T 2 MH 3 72 K 36 x ε
46 C Jia (e F > 2V 4 M 2, F 2W 2 H 6, 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4 T, V 2 M 2 T, W 2 H 6, U 4 W 2 K 2 H} U 2 V 2 W 2 (V 4 T 3 8 (V 2 M 2 T 24 (W 2 H 6 2 (U 4 W 2 K 2 H 2 = T 5 2 M 2 HK x ε (f F > 2V 4 M 2, F 2W 2 H 6, F > 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4 T, V 2 M 2 T, W 2 H 6, U 4 W 2 T, U 2 W 6 K 2 HT } U 2 V 2 W 2 (V 4 T 3 8 (V 2 M 2 T 24 (W 2 H 6 2 (U 4 W 2 T 2 = T 2 M 2 H 2 x ε (g F > 2V 4 M 2, 2W 2 H 6, F 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4 T, V 2 M 2 T, W 2 T, W 36 H 6 T, U 4 W 2 K 2 H} U 2 V 2 W 2 (V 4 T 24 (V 2 M 2 T 72 (W 36 H 6 T 36 (U 4 W 2 K 2 H 2 = T 2 M 36 H 2 3 K x ε (h F > 2V 4 M 2, 2W 2 H 6, 2U 4 W 2 K 2 H Then U 2 V 2 W 2 min{v 4, V 2 M 2, W 2, W 36 H 6, U 4 W 2, U 2 W 6 K 2 H}T U 2 V 2 W 2 (V 4 3 8 (V 2 M 2 24 (W 2 2 (U 4 W 2 2 T = T M 2 x ε, since M X 3 5 (he laer follows from M 70 H X 7 5 In every region, or condiions are saisfied So he proof of Lemma 2 is complee 4 Mean vale esimae (III Lemma 3 Sppose ha P QRK = X and ha P (s, Q(s, R(s and K(s are Dirichle polynomials Define G(s = P (sq(sr(sk(s Le b = / log X and T 0 = log B ε X Assme frher ha for T0 2X, P (b iq(b i log B ε x and R(b i log B ε x Moreover, assme ha X R Q
Shor inervals conaining prime nmbers 47 and ha P and Q lie in one of he following regions: (i X 33 580 P X 80, P X 33 20 Q P ; (ii X 80 53 P X 220, P X 33 20 Q P X (iii X 53 220 5 P X 580, P X 33 20 Q P X 53 (iv X 5 (v X 7 (vi X 344 860 37 P X 00, X Q P 5 67 X 34 ; X 37 (vii 40 ; 0 ; 580 P X 7 55, P X 87 72 Q P X 53 0 ; 55 P X 344 860, P X 87 72 Q P 67 X 5 00 P X 653 700, X Q P X 4 Then (7 holds for T 0 T X P r o o f Le m = pq, h = r (a On applying Lemma 8 wih region (iv, we see ha (7 holds nder he condiions 00 34 ; P X 87 72 Q P X 33 20, X R Q (P Q 5 X 2 00, which can be wrien as P X 87 72 Q P X 33 20, X Q P 6 X 2 20, X R Q In he regions: X 5 580 P X 344 860, P X 87 72 Q P X 33 20 ; X 344 860 P X 73 430, X Q P X 33 20, he above condiions on P and Q are saisfied (b On applying Lemma 8 wih region (vi, we see ha (7 holds nder he condiions P X 33 20 Q P X 40, X R Q (P Q 7 X 3 70, which can be wrien as P X 33 20 Q P X 40, X Q P 6 X 3 60, X R Q In he regions: X 33 580 P X 80, P X 33 20 Q P ; X 80 P X 73 430, P X 33 20 Q P X 40 ; X 73 430 P X 25 680, X Q P X 40, he above condiions on P and Q are saisfied (c On applying Lemma 8 wih region (vii, we see ha (7 holds nder he condiions P X 40 Q P X 53 0, X R Q (P Q X 5 70,
48 C Jia which can be wrien as P X 40 Q P X 53 0, X Q P 2 X 5 40, X R Q In he regions: X 53 220 P X 25 680, P X 40 Q P X 53 0 ; X 25 680 P X 703 870, X Q P X 53 0, he above condiions on P and Q are saisfied (d On applying Lemma 8 wih regions (viii and (x, we see ha (7 holds nder he condiions P X 53 0 Q P X 4 00, X R Q (P Q X 5 8, which can be wrien as P X 53 0 Q P X 4 00, X Q P 67 X 5 34, X R Q In he regions: X 7 55 P X 37 00, P X 53 0 Q P 67 X 5 34 ; X 37 00 P X 703 870, P X 53 0 Q P X 4 00 ; X 703 870 P X 653 700, X Q P X 4 00, he above condiions on P and Q are saisfied Ping ogeher he above regions, we ge Lemma 3 Lemma 4 Sppose ha P QRL = X and ha P (s, Q(s and R(s are Dirichle polynomials, L(s = l s, l L and F (s = P (sq(sr(sl(s Le b = / log X and T = L Assme frher ha for T 2X, P (biq(bi log B ε x and R(bi log B ε x Moreover, assme ha X R Q and ha P and Q lie in one of he following regions: (i X 2 P X 00, X Q P ; (ii X 2 00 3 P X 60, X Q P 2 7 3 X 20 ; (iii X 3 60 263 P X 4300, X Q P 2 6 X (iv X 263 4300 344 P X 860, X Q P X 87 20 ; 72
Then for T T X, we have ( ( min 2 η, 2T T T Shor inervals conaining prime nmbers 4 F (b i 2 η 2 log 0B x P r o o f Le m = pq and n = r An applicaion of Lemma 3 yields ha ( holds nder one of he following condiions: (a M X 32, 2 N M 3 X 0 ; (b X 32 53 M X 60, 23 N X 80 ; (c X 53 60 3 M X 32, N M 2 6 3 X (d X 3 32 M X 45, N M 2 X 2 20 ; (e X 45 87 M X 72, N M 2 X 2 20 Using he same discssion as in Lemma 8 wih regions (i and (ii, we dece ha ( holds nder he condiion 20 ; (f X 45 M X 87 72, M 2 X 2 20 N M 5 X 2 00 In he regions: condiion (a is saisfied In he regions: X P X 64, X Q P ; X 64 P X 477 2720, X Q P X 32, X 64 P X 53 320, P X 32 Q P ; X 53 320 P X 477 2720, P X 32 Q P X 53 60 ; X 477 2720 P X 63 2720, X Q P X 53 60, condiion (b is saisfied In he regions: X 53 320 P X 3 64, P X 53 60 Q P ; X 3 64 P X 63 2720, P X 53 60 Q P X 3 32 ; X 63 2720 P X 87 2720, X Q P X 3 32, condiion (c is saisfied
50 C Jia In he regions: X 3 64 P X 2 00, P X 3 32 Q P ; X 2 00 P X 3 60, P X 3 32 Q P 2 3 X 7 20 ; X 3 60 P X 87 2720, P X 3 32 Q P X 45 ; X 87 2720 P X 242 765, X Q P X 45, condiion (d is saisfied In he regions: X 3 60 P X 263 4300, P X 45 Q P 6 X 2 20 ; X 263 4300 P X 242 765, P X 45 Q P X 87 72 ; X 242 765 P X 344 860, X Q P X 87 72, condiions (e and (f are saisfied Ping ogeher he above regions, we ge Lemma 4 Lemma 5 Under he assmpion of Lemma 4, assme also ha X R Q and ha P and Q lie in one of he following regions: (i X 27 800 653 P X 700, P X 4 00 Q X 60 ; (ii X 653 700 5073 P X 320, X Q X 60 ; (iii X 5073 320 554 P X 340, X Q P 82 33 6 X 230 Then ( holds for T T X P r o o f Le m = pq and n = r An applicaion of Lemma 4 yields ha ( holds nder he condiions In he regions: Q P X 2 40, X R Q X 60 X 27 800 P X 653 700, P X 4 00 Q X 60 ; X 653 700 P X 5073 320, X Q X 60 ; X 5073 320 P X 554 340, X Q P 82 6 X 33 230, he above condiions on P and Q are saisfied So he proof of Lemma 5 is complee
Shor inervals conaining prime nmbers 5 5 The remainder erm in he sieve mehod Lemma 6 Sppose ha M X 2 40, H X 60 and ha a(m = O(, b(h = O( Then for real nmbers x (X, 2X, excep for a se he measre of which is O(X log B X, we have Σ = m M h H ( a(mb(h x<mhl xηx ηx = O(ηx log B x mh P r o o f If MH X 20, he conclsion is obvios In he following we sppose (0 MH > X 20 Le b = / log X and MHL = X Sppose ha M(s and H(s are Dirichle polynomials, L(s = l L /ls and F (s = M(sH(sL(s Perron s formla yields x<mhl xηx m M, h H a(mb(h = bix 2πi b ix F (s ( ηs x s ds O(x ε s If s = b i and c L, by Theorem on page 442 of [8], we have l s = (c 2L s (c L s ( O s L Moreover, c L<l c 2 L ( η s x s = ηx s O( s η 2 x, s Le T = L Then bit F (s ( ηs x s ds 2πi s b it where ( η s x s ηx s = η bit M(sH(s (c2l s (c L s x s ds O(S O(S 2, 2πi s b it S = ηx L S 2 = η 2 x T T M(b ih(b i, T T M(b ih(b i
52 C Jia A rivial esimae yields S ηx L ηx ε and S 2 η 2 x L ηx ε By Perron s formla again, bit η M(sH(s (c2l s (c L s x s ds 2πi s b it Now we have ( Σ = where x<mhl xηx m M, h H = ηx m M h H = ηx m M h H a(mb(h mh a(mb(h mh a(mb(h ηx m M h H ( ηx ε O T O(ηx ε a(mb(h mh O ( ηx ε MH = b it F (sϱ(sx s ds bix F (sϱ(sx s ds O(ηx ε, 2πi 2πi b ix bit ϱ(s = ( ηs s If s = b i and Im(s T, hen ϱ(s min(η, /T We proceed o esimae We have Here 2X X dx b2it bit Ψ = Ψ log 2 x 2X X bix dx F (sϱ(sx s ds 2 bit max T T X F (sϱ(sx s ds 2 2X X dx b2it bit F (sϱ(sx s ds 2
= 2X X By Lemma 4, b2it dx Shor inervals conaining prime nmbers 53 bit min 2 ( η, T x 3 min 2 ( η, T b2it ds bit b2it bit b2it bit x 3 log x min 2 ( η, T F (s F (s 2 ϱ(s ϱ(s 2 x s s 2 ds 2 b2it ds 2T Ψ x 3 log 3 x max T T X min2 bit b2it ds T bit F (s F (s 2 F (b i 2 ( η, 2T T T Hence, he measre of he se of x saisfying bix F (sϱ(sx s ds ηx log B x bit is O(X log B X In he same way, we can deal wih he inegral b it b ix F (sϱ(sx s ds So he proof of Lemma 6 is complee 2X X x s s 2 dx ds 2 F (s 2 F (s 2 2 ds 2 s s 2 F (b i 2 η 2 x 3 ε Lemma 7 Sppose ha a(p, b(q, c(r = O( and ha X R Q Moreover, assme ha P and Q lie in one of he following regions: X (i 2 P X 00, X Q P ; (ii X 2 00 3 P X 60, X Q P 2 7 3 X (iii X 3 (iv X 263 (v X 27 (vi X 653 700 5073 P X 320, X Q X 60 ; X 5073 (vii 20 ; 60 P X 263 4300, X Q P 6 X 2 20 ; 4300 P X 344 860, X Q P X 87 72 ; 800 P X 653 700, P X 4 00 Q X 60 ; 320 P X 554 340, X Q P 82 6 X 33 230
54 C Jia Le b = / log X, P QRL = X and T = L Assme frher ha for T 2X, P (b iq(b i log B ε x and R(b i log B ε x Then for real nmbers x (X, 2X, excep for a se he measre of which is O(X log B X, we have ( a(pb(qc(r ηx = O(ηx log B x pqr p P q Q r R x<pqrl xηx P r o o f Using Lemmas 4, 5 and he discssion in Lemma 6, we ge he asserion 6 Asympoic formla Lemma 8 Sppose ha X 76 M X δ and ha 0 a(m = O( If m has a prime facor < X δ, hen a(m = 0 Then for real nmbers x (X, 2X, excep for a se he measre of which is O(X log B X, we have Σ = a(m x<mp xηx m M ( ( = η O a(m log x x<mh xηx m M m M x m <p 2x m O(ηx log B x P r o o f Le b = / log X and MH = X Sppose ha M(s is a Dirichle polynomial, H(s = h H Λ(h/hs and G(s = M(sH(s Perron s formla yields Σ = a(mλ(h = bix G(s ( ηs x s ds O(x ε 2πi s b ix Le T 0 = log B ε X By ( and he discssion in Lemma 6, bit 0 G(s ( ηs x s ds 2πi s b it 0 = η bit 0 M(s (c2h s (c H s x s ds O(S O(S 2, 2πi s b it 0 where T 0 S = ηx log 2B ε x T 0 M(b i, S 2 = η 2 x T 0 T 0 M(b i
A rivial esimae yields By Perron s formla again, Shor inervals conaining prime nmbers 55 S ηx log 2B x and S 2 ηx log 2B x bit η 0 M(s (c2h s (c H s x s ds 2πi s b it 0 Hence, where and Σ = ηx m M = ηx m M = ηx m M a(m m a(m m ( ηx log 2 O x a(m m O(ηx log 2B x b ix T 0 O(ηx log 2B x O ( ηx log 2 x b it 0 G(sϱ(sx s ds bix G(sϱ(sx s ds, 2πi 2πi bit 0 ϱ(s = ( ηs s By Lemma and he discssion in Lemma 6, we have bix G(sϱ(sx s ds = O(ηx log 2B x 2πi bit 0 b it 0 G(sϱ(sx s ds = O(ηx log 2B x 2πi b ix for real nmbers x (X, 2X, excep for a se he measre of which is O(X log B X So, Σ = ηx a(m m O(ηx log 2B x, m M ( ( Σ = ηx O log x m M ( ( = η O a(m log x m M The proof of Lemma 8 is complee M a(m m log(x/m O(ηx log B x x m <p 2x m O(ηx log B x
56 C Jia Lemma Sppose ha X HK X 76, MHK = X and ha 0 b(h = O(, 0 g(k = O( If h has a prime facor < X δ, hen b(h = 0, and similarly for g(k Sppose ha H(s and K(s are Dirichle polynomials, M(s = m M Λ(m/ms and G(s = M(sH(sK(s Le b = / log X and T 0 = log B ε X If (7 holds for T 0 T X, hen b(hg(k x<hkp xηx h H, k K ( ( = η O log x h H, k K b(hg(k x 2x hk <p hk O(ηx log B x 7 Bchsab s fncion We define w( as he coninos solion of he eqaions { w( = /, 2, (2 (w( = w(, > 2 w( is called Bchsab s fncion and plays an imporan role in finding asympoic formlas in he sieve mehod In pariclar, w( = w( = log( w( =, 2 3, log( log( 3 2 2 log( 2 log(s s Lemma 20 We have he following bonds: (i w( 05607 for 247, (ii w( 05644 for 3, (iii 0562 w( 0567 for 4 log(, 3 4, ds, 4 5 P r o o f I is easy o see ha 05 w( for 2 Then we employ incion Sppose ha 05 w( for k k If k k 2,
hen (2 yields Shor inervals conaining prime nmbers (3 w( = (k w(k w( Hence, 05 w( for k k 2 By incion, we obain 05 w( for If > 2, (2 yields (4 w w( w( ( = If 2 3, by calclaion, we have max w(2 0 k 0 0 4 k 05676 4 From (4 and 05 w( for, i follows ha w ( 4 if > 2 Using Lagrange s mean vale heorem, we have w( 05672 for 2 3 By incion, we obain 05 w( 05672 for 2 If 3 4, by calclaion, we have max w(3 0 k 0 0 4 k 05643, 4 min w(3 0 k 0 0 4 k 05608 4 From (4 and 05 w( 05672 for 2, i follows ha w ( 00224 if > 3 The above discssion implies ha 05607 w( 05644 for 3 By he same discssion we can also ge 05607 w( for 247 If 4 5, by calclaion, we have max w(4 0 k 0 0 4 k 0566, 4 k min w(4 0 k 0 0 4 k 0563 4 The above discssion and he fac ha w ( 00224 for > 3 imply ha 0562 w( 0567 for 4 Gahering ogeher he above discssion, we ge Lemma 20 Lemma 2 Sppose ha E = {n : < n 2} and z Le P (z = p<z p Then for sfficienly large and z, we have ( ( log S(E, z = = w log z <n 2 (n,p (z= O(ε log z P r o o f See Lemma 5 of [0] If (2 2 heorem < z, i is he prime nmber
58 C Jia 8 Sieve mehod We proceed o show ha (5 π(x ηx π(x 00ηx log x for real nmbers x (X, 2X, excep for a se he measre of which is O(X log B X Le (6 Then A = {n : x < n x ηx}, P (z = p, S(A, z = p<z a A (a,p (z= (7 π(x ηx π(x = S(A, (2X 2 Bchsab s ideniy yields (8 S(A, (2X 2 = S(A, X X <p (2X 2 = S(A, X X <p (2X 2 X <p (2X 2 S(A p, p S(A p, X X <q<min(p,(2x/p 2 S(A pq, q The following lemmas always concern real nmbers x (X, 2X, excep for a se he measre of which is O(X log B X Le ( B = {n : x < n 2x} Lemma 22 P r o o f We have S(A, X 530022ηx log x (20 S(A, X = S(A, X δ S(A p, p Le r(a, d = A d ηx d, W (z = p<z X δ <p X ( = ( O(ε e γ p log z, where γ is Eler s consan
Shor inervals conaining prime nmbers 5 Le z = X δ and D = X 2 40 Applying Iwaniec s sieve mehod (see Theorem of [8], we have where S(A, X δ ηx log z f R = ( log D O(εηx log x R, log z m X 2 40 a(m r(a, m Lemma 6 yields R = O(ηx log 5 x By Theorem 8 on page 8 of [7], we have ( log D f = e γ O(ε 2, log z where γ is Eler s consan Ths, S(A, X δ e γ δ ηx log x O(εηx log x In he same way, S(A, X δ ηx ( log D log z F log z O(εηx log x e γ δ ηx log x O(εηx log x So, we have he asympoic formla m X 2 40 (2 S(A, X δ = e γ δ ηx log x O(εηx log x Now, X δ <p X S(A p, p = x<pq xηx, b(m r(a, m where X δ < p X and he leas prime facor of q is greaer han p Using Lemmas 8 and 2 and he prime nmber heorem, we have (22 S(A p, p X δ <p X = η = ηx X δ <p X X δ <p X S(B p, p O(εηx log x p log p w ( log(x/p log p O(δηx log x
60 C Jia = ηx log x = ηx log x δ δ 2 w { = ηx log x δ w ( O(δηx log x w( O(δηx log x ( δ = e γ δ ηx log x w ( w ( } O(δηx log x ηx log x O(δηx log x, since w(/δ = e γ O(ε 2 (see Lemma 2 on page 7 of [7] Hence, ( (23 S(A, X = w ηx log x O(δηx log x By Lemma 20, we ge S(A, X 0562ηx log x O(δηx log x 530022ηx log x So he proof of Lemma 22 is complee Lemma 23 X <p (2X 2 P r o o f Bchsab s ideniy yields S(A p, X X <p (2X 2 = X <p (2X 2 S(A p, X 82347ηx log x S(A p, X δ X <p (2X 2 X δ <q<x Using Lemma 6, in he same way as in Lemma 22, we have S(A p, X δ S(A pq, q X <p (2X 2 = X <p (2X 2 e γ δ p ηx log x O(εηx log x = e γ δ ηx log x 2 O(εηx log x
Shor inervals conaining prime nmbers 6 Using Lemmas 8 and 2, in he same way as in Lemma 22, we have X <p (2X 2 = η = ηx X δ <q<x X <p (2X 2 X <p (2X 2 O(δηx log x = ηx log x = ηx log x 2 2 = δ ηx log x S(A pq, q X δ <q<x X δ <q<x δ ( 2 ηx log x = e γ δ ηx log x 2 w S(B pq, q O(εηx log x ( log(x/(pq pq log q w log q ( δ ( O(δηx log x w(r dr O(δηx log x ( ( w ( δ 2 O(δηx log x 2 ( w ( O(δηx log x ηx log x 2 ( w ( Gahering ogeher he above discssion and applying Lemma 20, we have X <p (2X 2 S(A p, X = ηx log x 2 ( w ( O(δηx log x
62 C Jia 0567 ( log ηx log x O(δηx log x 8 82347ηx log x So he proof of Lemma 23 is complee We now se (24 Ω = where X <p (2X 2 4 i= X <q<min(p,( 2X p 2 4 S(A pq, q = Ω i, (p,q D i i= D = {(p, q : X < p X 2 00, X < q < p}, S(A pq, q D 2 = {(p, q : X 2 00 < p X 3 60, X < q < p 2 3 X 7 20 }, D 3 = {(p, q : X 3 60 < p X 33 580, X < q < p 6 X 2 20 }, D 4 = {(p, q : X 33 580 < p X 80, X < q < p 6 X 2 20 }, D 5 = {(p, q : X 33 580 < p X 80, p X 33 20 < q < p}, D 6 = {(p, q : X 80 < p X 53 220, X < q < p 6 X 2 20 }, D 7 = {(p, q : X 80 < p X 53 220, p X 33 20 < q < p X 40 }, D 8 = {(p, q : X 53 220 < p X 263 4300, X < q < p 6 X 2 20 }, D = {(p, q : X 53 220 < p X 263 4300, p X 33 20 < q < p X 53 0 }, D 0 = {(p, q : X 263 4300 < p X 5 580, X < q < p X 87 72 }, D = {(p, q : X 263 4300 < p X 5 580, p X 33 20 < q < p X 53 0 }, D 2 = {(p, q : X 5 580 < p X 70, X < q < p X 87 72 }, D 3 = {(p, q : X 5 580 < p X 70, p X 87 72 < q < p X 53 0 }, D 4 = {(p, q : X 70 < p X 3 0, X < q < p X 87 72 }, D 5 = {(p, q : X 70 < p X 3 0, p X 87 72 < q < p X 53 0 }, D 6 = {(p, q : X 70 < p X 3 0, p 6 5 X 00 < q < p 8 X 23 80 }, D 7 = {(p, q : X 3 0 < p X 7 55, X < q < p X 87 72 },
Shor inervals conaining prime nmbers 63 D 8 = {(p, q : X 3 0 < p X 7 55, p X 87 72 < q < p X 53 0 }, D = {(p, q : X 3 0 < p X 7 55, p 2 X 0 < q < p 8 X 23 80 }, D 20 = {(p, q : X 7 55 < p X 60, X < q < p X 87 72 }, D 2 = {(p, q : X 7 55 < p X 60, p X 87 72 < q < p 67 X 5 34 }, D 22 = {(p, q : X 7 55 < p X 60, p 2 X 0 < q < p 8 X 23 80 }, D 23 = {(p, q : X 60 < p X 247 770, X < q < p X 87 72 }, D 24 = {(p, q : X 60 < p X 247 770, p X 87 72 < q < p 67 X 5 34 }, D 25 = {(p, q : X 60 < p X 247 770, X 0 < q < p 8 X 23 80 }, D 26 = {(p, q : X 247 770 < p X 35 00, X < q < p X 87 72 }, D 27 = {(p, q : X 247 770 < p X 35 00, p X 87 72 < q < p 67 X 5 34 }, D 28 = {(p, q : X 247 770 < p X 35 00, X 0 < q < p X 25 44 }, D 2 = {(p, q : X 35 00 < p X 48 450, X < q < p X 87 72 }, D 30 = {(p, q : X 35 00 < p X 48 450, p X 87 72 < q < p 67 X 5 34 }, D 3 = {(p, q : X 35 00 < p X 48 450, X 0 < q < p 6 X 5 }, D 32 = {(p, q : X 48 450 < p X 443 305, X < q < p X 87 72 }, D 33 = {(p, q : X 48 450 < p X 443 305, p X 87 72 < q < p 67 X 5 34 }, D 34 = {(p, q : X 48 450 < p X 443 305, X 0 < q < p X 20 }, D 35 = {(p, q : X 443 305 < p X 55, X < q < p X 87 72 }, D 36 = {(p, q : X 443 305 < p X 55, p X 87 72 < q < p 67 X 5 34 }, D 37 = {(p, q : X 443 305 < p X 55, X 0 < q < p 2 X 2 0 }, D 38 = {(p, q : X 55 < p X 344 860, X < q < p X 87 72 }, D 3 = {(p, q : X 55 < p X 344 860, p X 87 72 < q < p 67 X 5 34 }, D 40 = {(p, q : X 55 < p X 344 860, p 4 X 8 < q < p 2 X 2 0 }, D 4 = {(p, q : X 344 860 < p X 843 5280, X < q < p 67 X 5 34 }, D 42 = {(p, q : X 344 860 < p X 843 5280, p 4 X 8 < q < p 2 X 2 0 },
64 C Jia D 43 = {(p, q : X 843 5280 < p X 37 00, X < q < p 67 X 5 34 }, D 44 = {(p, q : X 843 5280 < p X 37 00, p 70 5 X 7 25 < q < p 4 X 8 }, D 45 = {(p, q : X 843 5280 < p X 37 00, p 4 X 8 < q < p 2 X 2 0 }, D 46 = {(p, q : X 37 00 < p X 27 800, X < q < p X 4 00 }, D 47 = {(p, q : X 37 00 < p X 27 800, p 82 6 X 33 230 < q < p 4 X 8 }, D 48 = {(p, q : X 37 00 < p X 27 800, p 4 X 8 < q < p 2 X 2 0 }, D 4 = {(p, q : X 27 800 < p X 243 5620, X < q < p X 4 00 }, D 50 = {(p, q : X 27 800 < p X 243 5620, p X 4 00 < q < X 60 }, D 5 = {(p, q : X 27 800 < p X 243 5620, p 82 6 X 33 230 < q < p 4 X 8 }, D 52 = {(p, q : X 27 800 < p X 243 5620, p 4 X 8 < q < p 2 X 2 0 }, D 53 = {(p, q : X 243 5620 < p X 653 700, X < q < p X 4 00 }, D 54 = {(p, q : X 243 5620 < p X 653 700, p X 4 00 < q < X 60 }, D 55 = {(p, q : X 243 5620 < p X 653 700, p 82 6 X 33 230 < q < p 4 X 8 }, D 56 = {(p, q : X 243 5620 < p X 653 700, p 4 X 8 < q < p 2 X 5 76 }, D = {(p, q : X 653 700 < p X 5073 320, X < q < X 60 }, D 58 = {(p, q : X 653 700 < p X 5073 320, p 82 6 X 33 230 < q < p 4 X 8 }, D 5 = {(p, q : X 653 700 < p X 5073 320, p 4 X 8 < q < p 2 X 5 76 }, D 60 = {(p, q : X 5073 320 < p X 554 340, X < q < p 82 6 X 33 230 }, D 6 = {(p, q : X 5073 320 < p X 554 340, p 82 6 X 33 230 < q < p 4 X 8 }, D 62 = {(p, q : X 5073 320 < p X 554 340, p 4 X 8 < q < p 2 X 5 76 }, D 63 = {(p, q : X 554 340 < p X 363 860, X < q < p 4 X 8 }, D 64 = {(p, q : X 554 340 < p X 363 860, p 4 X 8 < q < p 2 X 5 76 }, D 65 = {(p, q : X 363 860 < p X 063 2640, X < q < p 2 X 5 76 }, D 66 = {(p, q : X 063 2640 < p X 45, X < q < p 2 X 5 76 }, D 67 = {(p, q : X 063 2640 < p X 45, p 2 X 5 76 < q < p 35 23 X 7 230 },
Shor inervals conaining prime nmbers 65 D 68 = {(p, q : X 45 < p X 44, X < q < p 2 X 5 76 }, D 6 = {(p, q : X 45 < p X 44, p 2 X 5 76 < q < p 4 27 X 42 540 }, D 70 = {(p, q : X 45 < p X 44, p 2 0 X 40 < q < p 5 X 2 00 }, D 7 = {(p, q : X 44 < p X 433 860, X < q < p 2 X 5 76 }, D 72 = {(p, q : X 44 < p X 433 860, p 2 X 5 76 < q < p 4 27 X 42 540 }, D 73 = {(p, q : X 44 < p X 433 860, p 2 0 X 40 < q < p 5 X 2 00 }, D 74 = {(p, q : X 44 < p X 433 860, p 6 5 X 25 < q < p 7 X 3 70 }, D 75 = {(p, q : X 433 860 < p X 26 550, p 35 2 X 40 < q < p 2 0 X 40 }, D 76 = {(p, q : X 433 860 < p X 26 550, p 2 0 X 40 < q < p 5 X 2 00 }, D 77 = {(p, q : X 433 860 < p X 26 550, p 6 5 X 25 < q < p 7 X 3 70 }, D 78 = {(p, q : X 26 550 < p X 87 72, X < q < p 2 0 X 40 }, D 7 = {(p, q : X 26 550 < p X 87 72, p 2 0 X 40 < q < p 5 X 2 00 }, D 80 = {(p, q : X 26 550 < p X 87 72, p 6 5 X 25 < q < p 7 X 3 70 }, D 8 = {(p, q : X 87 72 < p X 33 20, X < q < p 5 X 2 00 }, D 82 = {(p, q : X 87 72 < p X 33 20, p 6 5 X 25 < q < p 7 X 3 70 }, D 83 = {(p, q : X 33 20 < p X 40, X < q < p 7 X 3 70 }, D 84 = {(p, q : X 40 < p X 53 0, X < q < p X 5 70 }, D = {(p, q : X 53 0 < p X 33 700, X < q < p X 5 8 }, D 86 = {(p, q : X 53 0 < p X 33 700, X 0 < q < p X 5 70 }, D 87 = {(p, q : X 33 700 < p X 233 480, X < q < p X 5 8 }, D 88 = {(p, q : X 33 700 < p X 233 480, X 0 < q < p 6 X 63 20 }, D 8 = {(p, q : X 233 480 < p X 4 00, X < q < p X 5 8 }, D 0 = {(p, q : X 233 480 < p X 4 00, p X 5 8 < q < p 70 X 7 55 }, D = {(p, q : X 233 480 < p X 4 00, X 0 < q < p 6 X 63 20 }, D 2 = {(p, q : X 4 00 < p X 244 550, X < q < p 70 X 7 55 },
66 C Jia D 3 = {(p, q : X 4 00 < p X 244 550, X 0 < q < p X 7 0 }, D 4 = {(p, q : X 244 550 < p X 2, X 0 < q < p X 7 0 } Bonds of he sieve fncions Lemma 24 Ω 6 Ω Ω 22 Ω 25 Ω 28 Ω 3 Ω 34 Ω 37 Ω 40 Ω 42 Ω 45 Ω 48 Ω 52 Ω 56 Ω 5 Ω 62 Ω 64 Ω 65 Ω 66 Ω 68 Ω 7 034443ηx log x P r o o f We have Ω 6 = X 70 <p X 3 0 p 5 6 X 00 <q<p 8 X 80 23 S(A pq, q = x<pqr xηx where X 70 < p X 3 0, p 6 5 X 00 < q < p 8 X 23 80 and he leas prime facor of r is greaer han q Le h = q and k = r By Lemma 6 wih region (i, (7 holds Then Lemmas and 2 yield Ω 6 = η S(B pq, q O(εηx log x = ηx X 70 <p X 3 0 X 70 <p X 3 0 O(εηx log x = ηx log x = ηx log x 3 0 7 260 3 0 7 260 3 0 70 p 6 5 X 00 <q<p 8 X 23 80 ( 7 260 70 3 ( 00 6 5 23 80 8 3 ( p 6 5 X 00 <q<p 8 X 80 23 23 80 8 00 6 5 23 80 8 00 6 5 2 w ( ( ( ( O(ε pq log q w, ( log(x/(pq log q O(εηx log x ( ( log 2 (000056 00076 0035ηx log x = 0022663ηx log x
Shor inervals conaining prime nmbers 67 Using he above discssion and Lemma 20, we have Ω Ω 22 Ω 25 Ω 28 Ω 3 Ω 34 Ω 37 Ω 40 Ω 42 Ω 45 Ω 48 Ω 52 Ω 56 Ω 5 Ω 62 Ω 64 Ω 65 Ω 66 Ω 68 Ω 7 ( 60 = ηx log x 247 770 60 48 450 35 00 443 305 48 450 55 443 305 243 5620 55 363 860 243 5620 433 860 363 860 3 0 23 80 8 0 5 6 0 20 0 2 w 2 0 2 0 2 0 2 8 4 5 76 2 8 4 5 76 2 ( 60 ηx log x 60 3 0 3 0 23 80 8 3 ( 23 80 8 0 2 2 w 2 w 2 w ( 2 w 2 w 2 w 2 w ( ( ( ( ( ( 3 ( 0 2 ( ( 35 00 247 770 O(ε 25 44 0 2 w ( ( ( log ( 2
68 C Jia 247 770 60 247 770 60 35 00 247 770 35 00 247 770 28 35 00 28 35 00 48 450 28 443 305 48 450 55 443 305 65 83 55 05607 3 ( 0 23 80 8 3 ( 3 ( 0 25 44 3 ( 3 ( 0 5 6 3 ( 5 6 0 20 0 2 0 2 0 2 0 2 8 4 243 5620 65 83 ( ( log ( 2 ( ( ( log ( 2 ( ( ( log ( 2 ( ( ( log ( 2 ( ( log ( 2 ( ( log ( 2 ( ( log ( 2 4 ( 8 4 2