Almost all short intervals containing prime numbers

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1 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]

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,

3 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

4 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 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

5 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 M X 60, 23 N X 80 ; (iii X M X 32, N M X (iv X 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 ;

6 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

7 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 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 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

8 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 (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

9 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 (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

10 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

11 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

12 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 (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 and X 00 H Combining he above, we obain (8 Hence, Lemma 5 follows

13 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 M X 770, X 0 H M 23 8 X 80 ; (iv X M X 00, X 0 H M X ; (v X M X 450, X 0 H M 6 X 5 ; (vi X M X 305, X 0 H M X 20 ; (vii X M X 55, X 0 H M 2 2 X 0 ; (viii X M X 5620, M 4 X 8 H M 2 2 X 0 ; (ix X M X 860, M 4 X 8 H M 2 5 X 76 ; (x X 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 M X 55, M X 20 H M 2 X 2 0 ; X 55 M X , M 4 X 8 H M 2 X 2 0 ; X M X , M 4 X 8 H M 2 X 5 76 ; X M X , 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 ; X 3 0 M X 60, M 2 X 3 45 H M 8 X ; X 60 M X , M X 20 H M 8 X ; X M X 35 00, M X 20 H M X ; X M X , M X 20 H M 6 X 5, we apply Lemma 5 wih condiion 3 Ping ogeher he above regions, we ge Lemma 6

14 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 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

15 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 (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,

16 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

17 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

18 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

19 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 (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 M X 72, M 2 0 X 40 H M 2 5 X (iii X M X 72, M 6 5 X 25 3 H M 7 X 70 ; (iv X M X 20, X H M 2 5 X 00 ; (v X M X 20, M 6 5 X 25 3 H M 7 X 70 ; (vi X M X 40, X 3 H M 7 X 70 ; (vii X M X 0, X H M X 5 70 ; (viii X M X 700, X H M 5 X 8 ; (ix X M X 700, X 0 H M X 5 70 ; (x X M X 00, X H M 5 X 8 ; (xi X M X 00, X 0 H M 6 X ; (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 M X 40, M 2 X 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 M X 33 20, X H M 5 X 2 00 ; X M X 40, X H M 2 X ;

20 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 M X 40, M X 5 H M 7 X 3 70 ; X 40 M X , M X 7 0 H M X 5 70 ; X 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: M X 275, M X 25 H M (i 4 X X M X 860, X H M (ii 4 X 8 ; X 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

21 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

22 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 (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 M X 340, M X 230 H M 70 7 (i 5 X X M X 275, X H M 70 7 (ii 5 X 25 ; X M X 860, M 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 , 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:

23 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

24 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 M X 00, M X 25 H M 4 X 8 ; (ii X M X 340, M X 230 H M 4 X 8 ; (iii X M X 860, X H M 4 X 8 ; (iv X M X 45, M 2 5 X 76 H M X 230 ; (v X M X 860, M 2 5 X 76 H M 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 M X 550, M 35 2 X 40 H M 2 (i 0 X 40 ; X M X 72, X H M 2 (ii 0 X 40 ; X M X 00, M 5 X 8 H M 70 7 (iii X X M X 550, X H M 70 7 (iv X 55 Then (7 holds for T 0 T X 55 ;

25 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 , 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

26 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 (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

27 Shor inervals conaining prime nmbers 47 and ha P and Q lie in one of he following regions: (i X P X 80, P X Q P ; (ii X P X 220, P X Q P X (iii X P X 580, P X Q P X 53 (iv X 5 (v X 7 (vi X P X 00, X Q P 5 67 X 34 ; X 37 (vii 40 ; 0 ; 580 P X 7 55, P X Q P X 53 0 ; 55 P X , P X Q P 67 X 5 00 P X , 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 ; P X Q P X 33 20, X R Q (P Q 5 X 2 00, which can be wrien as P X Q P X 33 20, X Q P 6 X 2 20, X R Q In he regions: X P X , P X Q P X ; X P X , 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 Q P X 40, X R Q (P Q 7 X 3 70, which can be wrien as P X Q P X 40, X Q P 6 X 3 60, X R Q In he regions: X P X 80, P X Q P ; X 80 P X , P X Q P X 40 ; X P X , 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,

28 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 P X , P X 40 Q P X 53 0 ; X P X , 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 P X , P X 53 0 Q P X 4 00 ; X P X , 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 P X 60, X Q P X 20 ; (iii X P X 4300, X Q P 2 6 X (iv X P X 860, X Q P X ; 72

29 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 M X 60, 23 N X 80 ; (c X M X 32, N M X (d X 3 32 M X 45, N M 2 X 2 20 ; (e X 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 , X Q P X 32, X 64 P X , P X 32 Q P ; X P X , P X 32 Q P X ; X P X , X Q P X 53 60, condiion (b is saisfied In he regions: X P X 3 64, P X Q P ; X 3 64 P X , P X Q P X 3 32 ; X P X , X Q P X 3 32, condiion (c is saisfied

30 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 , P X 3 32 Q P X 45 ; X P X , X Q P X 45, condiion (d is saisfied In he regions: X 3 60 P X , P X 45 Q P 6 X 2 20 ; X P X , P X 45 Q P X ; X P X , 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 P X 700, P X 4 00 Q X 60 ; (ii X P X 320, X Q X 60 ; (iii X P X 340, X Q P 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 P X , P X 4 00 Q X 60 ; X P X , X Q X 60 ; X P X , X Q P 82 6 X , he above condiions on P and Q are saisfied So he proof of Lemma 5 is complee

31 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

32 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

33 = 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 P X 60, X Q P X (iii X 3 (iv X 263 (v X 27 (vi X P X 320, X Q X 60 ; X 5073 (vii 20 ; 60 P X , X Q P 6 X 2 20 ; 4300 P X , X Q P X ; 800 P X , P X 4 00 Q X 60 ; 320 P X , X Q P 82 6 X

34 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

35 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

36 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( log( 2 log(s s Lemma 20 We have he following bonds: (i w( for 247, (ii w( 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,

37 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 k From (4 and 05 w( for, i follows ha w ( 4 if > 2 Using Lagrange s mean vale heorem, we have w( for 2 3 By incion, we obain 05 w( for 2 If 3 4, by calclaion, we have max w(3 0 k k 05643, 4 min w(3 0 k k From (4 and 05 w( for 2, i follows ha w ( if > 3 The above discssion implies ha w( for 3 By he same discssion we can also ge w( for 247 If 4 5, by calclaion, we have max w(4 0 k k 0566, 4 k min w(4 0 k k The above discssion and he fac ha w ( 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

38 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 ηx 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

39 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

40 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 ηx 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

41 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

42 62 C Jia 0567 ( log ηx log x O(δηx log x ηx 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 , X < q < p 6 X 2 20 }, D 4 = {(p, q : X < p X 80, X < q < p 6 X 2 20 }, D 5 = {(p, q : X < p X 80, p X < q < p}, D 6 = {(p, q : X 80 < p X , X < q < p 6 X 2 20 }, D 7 = {(p, q : X 80 < p X , p X < q < p X 40 }, D 8 = {(p, q : X < p X , X < q < p 6 X 2 20 }, D = {(p, q : X < p X , p X < q < p X 53 0 }, D 0 = {(p, q : X < p X 5 580, X < q < p X }, D = {(p, q : X < p X 5 580, p X < q < p X 53 0 }, D 2 = {(p, q : X < p X 70, X < q < p X }, D 3 = {(p, q : X < p X 70, p X < q < p X 53 0 }, D 4 = {(p, q : X 70 < p X 3 0, X < q < p X }, D 5 = {(p, q : X 70 < p X 3 0, p X < q < p X 53 0 }, D 6 = {(p, q : X 70 < p X 3 0, p 6 5 X 00 < q < p 8 X }, D 7 = {(p, q : X 3 0 < p X 7 55, X < q < p X },

43 Shor inervals conaining prime nmbers 63 D 8 = {(p, q : X 3 0 < p X 7 55, p X < q < p X 53 0 }, D = {(p, q : X 3 0 < p X 7 55, p 2 X 0 < q < p 8 X }, D 20 = {(p, q : X 7 55 < p X 60, X < q < p X }, D 2 = {(p, q : X 7 55 < p X 60, p X < q < p 67 X 5 34 }, D 22 = {(p, q : X 7 55 < p X 60, p 2 X 0 < q < p 8 X }, D 23 = {(p, q : X 60 < p X , X < q < p X }, D 24 = {(p, q : X 60 < p X , p X < q < p 67 X 5 34 }, D 25 = {(p, q : X 60 < p X , X 0 < q < p 8 X }, D 26 = {(p, q : X < p X 35 00, X < q < p X }, D 27 = {(p, q : X < p X 35 00, p X < q < p 67 X 5 34 }, D 28 = {(p, q : X < p X 35 00, X 0 < q < p X }, D 2 = {(p, q : X < p X , X < q < p X }, D 30 = {(p, q : X < p X , p X < q < p 67 X 5 34 }, D 3 = {(p, q : X < p X , X 0 < q < p 6 X 5 }, D 32 = {(p, q : X < p X , X < q < p X }, D 33 = {(p, q : X < p X , p X < q < p 67 X 5 34 }, D 34 = {(p, q : X < p X , X 0 < q < p X 20 }, D 35 = {(p, q : X < p X 55, X < q < p X }, D 36 = {(p, q : X < p X 55, p X < q < p 67 X 5 34 }, D 37 = {(p, q : X < p X 55, X 0 < q < p 2 X 2 0 }, D 38 = {(p, q : X 55 < p X , X < q < p X }, D 3 = {(p, q : X 55 < p X , p X < q < p 67 X 5 34 }, D 40 = {(p, q : X 55 < p X , p 4 X 8 < q < p 2 X 2 0 }, D 4 = {(p, q : X < p X , X < q < p 67 X 5 34 }, D 42 = {(p, q : X < p X , p 4 X 8 < q < p 2 X 2 0 },

44 64 C Jia D 43 = {(p, q : X < p X 37 00, X < q < p 67 X 5 34 }, D 44 = {(p, q : X < p X 37 00, p 70 5 X 7 25 < q < p 4 X 8 }, D 45 = {(p, q : X < p X 37 00, p 4 X 8 < q < p 2 X 2 0 }, D 46 = {(p, q : X < p X , X < q < p X 4 00 }, D 47 = {(p, q : X < p X , p 82 6 X < q < p 4 X 8 }, D 48 = {(p, q : X < p X , p 4 X 8 < q < p 2 X 2 0 }, D 4 = {(p, q : X < p X , X < q < p X 4 00 }, D 50 = {(p, q : X < p X , p X 4 00 < q < X 60 }, D 5 = {(p, q : X < p X , p 82 6 X < q < p 4 X 8 }, D 52 = {(p, q : X < p X , p 4 X 8 < q < p 2 X 2 0 }, D 53 = {(p, q : X < p X , X < q < p X 4 00 }, D 54 = {(p, q : X < p X , p X 4 00 < q < X 60 }, D 55 = {(p, q : X < p X , p 82 6 X < q < p 4 X 8 }, D 56 = {(p, q : X < p X , p 4 X 8 < q < p 2 X 5 76 }, D = {(p, q : X < p X , X < q < X 60 }, D 58 = {(p, q : X < p X , p 82 6 X < q < p 4 X 8 }, D 5 = {(p, q : X < p X , p 4 X 8 < q < p 2 X 5 76 }, D 60 = {(p, q : X < p X , X < q < p 82 6 X }, D 6 = {(p, q : X < p X , p 82 6 X < q < p 4 X 8 }, D 62 = {(p, q : X < p X , p 4 X 8 < q < p 2 X 5 76 }, D 63 = {(p, q : X < p X , X < q < p 4 X 8 }, D 64 = {(p, q : X < p X , p 4 X 8 < q < p 2 X 5 76 }, D 65 = {(p, q : X < p X , X < q < p 2 X 5 76 }, D 66 = {(p, q : X < p X 45, X < q < p 2 X 5 76 }, D 67 = {(p, q : X < p X 45, p 2 X 5 76 < q < p X },

45 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 }, 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 , X < q < p 2 X 5 76 }, D 72 = {(p, q : X 44 < p X , p 2 X 5 76 < q < p 4 27 X }, D 73 = {(p, q : X 44 < p X , p 2 0 X 40 < q < p 5 X 2 00 }, D 74 = {(p, q : X 44 < p X , p 6 5 X 25 < q < p 7 X 3 70 }, D 75 = {(p, q : X < p X , p 35 2 X 40 < q < p 2 0 X 40 }, D 76 = {(p, q : X < p X , p 2 0 X 40 < q < p 5 X 2 00 }, D 77 = {(p, q : X < p X , p 6 5 X 25 < q < p 7 X 3 70 }, D 78 = {(p, q : X < p X 87 72, X < q < p 2 0 X 40 }, D 7 = {(p, q : X < p X 87 72, p 2 0 X 40 < q < p 5 X 2 00 }, D 80 = {(p, q : X < p X 87 72, p 6 5 X 25 < q < p 7 X 3 70 }, D 8 = {(p, q : X < p X 33 20, X < q < p 5 X 2 00 }, D 82 = {(p, q : X < p X 33 20, p 6 5 X 25 < q < p 7 X 3 70 }, D 83 = {(p, q : X < 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 , X < q < p X 5 8 }, D 86 = {(p, q : X 53 0 < p X , X 0 < q < p X 5 70 }, D 87 = {(p, q : X < p X , X < q < p X 5 8 }, D 88 = {(p, q : X < p X , X 0 < q < p 6 X }, D 8 = {(p, q : X < p X 4 00, X < q < p X 5 8 }, D 0 = {(p, q : X < p X 4 00, p X 5 8 < q < p 70 X 7 55 }, D = {(p, q : X < p X 4 00, X 0 < q < p 6 X }, D 2 = {(p, q : X 4 00 < p X , X < q < p 70 X 7 55 },

46 66 C Jia D 3 = {(p, q : X 4 00 < p X , X 0 < q < p X 7 0 }, D 4 = {(p, q : X < 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 ηx 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 S(A pq, q = x<pqr xηx where X 70 < p X 3 0, p 6 5 X 00 < q < p 8 X 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 p 6 5 X 00 <q<p 8 X ( ( ( p 6 5 X 00 <q<p 8 X w ( ( ( ( O(ε pq log q w, ( log(x/(pq log q O(εηx log x ( ( log 2 ( ηx log x = ηx log x

47 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 w ( 60 ηx log x ( w 2 w 2 w ( 2 w 2 w 2 w 2 w ( ( ( ( ( ( 3 ( 0 2 ( ( O(ε w ( ( ( log ( 2

48 68 C Jia ( ( 3 ( ( 3 ( ( ( ( log ( 2 ( ( ( log ( 2 ( ( ( log ( 2 ( ( ( log ( 2 ( ( log ( 2 ( ( log ( 2 ( ( log ( 2 4 ( 8 4 2