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3 ,,,,..,,., {. (, ),, {,.,.,..,,.,.,, {..

4 : N {, Z {, Q {, Q p { p{ {. 3, R {, C {. : ord p {. 8, (k) {.42,!() { {. 24, () { {. 24, () { {. 25,., () { {. 26,. 9, () { {. 27,. 23, '() { ( ) {. 28, () { {. 29, k () { () {. 30, () { {. 3, () { () {. 3, r f () { f(x) 0 (mod ) {. 38. : [x] { x {. 5, fxg { x {. 5, (x) {. 6, 2 (x) {. 7, e(x) {. 39, (x) {, x {. 44, (x) { {. 44, (x) { {. 44.

5 , Q: ord p r {. 0, jjrjj p { p{ r {. 2, p (r r 2 ) { p{ {. 59. : Re s { s, Ims { s, (s) { { {. 32. : c( a) { ( ) {. 40, S l ( a) {. {. 4, S l ( a) {. {. 4, K( a b) { {. 43. : p. P P Q Q P Q P Q : a<b f() { 2 Z, a< b, b f() { 2 N, b, a<b f() { 2 Z, a<b, b f() { 2 N, b, pb f(p) { p b, pb f(p) { p b, p f(p) {, f(p) {. p : f(x) =O ; g(x), c>0 x 0, x x 0 jf(x)jcg(x). f(x) g(x), x 2 R, f(x), {,, g(x) > 0.

6 [x] fxg { 42 6 { 47 7 { 60 8 { {

7 ., : ) M N, M {. ) M N ) 2 M, 2) 2 M =) +2 M, M = N. ). :. N,..,,,.. a b 2 Z, b 6= 0., a b b j a, c 2 Z, a = bc. a b, b a. a b b - a. : {, b j a, a b 2 Z b 6= 0. 2., : ) a 2 Z, a 6= 0, a j a. ) a b 2 N, a j b b j a, a = b. ) a b c 2 Z, a j b b j c, a j c. ) a b 2 Z, a j b b j a, jaj = jbj. 7

8 ) a b 2 N a j b, a b. ) a b 2 Z, ab 6= 0 a j b, jajjbj. ) a b 2 Z a j b, ;a j b. ) a b c 2 Z a j b, a j bc. ) a b c 2 Z, a j b a j c, a j (b c).. a 2 Z m 2 N. q r 2 Z, a = mq + r 0 r<m. 3.. : q 2 Z, a [qm (q +)m). a = mq + r r, 0 :::m;., [qm (q +)m), q 2 Z,. 2. a 2 Z, m 2 N a a = mq + r, q r 2 Z 0 r<m. q r,, a m. 4. : ) 56 5 )9 7 );42 5 ) , a m 2 Z m 6= 0, q r 2 Z, a = mq + r 0 r<jmj. 6., a m 2 Z m 6= 0, q r 2 Z, a = mq + r ; jmj jmj <r. 2 2 { 3. a a 2 ::: a 2 Z, ( ). a a 2 ::: a,. {, a a 2 ::: a, { a a 2 ::: a (a a 2 ::: a ). 8

9 4. a a 2 ::: a 2 Z, (a a 2 ::: a ) =., a a 2 ::: a,. : 3, 5, 7, 2, 29,. 3, 7, 29. {. { d a a 2 ::: a d = k a + k 2 a k a k k 2 ::: k 2 Z (k k 2 ::: k )=., (a a 2 ::: a )=, k k 2 ::: k 2 Z, d = k a + k 2 a k a (k k 2 ::: k )=. 7. {. : I = fk a + k 2 a 2 + :::k a : k k 2 ::: k 2 Zg : a i 2 I i = 2 :::. I, I. m {, I ( ). a 2 I,, q r 2 Z, a = mq + r 0 r<m. I, r = a ; mq 2 I, r 6= 0, r<m, m. r =0 m j a. a i 2 I i = 2 :::, m a a 2 ::: a m d., d I, d j m d m.,, d = m,, d d = k a + k 2 a k a k k 2 ::: k 2 Z. l =(k k 2 ::: k ), (i), ld j d, l =. 9 (i)

10 8., {. : 3 {. 9. a b c 2 Z., b j ac (a b) =, b j c. : (a b) =, { u v 2 Z, au + bv =., acu + bcv = c. b j ac, b j c. 0.,. : 9. 5., : ), ), ), ),..,,., {. 2. a a 2 ::: a 2 N, a = mi k a k. r i a i a (i =2 3 ::: )., a, r 2, r 3, :::, r. 3.,, { -. 0

11 : 5, ( ),., 2. ( ),.. : 3 { -,. =2. 4. { 72, ;96, 80, 240, 360, ;504. : 3 (72 ; ;504) = ( ) = =( ) = ( ) = ( ) = =( ) = (24 2) = (2 24) = (2 0) = 2: 5. 2 a a 2 ::: a 2 N.,. : =2. =3,., a a 2 a 3 2 N. d =(a a 2 a 3 ). d j a, d j a 2 a 3 (a a 2 )=, (d a 2 )=. 9 d j a 3 (a a 3 )=, d =. 6. a a 2 ::: a 2 N k 2 N., a a 2 :::a k. : =2..

12 a j k, a 2 j k (a a 2 )=. k = a l l 2 N. a 2 j a l, 9, a 2 j l. l = a 2 s s 2 N. k = a a 2 s,..a a 2 j k. 7. a a 2 ::: a b b 2 ::: b m 2 Z a a 2 ::: a., (a a 2 ::: a b b 2 ::: b m )=((a a 2 ::: a ) b b 2 ::: b m ) : : a a 2 ::: a b 2 N., (a b a 2 b ::: a b)=(a a 2 ::: a )b: : (a a 2 ::: a )=d (a b a 2 b ::: a b)=h. b a b ::: a b, 8) b j h. {, h j a i b, h j a b i, i = 2 :::. 8), h j d,, h j db. b, d j a i. db j a i b, i = 2 ::: 8), db j h, db = h. 9. a a 2 ::: a b b 2 ::: b m., { a i b j, i, j m (a a 2 ::: a )(b b 2 ::: b m ) : : = m =2( )., a a 2 b b 2 2 N. 7 8, (a a 2 )(b b 2 )=(a (b b 2 ) a 2 (b b 2 )) = ((a b a b 2 ) (a 2 b a 2 b 2 )) =(a b a b 2 a 2 b a 2 b 2 ) : 20. a a 2 ::: a 2 N b 2 N., (b a a 2 :::a )=(b a )(b a 2 ) :::(b a ) : 2

13 : =2.. b i =(b a i ), i = b b 2 =(b 2 ba 2 ba a a 2 )=(b 2 (ba 2 ba ) a a 2 )=(b 2 b(a 2 a ) a a 2 ) =(b 2 b a a 2 )=(b a a 2 ) : 2. a a 2 ::: a 2 N., a a 2 :::a d d 2 :::d, d i a i, i = 2 :::. : =2.. a a 2 2 N (a a 2 )=. d d 2 a a 2, d d 2 j a a 2., d j a a 2, d 2 N, 20, d =(d a a 2 )=l l 2, l i =(d a i ) j a i, i = 2. d, , F =2 2 + = 2 :::. :, k 2 N, F +k ; 2=F +k; F +k;2 ::: F + F (F ; 2) : : F,, f(x) =a 0 x + a x ; + + a ; x + a., k l f(x), k j a l j a 0. :, a 0 a 6=0,. f ; k l =0 a 0 k + a k ; l + a 2 k ;2 l a ; kl ; + a l =0: 3

14 , l j a 0 k. 9 (k l) =, l j a 0. k j a l, k j a. 24., p 2+ 3p 3. :, p 2+ 3p 3 23,. { 6. a a 2 ::: a 2 Z,. a a 2 ::: a,. {, a a 2 ::: a, { a a 2 ::: a [a a 2 ::: a ]. 25.,, {. : , {, {. : a b 2 N., a b ab s, s 2 N. (a b) [a b](a b)=ab. : k 2 N a b (a b) =d. k = al l 2 N. b j al, b j a d d l ; b d d a =, 9 b j l. l = b s s 2 N, d d a b 2 N. ([a b] a)=a [a (a b)] = a. 4

15 29. a b 2 N., : ) a j b, ) [a b]=b, ) (a b) =a. 30. a b c 2 N. : ) ([a b] c)=[(a c) (b c)], ) [(a b) c]=([a c] [b c]), ) ([a b] [b c] [c a]) = [(a b) (b c) (c a)], ) [a b c](a b)(b c)(c a)=(a b c) abc, ) (a b c)[a b][b c][c a]=[a b c] abc. :,, ).. d =([a b] c), k =[(a c) (b c)]. 7, 9 27, d(a b)(a b c)=([a b](a b) c(a b))(a b c)=(ab ac bc)(a b c) k(a b)(a b c)= =(a 2 b a 2 c ab 2 ac 2 b 2 c bc 2 abc) : (a c)(b c) (a b)(a b c)=(a c)(b c)(a b) ((a c) (b c)) =(ab bc ac c 2 )(a b) =(a 2 b a 2 c ab 2 ac 2 b 2 c bc 2 abc) :, d = k. 3. ) ) a a 2 ::: a 2 Z, d =(a a 2 ::: a ) b 2 Z., a x + a 2 x a x = b x x 2 :::x 2 Z, d j b. 5 (i)

16 : (i), d, d j b. d j b. { k k 2 :::k 2 Z, d = a k + a 2 k a k. (i). x = b d k x 2 = b d k 2 ::: x = b d k :,,. (i) 32. f(x ::: x )=0 f, {.,, f.,,,. 33. (i) 32 (, ). : d =(a a 2 ::: a )., a i =0. x i. c = mi i a i 6=0 ; ja j ::: ja j,, c = ja j. 5, q i r i 2 Z, 2 i, a i = a q i + r i 0 r i <c 2 i : 6

17 x = y ; ; q 2 x q x. a y + r 2 x r x = b d =(a r 2 ::: r ). ( N), d, ( b) d. d,.,,. :. (a a 2 ::: a ) x +26x 2 +20x 3 ; 35x 4 =473: : ( ) = 3 3 j x +42x 2 +70x 3 ; 05x 4 =49: mi( ) = = 30 +2, 70 = , ;05 = 30 (;4) + 5. x = y ; x 2 ; 2x 3 +4x 4. 30y +2x 2 +0x 3 +5x 4 = 49 : {, mi( ) = 0 30 = 03+0, 2 = 0+2, 5 = 0+5. x 3 = y 2 ;3y ;x 2 ;x 4 2x 2 +0y 2 +5x 4 =49: 7

18 , mi(2 0 5) = 2, 0 = =22+. x 2 = y 3 ;5y 2 ;2x 4 2y 3 + x 4 =49. x 4 y 3,, x = 964 ; 7y 3 ; 7y 2 +7y x 2 = ; y 3 ; 5y 2, x 3 = 49 ; 3y 3 +6y 2 ; 3y x 4 =49; 2y 3. y y 2 y ) 6x ; 0x 2 +5x 3 =, ) 9x +5x 2 ; 2x 3 =33, ) 6x +5x 2 ; 2x 3 +3x 4 =55, ) 4x +6x 2 +8x 3 =0, ) 3x +4x 2 ; 5x 3 =7. : {. 36. a b c 2 Z, ab 6= 0, (a b) =d d j c. ), x 0 y 0 ax + by = c x = x 0 + b d t y = y 0 ; a d t t 2 Z: ) x 0 y 0. : ) a (x;x d 0)=; b(y;y d 0)., ; a b d d = 9. ), b 2 N. b =, x 0 =0, y 0 = c. b>. r ; = a r 0 = b 8

19 (a b), r i = q i+2 r i+ + r i+2 ; i s ; 2 r s; = q s+ r s r i q i 2 Z r s <r s; < <r <r 0., j s r s = P j r s;j; + Q j r s;j, P j Q j P = Q = ;q s P j+ = Q j Q j+ = ;q s;j Q j + P j :, r s =(r ; r 0 )=(a b) =d, d = P s a + Q s b., x 0 = c d P s, y 0 = c d Q s. 37., a b 2 N (a b) =d, a b 2 N, d = aa ; bb. : 36 ). 38. ) 735x +20y = ;25, ) 26x ; 4y =8, ) 2x ; 30y =46. : ) 735 = = = = 2 5, 36, s =3 r ; = 735, r 0 =20, r =55, r 2 =0, r 3 =5 q =4, q 2 =2, q 3 =5., (735 20) = 5 5 j;25,. { P = Q = P 2 = ;5 Q 2 = P 3 = Q 3 = ;59, x 0 = ;55, y 0 = 795., 36 ), x = ;55+24t y =795; 347t t 2 Z: ). ) x = ;92 ; 5t, y = ;345 ; 56t, t 2 Z. 9

20 2 7. >,. >,.,. 39., 2 N, >. :. 40., 2 N, p. : 7 = km, k m 2 N, <k m<.,, k m, p k, = km k 2 p 2,. 4. ), x 2. ), 00. : ) 40, 2 N, p x, p x< x,.,, p x, " \, ( p x x]. p x,. ) 2, 3, 5, 7,, 3, 7, 9, 23, 29, 3, 37, 4, 43, 47, 53, 59, 6, 67, 7, 73, 79, 83, 89, 97. : ( 238). x,,. 20

21 42. { 2 N, : p p ; j, p j. : , k 2 N m 2 N, m + m+2 ::: m+ k. : m =(k +)!;. 44., p, p. :, ( ). a b 2 N p j ab. p - a, (p a) = 9, p j b. 45. p., k p ;, ; p k p. : ; p 2 N, k! j p(p ; ) :::(p ; k +). k (k! p)=. 9 k! j (p ; ) :::(p ; k +)., ; p p. k. 2 N, > :..,, 2 N, > {,.,, k, m 2 N, = km <k m<., 2

22 k m.,.., p p 2 :::p l = q q 2 :::q s p i, q j. p q q 2 :::q s, 44 p q j. q j, p., p 2 q j., k = l p i q j (, N) :, 2 N, >, = p p 2 :::p l = q q 2 :::q s p i, q j., {., l>, s> p i 6= q j i, j., p 6= q., p <q. k =(q ; p )q 2 :::q s, <k<., k. k = q q 2 :::q s ; p q 2 :::q s = p p 2 :::p l ; p q 2 :::q s = p (p 2 :::p l ; q 2 :::q s ), p k., k, p q 2 :::q s,, p 22

23 q ; p,.. p q,.,. :, 47, k 2 N, k 2 m 2 N k{., kp m. : 0. : p 2., e., p 2 p 2+ 3p 3, e,... e, {. 8. p. 2 N ord p. p - ord p =0. p j, ord p = l, l 2 N {, p l j N = Y p p ordp p. : N, > = p l p l 2 2 :::p ls s, p i l i 2 N., p p 2 2 :::p s s, i 2 Z, 0 i l i, i = 2 :::s. 23

24 :. 5. p, 2 ord p ( 2 )=ord p +ord p 2. 2 N., :. 52. a a 2 :::a 2 N d =(a a 2 ::: a ) k =[a a 2 ::: a ] :, p ) ord p d =mi ; ord p a ord p a 2 ::: ord p a, ) ord p k = max ; ord p a ord p a 2 ::: ord p a. : 3 6. : 46 ( ) 47, {. d k, ), ), 52, {,, { a ::: a { :,, 30 ). p. ord p a =, ord p b =, ord p c =.,. 52 ord p (a b c) =, ord p [a b c] =, ord p [a b] =, ord p [a c] =, ord p [b c] =. 5, p.. 24

25 p{ 0. p. ord p r r 2 Q, r 6= 0. m 2 Z, m 6= 0, ord p m =ord p jmj. {, r 2 Q, r 6= 0 r = m, m 2 Z, m 6= 0 2 N, ord p r = ord p m ; ord p. 54., ord p r r 2 Q, r 6= , r 2 Q, r 6= 0 r = " Y p p lp(r), " =, l p (r) 2 Z, l p (r) 6= 0 p. l p (r) = ord p r. :.. 55 r 2 Q, r 6= , ;68,, ; r r r 2 2 Q, rr r 2 6=0 2 Z. ) ord p (r r 2 ) = ord p r + ord p r 2, ) ord p ; r r 2 =ordp r ; ord p r 2. ) ord p (r )= ord p r. 2. p r 2 Q. p{ - jjrjj p r jjrjj p = ( p ;ordpr r 6= 0 0 r =0: 25

26 58. p r r 2 2 Q. : ) jjr r 2 jj p = jjr jj p jjr 2 jj p, ) jjr + r 2 jj p max ; jjr jj p jjr 2 jj p, ) jjr jj p < jjr 2 jj p =) jjr + r 2 jj p = jjr 2 jj p. : ) ) ). r i,. r i = p l i m i i, l i =ord p r i, m i 2 Z, i 2 N, p - m i i i = 2., l 2 l. r + r 2 = p l 2 p l ;l 2 m + m 2 2 = p l 2 m m = p l ;l 2 m 2 + m 2 = 2 : ord p =0, ord p m 0. ord p (r + r 2 ) l 2 = ord p r 2 ) 2. l 2 <l,, ord p m =ord p (m 2 )=0. ord p (r + r 2 )=ord p r ) p. p{ p Q. r r 2 2 Q p (r r 2 )=jjr ; r 2 jj p., Q. : 58. :.,, {. 60., Q 2 : = ; : jj jj 2 = jj2 + jj 2 =2 ;;! 0!: 26

27 3. p. p{ Q p Q 59. :, p, Q Q p,., Q p 59,.. Q 6= Q p., p{ a 0 + a p + a 2 p 2 + :::, a i 2 Z, 0 a i p ; i =0 2 ::: ( p ). Q p R, ( p). Q p p{.,., Q p,. ) 58, { R. p{ {. {., {, F (x ::: x ) Q (.. 2 Q, F (x ::: x )=0 x ::: x ), R Q p p. Q p p{ :, p p 2 :::p s. =+p p 2 :::p s. 39 p., 27

28 p p i, p j p p 2 :::p s, p j ( ; )., p j,,. : : p F = , p, = 2 3 :::,,. :. 63., x 2 R, x>2 Y ; ; > p : px : Q ; P (x) = px ; ; p. Q ; P (x) = px + + +:::. (, p p 2 ),, P P (x) = +P x 0>x. P0,, x : ( 63), P (x)!, x!.. :. {,, P p, p,. { x

29 4., 2 N ( ),. : 5, , , 2 N = k 2 q, k q 2 N q. : : = ; ; 2 3 : :, s. m 2 N, m>4 s, 2 N, m., m., 65, m = k 2 q, k q 2 N q., k p m, q { 2 s. p m 2 s., m p m 2 s, m. : 66 P p, p. p 67., Z,, H a b = a + bz= fa + kb : k 2 Zg a b 2 Z, b 6= 0( ). 29

30 ), Z. ), H a b, a b 2 Z, b 6= 0. : ),, H a b. ) : H 4 = Z ; H 0 4 [ H 2 4 [ H 3 4. :, " \, { : A H 0 p p. 62 A, ; 62 A., k 2 Z, k 6=, k, k 2 A. Z A = f ;g.,. A, H 0 p. f ;g, Z. :, 68,. 3 [x] fxg 5. x 2 R. { 2 Z, x x ( x) [x]. x fxg = x ; [x]. : [2 4] = 2, [6] = 6, [;5 7] = ;6 f2 4g =0:4, f6g =0, f;5 7g =0:3. 30

31 69., x 2 R : ) x ; < [x] x, ) 0 fxg <, ) [x]=x () x 2 Z, ) [x + a] =[x]+a () a 2 Z, ) fxg = x () 0 x<, ) fx + ag = fxg() a 2 Z. 70., m 2 N, a q r 2 Z, a = mq + r 0 r<m, q = a m, r = m a m. 7., x 2 R 2 N, ; k=0 h x + k i =[x] : : P ; h(x) = k=0 x + k ; [x]. 5, 0 x< h(x) =0. {, 69,, h(x). h(x) =0 x 2 R. 72. a b 2 R, a<b f(x), [a b].,, f (x y) 2 R 2 : a<x b 0 <y f(x) g (i) S = a<b f() : : 5. :. (i),, S (ii). 3 (ii)

32 , S = S ; S 2, S = f() a<b S 2 = a<b ff()g : S, - f(x) ( x 4). {, S 2 0 S 2 [b] ; [a]. S., f(x), S 2 {., fxg e 2ix, = 0 2 :::, S 2, 39 x R 2 R, R L(R), ) L(R) = R ) L(R) =2 f (x y) 2 R 2 : x>0 y>0 xy R g : R p R R h p i 2 ; R : : ) 72. ),,, y = x. : L(R) ),,, { ), {,. 32

33 74. R 2 R, R K(R), ) K(R) =4 f (x y) 2 R 2 : x 2 + y 2 R g : p R ) K(R) =8 p R 2 h pr ; 2 i h pr i +4 + h pr ; 2 i ; 4 hp R=2 i 2 +4 h pr i +: :. 75., l 2 N (l ) =, ; k= kl = (l ; )( ; ) : (i) 2 :, l> > ( ). O(0 0), A(0 l), B( l), C( 0). (l ) =, OB,. {, 72, (i), OCB,,, OABC,.. (l ; )( ; ) , l 2 N, 2 - l (l ) =, kl k + = (l ; )( ; ) : l 4 k ; 2 k l; 2 : , k 2 N, x 2 R x 0, 2 N, x, k, x k. 33

34 78., k 2 N x 2 R, x k ; [x] k : x k k 0 x<k. = [x] k. x 79., 2 N p, ord p (!) = h pi + h p 2 i + h p 3 i + ::: : : =. >, k2n, k<., p, p 2p 3p ::: tp, t = p. v u, p.! =uv., p - u, ord p u =0. { v = p t t!,, ord p (t!) = t p + t 5,, ord p (!) = ord p v = t+ t p + t p 2 + :::. p 2 +:::. t 78,,. 80., x x 2 ::: x 2 R [x ]+[x 2 ]++[x ] [x + x x ] : : = 2( ). [x ]+[x 2 ] [x +x 2 ] fx +x 2 gfx g+fx 2 g. fxg, 0 x x 2 <. x + x 2 < x + x 2. 8., a a 2 ::: a 2 N, (a + a a )! a! a 2! ::: a! 2 N : : 79 80, x a :::x a (x + + x ) a ++a. 34

35 82., k 2 N, k k!. : 8. :, R,., a q 2 Z, ; a q q q (a q) =: (i) : k m 2 Z, jm ; kj m : (ii) =[] fg f2g ::: fg : (iii), fsg, 0 +, (ii) m = s, k =[s]. fsg (iii), (ii) m = s, k =[s]+. +, (iii) + +. ; Il = l l+ + +, l = 2 ::: ; (iii),, fs g fs 2 g, s <s 2, I l., fs2 g;fs g < + (ii) m = s 2 ; s, k =[s ] ; [s 2 ] 0, k m a, q 2 N., a q q (i). : 83.,, m 35

36 m,..,,, x 2 R (x) = ;fxg , (x) : ) x 2 R ; < 2 (x). 2 ) x 2 R Z (x) 0 (x) =;. ) k 2 Z, lim x!k x<k (x) =; 2 lim (x) = (k) = 2 : x!k xk ) (x). 7. x 2 R 2 (x) = Z x 0 (t) dt : 85., 2 (x) : ) x 2 R 0 2 (x) 8. ) 2 (x) x 2 R x 2 R Z, 0 2 (x) = (x). ) k 2 Z, 2 (k) =0. ) 2 (x). 36

37 86. a b 2 R, a<b f(x) [a b]. a<b f() = Z b a f(x) dx + (b)f(b) ; (a)f(a) ; : t 2 [a b] F (t) = Z t a f(x) dx ; Z t a (x)f 0 (x) dx H(t) =F (t) ; G(t) : G(t) = a<t Z b a (x)f 0 (x) dx : f() ; (t)f(t) F (t). [a b], { t 2 (a b), F 0 (t) =f(t) ; (t)f 0 (t). G(t)., [a b]. t 0 2 (a b) Z. 84, (t)f(t) t 0, P a<t f(), t t 0. G(t) (a b) Z. k 2 (a b)\z, 84 lim t!k t<k a<t f() = a<k; f() lim t!k tk a<t f() = a<k f() lim t!k G(t) =lim t!k G(t) =G(k), G(t) t<k tk (a b) \ Z. G(t) a b. F (t) G(t) [a b], H(t). k 2 Z, I k =(a b)\(k k +). t 2 I k G 0 (t) =f(t); (t)f 0 (t) =F 0 (t). H 0 (t) =0 t 2 I k H(t) I k. k H(t) [a b]., H(t) [a b]. H(b) =H(a) = (a)f(a),

38 : Z b a (x) f 0 (x) dx = a;b = a;b. Z Z (a b)\( +) (a b)\( +) (x) f 0 (x) dx = 2 + ; x f 0 (x) dx 88. a b 2 R, a<b f(x) [a b]. a<b f() = Z b a f(x) dx + (b)f(b) ; (a)f(a) ; ; 2 (b)f 0 (b) + 2 (a)f 0 (a) + Z b a 2 (x)f 00 (x) dx : :, : a b 2 R, a<b 2 N, 2 (a b] c 2 C., f(x) [a b], a<b c f() =; Z b a a<x : F = f(b) c ; c f() = a<b a<b c f 0 (x) dx + f(b) 38 a<b a<b c : c ; f(b) ; f() :

39 {, Z b Z b F = c f 0 (x) dx = a<b H(x ) = a<b ( x b 0 a x<: a c H(x ) f 0 (x) dx, F = Z b a a<b. c H(x ) f 0 (x) dx = Z b a a<x c f 0 (x) dx : , x 2 R, x x. x =lx + + (x) + O x x 2 : 84, =+ <x Z x =+ =lx + + (x) +(x) x = 2 + Z 85, j(x)j j 2(x)j x 2 + dt t + (x) ; x 2 ; 2(x) + x 2 2 (t) t 3 dt (x) =; 2(x) x 2 ; Z x Z x Z x 2 (t) t 3 dt : Z j 2 (t)j dt t 3 8x + dt 2 8 x t 3 4x : (t) t 3 dt =

40 :, 9,., = :::. 2: { x =lx + + O x x = O(l x) R 2 R, R> L(R) 73. ), L(R) =R l R +(2 ; ) + (R)+O() (R) =2 p R R ) : L(R) =R l R +(2 ; )R + O ;p R : : 73 ), 84, 9 6. : ). { ( 39 x 4),, O ; R 3 +", ">0.. {., O ; R 4 +", ">0. ( 263). 40

41 93. R 2 R, R> K(R) 74. ), K(R) =R + 0 (R)+O() 0 (R) =8 p R 2 ;p R ; 2 : ) K(R) =R + O ;p R : : 74, : )..,, O ; R 4 +", ">0. 94., x 2 R, x 2 x l = x l x ; x + O(l x) : : , x 2 R, x 2 : ) x = x+ + + O; max( x ) 2 R, >;, O ) >x = O; x ; 4

42 2 R, >, O ) x l c. = 2 l2 x + c + O l x x 96. ) x l 2 ) 2x l ) x p +l : N a = k= k l( +2; k) :, a, = 2 :::. :, k, 2 k. : lim! a =. 5 { 8., N C,. {. 42

43 9., f(), f() =,, 2 2 N, ( 2 )=, f( 2 )=f( )f( 2 ). 20., f(),, 2 2 N f( 2 )=f( )f( 2 ). 2., h(), 2 2 N, ( 2 )=, h( 2 )=h( )+h( 2 ). 22., h(), 2 2 N h( 2 )= h( )+h( 2 ). 98., h() ( ) a 2 C, a 6= 0, f() =a h() ( ). :,. 99. f() g() ( ). ), f()g() ( ). ), g() 6= 0 2 N, f () g() ( ). 00., f() ; k 2 N k = p l p l 2 ; 2 :::p ls s, f(k) = f p l f p l 2 ; 2 :::f p l s s. 0., f() g(), f ; p l = g ; p l p l 2 N, f() =g() 2 N. 43

44 02., f() g() f(p) =g(p) p, f() =g() 2 N. 23. f(). F () = dj f(d) ( d ) f(). 03., F () f(), F (). : : f(). f() = F ()., > f(k) k<. f() =F () ; kj k< f(k) :, F () f(). :, f() f (), 2 N F () = dj f(d) = dj f (d) : (i) f() = f () = F (). 2 N, >, k 2 N, k<, f(k) =f (k). (i), f() =F () ; dj d< f(d) =F () ; dj d< f (d) =f () :, f() =f () 2 N. 44

45 : F () f() 23. f() F () , f(), F (). : 2 2 N ( 2 )=. 2, F ( 2 )= f(d) = f(d d 2 )= dj 2 d j d 2 j 2 = dj d 2 j 2 f(d )f(d 2 )= d j f(d ) d 2 j 2 f(d 2 )=F ( )F ( 2 ) : 05., F () f() F (), f(). : P F () = dj f(d) F (), f() = F () =., f(). 2 2 N, ( 2 ) = f( 2 ) 6= f( )f( 2 )., 2 ( N)., 2 >, 2 >., F () f(), 2, 2, F ( 2 )= f(d) =f( 2 )+ f(d) = dj 2 dj 2 d< 2 = f( 2 )+ d j = f( 2 )+ d j f(d d 2 )= d 2 j 2 d d 2 < 2 f(d )f(d 2 ) : d 2 j 2 d d 2 < 2 45

46 F ( )F ( 2 )= f(d ) f(d 2 )= f(d )f(d 2 )= dj d2j2 dj d 2 j 2 = f( )f( 2 )+ d j f(d )f(d 2 ) : d 2 j 2 d d 2 < 2 {, F (), F ( 2 )=F ( )F ( 2 )., f( 2 )=f( )f( 2 ), 2. f(). 06. f() F ()., k 2 N, k> k = p l p l 2 2 :::p ls s, F (k) = sy i= +f ; p i + f ; p 2 i + + f ; p l i i : : k. F (). 07. f()., 2 2 N f ; [ 2 ] f ; ( 2 ) = f( ) f( 2 ) : : f() P = f()., = f() = Y p +f(p)+f(p 2 )+f(p 3 )+:::. 46 (i)

47 f(), = f() = Y p : p p =+f(p)+f(p 2 )+f(p 3 )+::: : ; ; f(p) ; : (ii),, p, P f(). = x 2 R, x>2 P (x) = Q px p. f(). P (x) = x f()+(x) (iii) (x) = P 0>x f(), P0,, x. j(x)j P >x jf()j, lim (x) =0: x! (iii), (i). f(), p f(p) 6=,, p,,. (ii), p = ; ; f(p) ; (i). : (i). 6 { N.!(), () {. 47

48 :!(6) = (6) = 2,!(8) =, (8) = ,!(), () {. 25. () ( = (d) = 0 > : dj 0.,. : 05, 25.., ) () =, ) () =0 p 2 j p. ) () = (;) s s. : () =. 0 (), ) ), p, (p) =; (p l )=0 l 2 N, l>. 0= (d) =() + (p) =+(p) djp (p) =;. {, l =2 0= djp 2 (d) =() + (p)+(p 2 )=; +(p 2 )=(p 2 ) : 48

49 l 3, (p s )=0 s 2 N, 2 s l ;, 0= djp l (d) =()+(p)++(p l )=;+0++0+(p l )=(p l ) :, (p l )=0 l 2 N, l 2,. 2. () ), ) ) F () f()., : ) F () f(). ) 2 N f() = dj (d) F : d P : ),.. 2 N F () = f(d). 2 N dj () = dj (d) F d = dj (d) tj d f(t) : 25, () = f(t) (d) =f() tj dj t )., ) ). 4. F (x) f(x), x 2 R, x>0 x., : 49

50 ) x>0 F (x) = k= f(kx) : ) x>0 f(x) = k= (k) F (kx) : : ). x>0 H(x) = k= (k) F (kx) = k= (k) l= f(lkx) : 25, H(x) = m= k= l= kl=m (k) f(lkx) = m= f(mx) kjm (k) =f(x) )., ) ). 5. F (x) f(x), x 2 R, x>0., : ) x>0 F (x) = kx f x k : ) x>0 f(x) = kx (k) F x k : : 3 4. : 3 { 5. 50

51 6., x 2 R, x = kx (k) h x ki : : 5 f(x), f(x) = x>0. 7., k 2 N, ( 0 p p k j (d) = d k j ( d 2 N, d k j ). :,., = p l, p l 2 N N () ( (d) = 0 : dj 8.,. : , 2 N () =(;) (). : 0, = p l, p l 2 N. l , x 2 R, x [ p x]= kx (k) h x ki : 5

52 : N () dj (d) =l: 2., 2 N () = dj (d) l d = ; dj (d) ld: : , 2 2 N 2 > ( 2 )=, ( 2 )=0. : 2 2, ( 2 )=; (d) ld = ; (d ) (d 2 )l(d d 2 )= dj 2 d j d 2 j 2 = ; d j d 2 j 2 (d ) (d 2 )(ld +ld 2 )= = ; d j (d )ld d 2 j 2 (d 2 ) ; d 2 j 2 (d 2 )ld 2 d j (d ) : 25,. 23., 2 N () = ( l p = p l p l 2 N 0 : : 2, () = 0. >. p p 2, = p l p l 2 2 m, l l 2 m 2 N (p p 2 m)=. 22 = p l, 2 = p l 2 2 m, () =0. 52

53 , = p l p l 2 N. 50, 2, ; p l = ; djp l (d) ld = ; l ; p l p =lp: 24. () , x 2 R, x 2 h x (k) ki kx =0 = x l x ; x + O(l x) : : 77, 94 27, kx (k) h x ki = kx = x (k) x kj = x kj (k) = l = x l x ; x + O(l x) : N '() k 2 N, k, (k ) =. '(). 26. '() = dj (d) d : : 8) 25, 28, '() = = (d) = (d) : k (k )= k dj(k ) k, '() = (d) : dj k djk k,. d 53 djk dj

54 27., 2 N = dj '(d) : : : 2 3 ::: ;.., d j d>0, '(d) d. 29.,. : 04, 26 (), , ; : p '() = Y pj : 06, :, p =2 p 2 =3 ::: p s. = p p 2 :::p s m = '(). m =, k = k 2 p p 2 ::: p s,., 30 Q s m = (p i= i ; ) >.,,... 54

55 32., 2 N '() = dj 2 (d) '(d) : : 99, 0 29, 2 (). 04, '().,., '() 0, = p l, p l2n., 30, '() = ; p = p p ; :, 50, 30, dj 2 (d) '(d) = l =0. 2; p ' ; p =+2 (p) '(p) =+ p ; = p p ; 29. P () = dj { 2 N. 33., (). : , 2 N, >, = p l p l 2 2 :::ps ls, () =(l +)(l 2 +):::(l s +). : , 2 2 N ( 2 ) ( ) ( 2 ) : 55

56 : , 2 N dj 3 (d) = 2 (d) : dj : 0, 04, 33 P m k= k3 = ;P m k= k2. 37., " 2 R, ">0 C(") > 0, 2 N () C(") " : : " (), 0 <"<., >. = p l p l 2 2 :::p ls s, l i 2 N. () " = sy i= l i + p "l i i = UV U, i, p i 2 =", V i. V. l 2 N 2 l l +, p i > 2 =" l i + p "l i i l i + 2 l i V. {, U { 2 =" l i + p "l i i l i + 2 "l i 2l i 2 "l i = 4 " (, x 2 R, x>0 2 x = e x l 2 > +xl 2 > x ). 2 U 4 " 2 =" U V. 56 :

57 38., A>0 2 :::, ( m ) lim m! (l m ) = : A : l =[A]+2 H l l. m = H lm, m = 2 :::. 34, ( m )=(m +) l. ( m ) (l m ) A = (m +)l (m l H l ) A ml;a (l H l ) ;A m (l H l ) ;A. :, 37, 38. {,, " 2 R, ">0 () < 2 (+") l l l., () > 2 (;") l l l : 39. k m 2 N, k 2 t(k m) x y + x 2 y x k y k = m x y ::: x k y k 2 N., " 2 R, ">0 k t(k m) lim =0: m! mk;+" " =0? 57

58 : 29 37,, >0 t(k m) = ::: k 2N ++ k =m C() k m k ( ) ::: ( k ) C() k ::: k 2N ++ k =m ::: k 2 ++ k =m C() k m k ::: k;m ::: k (k ; ) C() k m k;+k : = "(2k) ; t(k m) (k ; ) C()k mk;+" m "=2. " =0, m t(k m) ::: k 2N ++ k =m ::: k; m k m k; : 2k 30. k 2 N. k () k- m m 2 :::m k 2 N, m m 2 :::m k =. 40., () =, 2 () =(), k 2 N, k 2 k () = P dj k;(d). :, , k 2 N k (). : k (),. 58

59 : = p l, p l 2 N, 4. : = p l p l 2 2 :::p ls s, k () = sy i= (l i + )(l i +2):::(l i + k ; ) (k ; )! 43., k 2 N, k 2 " 2 R, ">0 C(k ") > 0, 2 N k () C(k ") " : : k () () k; N 2 C. () = P dj d.,, () = (). 44., 2 C (). 45. () 2 C, 6= 0,. : = p l :::p ls s, () = sy i= p (l i+) i ; p : i ; 46., A>0, 2 N A () '() 2 : :, > = p l :::p ls s, l i 2 N. f() =() '() ; , Q s ; f() = i= ;p ;l i ; i. f() sy i= ; p 2 i Y i=2 : ; i 2 :,. 59

60 7 { 32. s 2 C, Re s >, { (s) = = s : : { {., (s) C, s =., (s) s = ;2 ;4 ;6 ::: ( )., (s) 0 Re s., Re s =.. 2, { {. (s),, R, >.,, (s), Re s { Re s >. :,. 48., Re s > (s) = Y p ; p s ; : : , (s) 6= 0 Re s >. 60

61 : Re s = >, 08, Y p ; Y + p s p p = = () : 48. :, (2) = 2 6 k 2 N (2k) =c k 2k, c k 4, (4) =, 90 2 Q.,,, k 2 N (2k). { (2k +), k 2 N., (3) ( ), (2k +) , (5), (7), (9), (). 50., Re s > (s) = () : s = : 25, (s) = () s = = k= = m= () (k) = s m= () =: m s jm m s k 2N k=m 5., Re s > 0 (s) =; = l s : () :. 6

62 52., Re s > ; 0 (s) (s) = = () s : :, P () = s Re s >, (s). {, 50, , Re s > (2s) (s) = Y p + p s ; : : k 2 N, k 2., Re s > k (s) = = k () s : : , Re s > 2 (s ; ) (s) = = '() s : 56., Re s > 2 (s ; )(s) = = () s : : { 56. = a s 62

63 a, = 2 3 :::.., a b 2 N (a b) =, a+b, = 2 3 :::. 8 { 33. m 2 N a b 2 Z., a b m a b (mod m), m j (a;b)., a b m,, a b m a 6 b (mod m). : 33 8 (mod 5), ;29 (mod 8), (mod 7), 46 6;7 (mod 7). 57. a b c 2 Z m 2 N. : ) a a (mod m). ) a b (mod m) =) b a (mod m). ) a b (mod m) b c (mod m) =) a c (mod m). 58. a b c d 2 Z, m 2 N. : ) a b (mod m) =) a c b c (mod m), ) a b (mod m) =) ac bc (mod m), ) a b (mod m) c d (mod m) =) a c b d (mod m), ) a b (mod m) c d (mod m) =) ac bd (mod m), ) a b (mod m) =) a b (mod m). : ) ) 33. ) (ac);(bd) =(a;b)(c;d). ) ac;bd =(a;b)c+b(c;d). ) ). 59. a b 2 Z, m 2 N d 2 N a, b m., a b (mod m), a d b d (mod m d ). 63

64 : 33 b d ; a d m d = b ; a m : 60. a b 2 Z, m 2 N a b (mod m)., d 2 N a b, (d m) =, a d b d (mod m) : : m j ; b d ; a d d (d m) =, 9, m j ; b d ; a d. 6. a b 2 Z ::: s 2 N., a b (mod ::: s ) a b (mod ) ::: a b (mod s ) : : m 2 N MZ., M m ( {... (mod m)), : ) a b 2M a6= b =) a 6 b (mod m), ) x 2 Z c 2M, x c (mod m). 62.,... (mod m) m. 63., m, m,... (mod m). 64., m 2 N, ) 2 ::: m...(mod m), ) m, 0 2 ::: m;... (mod m), 2 ) m, ; m+ ; m+2 ::: ; 0 ::: m... (mod m)

65 65. m 2 N, a b 2 Z, a 6= 0 (a m) =., x... (mod m), ax + b... (mod m). : 58, N, ( 2 )=., x x 2 2, x 2 + x 2... (mod 2 ). :, x 2 + x 2 x x 0 2 (mod 2 ) x 2 x 0 2 (mod ), x x 0 (mod ) 60. x x 0... (mod ), x = x 0., x 2 = x 0 2., x 2 + x m 2 N., x y m, x + y... (mod m). : 59, m 2 N, x 0 2 Z M...(mod m)., m y 2 Z, x 0 + y 2M. 35. m 2 N RZ., R m (...(mod m)), : ) a 2R =) (a m) =, ) a b 2R a6= b =) a 6 b (mod m), ) x 2 Z, (x m) = c 2R, x c (mod m). : (mod 0). 65

66 69. m 2 N.,... (mod m) '(m). 70.,... (mod m)... (mod m).,... (mod m)... (mod m). 7. m 2 N., '(m), m m,... (mod m). 72. m 2 N, a 2 Z, a 6= 0 (a m) =., x... (mod m), ax... (mod m). : N, ( 2 )=., x x 2 2, x 2 +x 2... (mod 2 ). : 66, 7. : m 2 N, a 2 Z (a m) =., a '(m) (mod m) : : a a 2 ::: a '(m)... (mod m). 72 aa aa 2 ::: aa '(m)... (mod m). a a 2 :::a '(m) (aa )(aa 2 ) :::(aa '(m) ) a '(m) a a 2 :::a '(m) (mod m) : ( a a 2 :::a '(m) )=,

67 : a 2 Z p., a p a (mod p) : : p j a,. p - a a p; (mod p), 74. : a ::: a s 2 Z p., (a + + a s ) p a p + + a p s (mod p) : : 75. :, :, a 2 N. a =. a>, a ;. ; a p = (a ; p; p ) + =(a ; ) p + k= p (a ; ) p;k +: k 45, a p (a ; ) p + (a ; ) + a (mod p) : 67

68 78. p l 2 N., m 2 Z m (modp l ), m p (modp l+ ). : m = + kp l k 2 Z. m p =(+kp l ) p : p, a 2 Z p - a. 77 a p; (mod p). l 2 N, l ; 78 a p(p;) (mod p 2 ) ::: a pl; (p;) (mod p l ).. a '(pl ) (mod p l ). 2 N, (a ) = > ( = )., = p l :::p ls s, l i 2 N., a '(pl i i ) (mod p l i i )., '(p l i i ) j '(). 58 ) a '() (modp l i i ) is, , a 2 N, a, a 2;2 (mod 2 ) : : =3. >3, a m 2 N a m; (mod m)., m? :.,, m =56=3 7 a 2 N, 56 - a. a 2 (mod 3) a 0 (mod ) a 6 (mod 7) : 2 j 560, 0 j 560, 6 j 560, a 560 (mod 3) a 560 (mod ) a 560 (mod 7) : 68

69 6 a 560 (mod 56). :, m 2 N { a 2 N, a m; (modm). m 2 N, { a 2 N, (a m) =,., 56., a 2 N, a>, { a. { 994., N f 2 Z[x]. x 2 Z, f(x) 0(mod),.. :, Z[x] x. 37. m 2 N f g 2 Z[x]., f(x) 0(mod) g(x) 0(modm), N f 2 Z[x]. ) x 2 Z f(x) 0 (mod ) (i) x 2 x (mod ), x 2 (i). ) M...(mod ), (i), M,. 69

70 ) M M 0... (mod ), M, (i), M 0,. : 82, (i),... (mod ). 2 N f 2 Z[x] (i). f,. 83. ) x 8 ; 0 (mod 5), ) x (mod ). : ) x (mod 5) ) N f 2 Z[x]. r f () f(x) 0(mod),... (mod ). r f (). : : ) 8 ) , f 2 Z[x] r f (). : r f () =. 2 2 N ( 2 )=. M M , 6, 65, 66 38, r f ( 2 ) = = = r f ( ) r f ( 2 ) : x 2M x 2 2M 2 f (x 2 +x 2 )0 (mod 2 ) x 2M x 2 2M 2 f (x 2 )0 (mod ) f (x 2 )0 (mod 2 ) 70

71 86. 2 N, a b 2 Z d =(a ). : ) d - b, ax b (mod ). ) d j b, ax b (mod ) d. : ). ) a = a d, b = b d = d. (a )=. 59,, a x b (mod ). 65, x 2 :::, a x... (mod ), 2 ::: x, a x b (mod ). 2 ::: d, x x + x +2 ::: x +(d ; ) ax b (mod ) N, a b 2 Z (a ) =., ax b (mod ) x a '(); b (mod ). : 74. : 86 87,., " \... (mod ), x 23 (mod 32),... (mod 32), 2 ::: 32. : (09 32) =,. 32= , '(32) = '(8) '(3) '(3) = 422 = , x (mod 32)., (mod 32). 7

72 (mod 32). { 25 2 =625 (mod 32), (mod 32). x (mod 32). :, {,. 89. ) ax b (mod ), " \... (mod ) ) 88. :, (a ),,.,, (a ) =( )., <a<. r ; =, r 0 = a., q r 2 Z, s +2, r = q +2 r + + r +2 ; s (i) =r s+2 <r s+ < <r <r 0 = a. P 0 =, P = ;q. P a r (mod ) (ii) =0. j s +., P 2 Z, 0 j (ii). : P j; a r j; (mod ) P j a r j (mod ) :, ;q j+. (i) P j+ a r j+ (mod ), P j+ = P j; ;q j+ P j, P. P s+2 a r s+ (mod ), x P s+2 b (mod ). ) 09x 23 (mod 32), 32 = , 09 = , 94 = , 5 = 3 4+3, 4=3+. 72

73 , (32 09) =., ), (;2) (mod 32), (mod 32), (;20)09 4 (mod 32), (mod 32), (;83) 09 (mod 32). x ;83 23 ; (mod 32). : {, ,. 9. ) 90x 68 (mod 32), ) 42x 32 (mod 98), ) 27x 9 (mod 94). : ) x (mod 32) ) ) x 39 (mod 94). 92. ) (x ; 5)(x ; 7) x 2 + x + (mod ), ) (x +)(x ; 2) (x +3)(x +2)+x + 7 (mod 6), ) 2(x +3)+5 5x + (mod 80) ::: k 2 N c ::: c k 2 Z., x c (mod ) :::::: x c k (mod k ) (i) c i c j (mod ( i j )) i<j k: (ii) 73

74 x c (mod [ ::: k ]) (iii) c 2 Z, ::: k c ::: c k. : k =,, k>., (i), (ii). (ii). c (l) 2 Z, l k,, c (l) c i (mod i ) i l: (iv) c () = c. l 2 N, l<k c (l) 2 Z, (iv). c (l+) = c (l) + z[ ::: l ] (v) z 2 Z {., c (l+) c (l) c i (mod i ) i l: (vi) z, c (l+) c l+ (mod l+ ) (vii) z[ ::: l ] c l+ ; c (l) (mod l+ ) : (viii) 86, (viii) z c l+ ; c (l) 0 (mod ([ ::: l ] l+ )) : (ix) ([ ::: l ] l+ )=[( l+ ) ::: ( l l+ )] (x) ( 30 ) ). (ii), c (l) (iv) c l+ ; c (l) c l+ ; c i 0 (mod ( l+ i )) i l: (xi) 74

75 (ix) (x), (xi) 25. c (l+) (v), z (viii),, (vi) (vii). c (l), l k, (iv),. c = c (k) (i),., x 2 Z, (iii), (i)., x 2 Z (i), i j (x ; c) i k, x (iii). :, ::: k, (i) ::: k.. 2: 93 (i). 3: { a i x b i (mod i ), i k.,,.. (i) ( ), { x 2 (mod 0) 2x 6 (mod 8) 6x 20 (mod 28) : (i) : 59,, (i) 3x (mod 5) 2x (mod 3) 4x 5 (mod 7) x 2 (mod 5) x 2 (mod 3) x 3 (mod 7) : (ii) (ii) x =2+5y y 2 Z. 2+5y 2 (mod 3), y 0(mod3),.. y =3z z 2 Z. x =2+5z (ii), 2+5z 3 (mod 7), z (mod7). z =+7t t 2 Z, x =7+05t., (i) x 7 (mod 05). 75

76 95. ) 36x 68 (mod 308) 66x 84 (mod 390) ) 42x 66 (mod 90) 44x 20 (mod 84) ) 5x (mod 6) 7x 9 (mod 0) x 7 (mod 5) ) x 3 (mod 5) x 4 (mod 2) x 8 (mod 35) : 96. {, 2, 3, 4, 5, 6 7. : a 2 Z, x a (mod 2) x 2a (mod 5) x 3a (mod 20). 98., k 2 N x 0 y 0 2 N, (x 0 + h y 0 + l) > h l 2 N, h l k. : k 2. m i M j, i j k, i{, j{. x 0 2 N x ;i (mod m i ) i = 2 ::: k (, 93, m ::: m k ). y 0 2 N y ;j (mod M j ) j = 2 ::: k ( )., h l k p h l, h{ l{, p h l j (m h M l ), p h l j (x 0 + h y 0 + l),. 76

77 3 {. 99. p, 2 N f 2 Z[x]., f(x) 0 (mod p), f(x) p. :. = f(x) =a 0 x+a. x x 2 2 Z, x 6 x 2 (mod p) a 0 x i +a 0 (mod p), i = 2,, p j a 0 (x ; x 2 ) 44, p j a 0., p j a. >, k 2 N, k<. f(x) x i 2 Z, i + p f(x) 0 (mod p). f(x) =f(x + )+(x ; x + )h(x) (i) h(x) 2 Z[x] ;. p j f(x + ), i 0 f(x i )=f(x + )+(x i ; x + )h(x i ) (x i ; x + )h(x i ) (mod p) : 44 x i, p j h(x i ) i,,, h(x) p. (i), f(x). : 99, f(x) p, f(x) 0(modp) f(x). {, ,? :., x 2 (mod8) 4. 77

78 20. p>2 f(x) =(x ; )(x ; 2) :::(x ; (p ; )) ; x p; +:, f(x) p. : 75, 2 ::: p; f(x) 0(modp). f(x) p ; 2, , p, (p ; )! ; (mod p). : p =2, p>2 20,,, (p ; )! +. : , (m ; )! 6 ; (mod m), m 2 N. : k j m <k<m, k j (m ; )!, k - (m ; )!+. : ,, (m ; )! m P, p>3, p; (p;)! k= 0(modp 2 ). k : f(x), 20, f(x) =c 0 x p;2 + c x p;3 + + c p;4 x 2 + c p;3 x + c p;2 : (i), c p;3 = ; P p; (p;)! k= k f(x), c p;2 =(p ; )! +. f(p) =(p ; )! ; p p; +=c p;2 ; p p; c p;2 (mod p 3 ) : (ii) 78

79 , (i) f(p) c p;4 p 2 + c p;3 p + c p;2 (mod p 3 ) : (iii) (ii) (iii), c p;4 p 2 + c p;3 p 0 (mod p 3 ) 59, c p;4 p + c p;3 0(modp 2 ). 20 p j c p;4. c p;3 0(modp 2 ),. : p f 2 Z[x]., r 2 Z[x] { p, f(x) 0(modp) r(x) 0(modp). : x 2 +6x 9 ; x (mod7) { p, l 2 N, l 2 M... (mod p l ). f 2 Z[x] x 0 2 Z f(x) 0 (mod p l; ). : ) f 0 (x 0 ) 6 0 (mod p), y 2M, f(y) 0 (mod p l ) y x 0 (mod p l; ) : (i) ) f 0 (x 0 ) 0(modp) f(x 0 ) 6 0 (mod p l ), y 2M, (i). ) f 0 (x 0 ) 0(modp) f(x 0 ) 0(modp l ), p y 2M, (i). : f(x + h) =f(x)+ f0 (x)! h + f00 (x) 2! h f () (x) h :! 82, k! f (k) (x) 2 Z[x] k 2 N. (ii) x = x 0, h = tp l;, t 2 Z. f(x 0 )=mp l; 79 (ii)

80 m 2 Z., 2(l ; ) l, f(x 0 + tp l; ) (m + f 0 (x 0 ) t) p l; (mod p l ) : (iii) ). p - f 0 (x 0 ), 86, t 0 2 Z, m + f 0 (x 0 ) t 0 0(modp). y 0 2M, y 0 x 0 + t 0 p l; (mod p), (i)., y 2M (i), y = x 0 + tp l;, t 2 Z. (i), (iii) 59, m + f 0 (x 0 ) t 0(modp) 86 t t 0 (mod p). 34 y = y 0, ). ). y 2M, (i), y = x 0 + tp l; t 2 Z, (iii). ). t 2 Z y = x 0 +tp l; (i). 68. : p l, p. p ( ).,, p 2 p 3 ::: p l f(x) =x 4 ;4x 3 ;8x 2 ;9x;4. f(x) 0 (mod 25). :, f(x) 0(mod5) x 2 (mod 5) x 3(mod5). f 0 (x) =4x 3 ; 2x 2 ; 6x ; 9, f 0 (2) 3(mod5) f 0 (3) 3(mod5). 207,., x 2 (mod 5). x =2+5t t 2 Z., 207, f(2) + 5f 0 (2)t 0 (mod 25), 5+5t 0 (mod 25). +3t 0(mod5), t 3(mod5). x 7 (mod 25). 80

81 , x 3(mod5). x = 3+5t, t 2 Z, f(3)+5f 0 (3)t 0(mod25), t 0 (mod 25), 4+3t 0(mod5), t 2(mod5). x 3 (mod 25) f(x) 0 (mod 49) f(x) = x 5 + x ; 5. :, f(x) 0(mod7) x 3(mod7) x 5 (mod 7). 7 j f 0 (3) 49 - f(3), 207 x 3(mod7). {, 7 - f 0 (5),, 208,, x 47 (mod 49). 20. ) x 4 +67x ; 29 0 (mod 2) ) x 2 ; 54x (mod 69) ) x 3 ; 3x 2 + x +2 0 (mod 25). : ) f(x) =x 4 +67x ; 29 f 0 (x) =4x f(x) 0 (mod ) x (mod )., x 2 (mod )., j f 0 (2) 2 j f(2). 207 ),, x 2+t (mod 2), t =0 2 ::: 0. {, - f 0 (3). 207 ),, x 3 (mod ), x 4 (mod 2)., - f 0 (4), x 4 (mod ), x 92 (mod 2)., f(x) 0 (mod 2) x (mod 2) : ) x 23 3 (mod 69). ) x 2 (mod 25). 8

82 2. f(x) 0 (mod m) (i) f 2 Z[x] m 2 N. : m m = p l :::p ls s. 6, (i) f(x) 0 (mod p l i i ) i s: (ii), 207. (ii), (i). (ii) i{ x c () i c (2) i ::: c ( i) i (mod p l i i ) : j ::: j s 2 N, j ::: j s s (iii) x c (j i) i (mod p l i i ) i s:, 93, (i). j ::: j s, (iii), (i). 22. ) x 3 ; 6x 2 +9x ; 5 0 (mod 360) ) x 4 +7x 3 +4x 2 +3x +2 0 (mod 2520) ) x 2 ; 80x +3 0 (mod 7920). : ) 360 = , f(x) 0 (mod 2 3 ) f(x) 0 (mod 3 2 ) f(x) 0 (mod 5) 82

83 f(x) =x 3 ;6x 2 +9x;5. 207, 93, 48, x a +30b (mod 360), a = 3 7 b =0 2 :::. ) 2520 = , 5. ) 32, x a b (mod 7920), a 83, 457, 73, 073, 307, 483, 667, 843, 2297, 2473, 2657, 2833, 3067, 3427, 3672, 3683, b x 2 R,, e(x) =e 2ix. P a<kb e; f(k), a b 2 R, a<b f(x),. 23., ) x 2 Z e(x) =. ) x 2 R je(x)j =. ) x y 2 R e(x + y) =e(x) e(y). ) 2 N ; x e x. 24., a<kb e ; f(k) [b] ; [a] : : 23 ). : 24. f(k), a<k b, [0 ) ( ),, e(f(k)) 83

84 " \. { [b] ; [a], m 2 N 2 R Z., m ; e k = e(m) ; e() ; e() : k= :. :, 25, 26 27,.,, { 25,.,,, 40.,. 26. a m 2 N., m k= ak e = m ( m m j a 0 m - a: : 23 ) m 2 N 2 R Z., m ; e k mi m k= 2jjjj jjjj {. 84

85 : 25 e(m) ; e() ; j si j = si(jjjj) 2jjjj : : 27 2 Z, ; mi m 2jjjj = m. 40. c( a) 2 N a 2 Z c( a) = k (k )= ak e : 28.,, k... (mod ). : 23 ). 29., 2 N, a b 2 Z ( b) =, c( a) =c( ab). : , 2 N, a 2 Z ( a) =, c( a) =(). : 29, a =. 25, c( ) = k (k )= e k = k e k dj(k ) (d) :, 26 k c( ) = (d) e = (d) e l = () : dj dj d k k0 (mod d) 85 l d

86 22., a. : 2 2 N ( 2 )= a(k 2 + k 2 ) c( 2 a)= k (k 2 )= = k (k )= ak e = 2 k e ak k 2 2 (k 2 2 )= k 2 2 (k )= (k 2 2 )= e ak2 2 e 2 = c( a) c( 2 a) : 222. p, l 2 N, a 2 Z, a 6= 0 ord p a =., c(p l a)= 8 >< >: p l ; p l; l ;p l; l = + 0 l +2: : 29,, a = p. l, l>,, c(p l p )=p c(p l; ) , 2 N a 2 Z c( a) = '() : (i) ' ( a) ( a) : a =0. a 6= 0. (i) h( a). c( a) h( a) 20, 99, 0, , c(p l a)=h(p l a), p l 2 N. {, ord p a =., c(p l p )=h(p l p ),, =

87 4. l 2 N a 2 Z.. S l ( a) S l ( a) ak l ak l S l ( a) = k e S l ( a) = k (k )= : l =. 26,,,.. l 2 {. 2: ak l f(k), f 2 Z[x], {,.. f(x),. 224.,., k... (mod )... (mod ). 225., l 2 2 N, a a 2 2 Z ( 2 )=, ) S l ( 2 a 2 + a 2 )=S l ( a ) S l ( 2 a 2 ). ) S l ( 2 a 2 + a 2 )=S l ( a ) S l ( 2 a 2 ). : ). k k 2 2 Z, (a 2 + a 2 )(k 2 + k 2 ) l (a 2 + a 2 )(k l l 2 + k2 l l ) a l+ 2 k l + a 2 l+ k l 2 (mod 2 ) : 4, 65, 66, 23, 224 e : 87

88 ,, S l ( 2 a 2 + a 2 )= e k k 2 2 = e k k 2 2 = k e a ( 2 k ) l = S l ( a ) S l ( 2 a 2 ) : (a 2 + a 2 )(k 2 + k 2 ) l 2 a l+ 2 k l + a 2 l+ k l 2 = 2 a2 ( k 2 ) l k 2 2 e 2 = = ) N, a 2 Z (a ) =., S = P k e; ak 2 jsj ( p 2 - p 2 2 j : : jsj 2 = S S ( S S). a(k 2 ; k2) 2 jsj 2 = k e ak 2 k 2 e ;ak 2 2 = e k k 2 k ; k 2 h (mod ) h 2 N,, : k 2 ; k 2 2 h(h +2k 2 ) h 2 +2hk 2 (mod ), 23, a(k 2 e ; k 2) 2 ah 2 = e e 2ahk2, h k 2 k,, k, k k 2 + h (mod ). : 88

89 26 jsj 2 = h k k 2 k 2 ;k h (mod ) = = h ah 2 e h 2ah0 (mod ) k 2 ah 2 e e 2ahk2 ah 2 e : 2ahk2 e (a ) =,, 2 j, 2 -. jsj 2. ( j : 226..,,,. {,. 2: 226 a =, k k 2 e = +i; +i ; p = 8 >< >: = = ( + i) p 0 (mod 4) p (mod 4) 0 2 (mod 4) i p 3 (mod 4) : N. k 2 Z, (k ) =, (k) 2 N k(k) (mod ), (k). 89

90 :, k (k)., k, k = (k) , k.. (mod ), (k).. (mod ). 229., 2 N, k 2 Z (k ) =, (;k) ;k (mod ) N k k 2 2 Z ( k k 2 )=., k k 2 (mod ), k = k N k k 2 2 Z ( k k 2 )=., (k k 2 ) k k 2 (mod ) N k k 2 2 Z,, ( k )=( 2 k 2 )=( 2 )=: (k 2 + k 2 ) 2 (k 2 2) 2 + (k 2 2 ) 2 (mod 2 ) : :, (k 2 + k 2 2 )=(k 2 2 )=(k )=. 42, (k 2 + k 2 ) ; (k 2 2) 2 + (k 2 2 ) 2 (mod 2 ), 6,, 2., 42, (k 2 + k 2 ) ; (k 2 2) 2 + (k 2 2 ) 2 (k 2 2) (k 2 2) (mod ) : 2,. 90

91 43. 2 N a b 2 Z. ak + bk K( a b) = k ( k)= : k (k). 2:. {, c( a) =K( a 0). 3:.,,., ( ab) =, " 2 R, ">0 C(") > 0, jk( a b)j C(") +" ,, k... (mod ). e : : 23 ) , 2 N a b 2 Z K( a b). : 72, ,. 235., 2 N a b 2 Z, K( a b) =K( b a) : : , 2 N, a b l 2 Z (l ) =, K( a bl)=k( al b) : 9

92 : 72, N, ( 2 ) = a a 2 2 Z. K( 2 a a 2 2 ) = K( a ) K( 2 a 2 ) : : K( 2 a 2 + a ) = = k k 2 2 (k )= (k 2 2 )= e (a a 2 2 )(k 2 + k 2 )+(k 2 + k 2 ) 2 2 (a a 2 2 )(k 2 + k 2 ) k a k 2 a 2 3 (mod 2 ) , K( 2 a a 2 2 ) = = k k 2 2 (k )= (k 2 2 )= e a k (k 2 2 ) a2 k e (k 2 2) 2 : ,, K( i a i ), i = 2,. : x 2 R, x>0, (x) = px (x) = px l p (x) = kx (k) (x) (x), x, (x) (k). 92

93 : (x) (x). : () = ( 7) = (0 96) = 0, (3) = 2, (7 2) = 4, (6 3) = l 2 + l 3 + l 5, (0)=l2+l3+l2+l5+l7+l2+l x 2 R, x. Q P = p p p, x x 4 P =, x 2 [ 4). (x) =( p x) ; + djp (d) h x di : :, x 2 [ 4). x 4. S 2 N, x, ( P ) = , S = (d) = x dj( P ) djp (d) x 0 (mod d) = djp h x (d) : di, 40, 2 N, x, P, ( p x x], =. S = (x) ; ( p x)+,. : (x) 238. x,, P x. 2: 238,,. 20{ {,.. c > 0, c 0 > 0, x 2, x x c x l x (x) c0 93 x l x :

94 : c c 0,. { c c 0.., (x) c c 0,, (x) lim x! x =: (i) l x 896. {., (x) { R x dt x,. 2 l t l x {, (x) = Z x 2 dt l t +(x) (x) (x) =O ; xe ;c p l x c>0., lim x! R x dt 2 l t x l x = (ii) (iii) (ii) (iii) (i). { (x),, (x) =O xe ;c (l x)3=5 (l l x) ;=5 (iv) c>0., (x) =O ;p x l 2 x { (iii) (iv). 94

95 239., : c 2 > 0 c 0 2 > 0 x 2 2, x x 2 c 2 x (x) c 0 2x: : (x) (x) (x)lx: p x<px l p ; (x) ; ( p x) l p x = 2 (x)lx + O(p x l x) :. 240., 2 N, 2 Y p p 4 : :. =2 3, m 2 N, 2 m<.,, Y p p = Y p; p 4 ; < 4 :, =2m +. 2m + (2m + )(2m)(2m ; ) :::(m +2) M = = : m m!, 2 2m+ =(+) 2m+ 2M, M 4 m. {, N = Y m+<p2m+ 95 p:

96 N j (2m +)(2m)(2m ; ) :::(m +2) (N m!) =, 9 N j M. N M 4 m :, Y Y p = N p 4 m 4 m+ 4 2m+ p2m+ pm+. 24., x 2 R, x (x) x l 4 : : , x 2 R, x (x) =(x)+( p x)+( 3p x)+::: log 2 x. : 23. {, k 2 N ( kp x) 6= 0, kp x 2, k log 2 x. 243., x 2 R, x 2 (x) =(x)+o( p x) : : (x) ; (x) =( p x)+( 3p x)+::: : 24 O( p x). log 2 x O( 3p x). 96

97 244., : c 3 > 0 c 0 3 > 0 x 3 2, x x 3 c 3 x (x) c 0 3x: : , x 2 R, x 2 (x) =O(x) : : , x 2 R, x 2 (k) =lx + O() : k kx : 25, h x x l x + O(x) = (k) ki = (k) kx kx = x kx = x kx (k) k (k) k + O( (x)) = + O(x) : x. x k ; xo 247., x 2 R, x 2 px l p p : kx (k) k =lx + O() : = px 97 l p p +(x) k =

98 (x) = k2 p p k x = px l p p k px l p p(p ; ) p 2 + p 3 + ::: k=2 l k k(k ; ) : l p =, (x) =O() , x 2 R, x 2 c. px : f(t) = (l t) ; c = p =llx + c + O l x ( ; l P pqx ( p q, pq x). : pq ; pqx pq =2 p p x q x p 248. p q p x pq 250., x 2 R, x 2 Y px c>0. ; = c +O p l x l x 98 pq

99 :, 248. :, c e ;,. 25., c>0, x 2 R, x 2 (x) cx: (i) : c 0 > 0, O() 247, pt l p p ; l t c0 t 2 : (ii) 2 R, 0 <<, l > 3c 0 : (iii) x 2 R, x 2 ; S = x<px (ii), S ; l l p p : 2c 0 (iii), S c 0. c 0 S x x<px l p (x) x : x 2 ; (x) c 0 x. (x) c = mi 2x2; c = mi(c x 0 c ). c>0 (i) x 2. 99

100 252.. : 239, : 25, (x),., N, 2 d =[ 2 3 ::: ]. d () : : p, p p =ord p (d ). 52, m 2 N, m, p = ord p m. p p m, d = Y p p p Y p = () : 254., 2 N, 6 d 253, d 2 ;2 : : m 2 N, m I m = = = I m = Z 0 ;m k=0 ;m k=0 Z 0 ;m x m; k=0 ; m k ; m k x m; ( ; x) ;m dx : ; m k (;) k Z (;) k x k dx = 0 (;) k m + k : x m+k; dx = 00

101 { d,, I m = a d a 2 Z: (i) {, J (y) = = = J (y) = Z ; 0 ; k=0 ; k=0 Z 0 ( ; x + xy) ; dx : ; (xy) k ( ; x) ;;k dx = k k=0 ; k ; k Z y k x k ( ; x) ;;k dx = 0 I k+ y k :, u =+(y ; )x, y 6= J (y) = y ; Z y u ; du = y ; y ; = ; y J (y) k=0 y k : I m = ; m ; = m m m 2 N m : (ii) (i) (ii), m 2 N, m d m m : (iii) 0

102 m =[=2]. ( +) [=2] k=0 m k 2 [=2] + : {, =2 (iii) m =[=2] d [=2] [=2] 2 ; 2 + 2;2 : 255., c>0 x 0 2, x x 0 (x) c x l x : : 2 N, 6, , () l d l l 2;2 l = ( ; 2) l 2 l l 2 2 l :, x 2 R, x 6 (x) =([x]) l 2 2 [x] l[x] l 2 x ; 2 l x l 2 4 x l x : 256. p {., c>0, c 0 > 0 0, cl p c 0 l 0 : :, p. (x),, = (p ) c 0 p l p c 0 02 p l

103 , (c 0 ) ; l p : {, (x) lim! p =,, p = (p ) c p p : l p p 2, p = (p ) c c p l p 2 l : p 2c ; l. 257., c>0, 2 N '() c l l(0) : :, >. = q l q l 2 :::qs ls, l i 2 N q <q 2 < <q s., q q 2 :::q s 2 s, s = O ; l(0) : p <p 2 < <p s s, p i q i i = 2 ::: s. 30 sy '() = ; ; sy ; ; Y ; = : q i p i i= i= pp s ; p 250, 256 (i), '() = O(l p s)=o(l s) =O(l l(0)). 258., c>0, 2 N () c l l(0) : : (i)

104 6 259., x 2 R, x 2 '() = cx + O(l x) c = (2) ;. x : 5 77, 26, x '() = x = dx js 2 j P dx dj (d) d = dj (d) h x = d di dx (d) d (d) d x 0 (mod d) x d ; x d o = (d) (d) x S d 2 2 = : d do dx S = dx 9, d S 2 = O(l x) : {, 50,, js 0 j P d>x d 2 S = (2) ; ; S 0 S 0 = (d) : d 2 d>x 95 ) S 0 = O x = xs ; S 2 (i) (ii) (iii) : (iv) (i) {(iv). : 49,, ;2. 04

105 260., x 2 R, x 2 '() =cx 2 + O(x l x) c = 2 (2);. x : 259, 90 c = '() f(t) =t. 26., x 2 R, x 2 x '() = cx + O(x" ) P c = 2 (k) k= " 2 R, ">0, k'(k) O ". : 77 32, x '() = x = dx dj S = dx 2 (d) '(d) 2 (d) '(d) = 2 (d) '(d) dx h x di = x d ; xo = xs + O(S 2 ) (i) d 2 (d) d'(d) 95 ) 257, S 2 = O dx S 2 = dx '(d) : S. = O(x " ) : (ii) d ;" S = c + O ; S 3 (iii) 05

106 P S 3 = d>x c d'(d) (,,, 257). 95 ) 257, S 3 = O d>x (i) {(iv). = O(x ;+" ) : (iv) d 2;" 262., x 2 R, x 2 '() = c l x + c0 + O ; x ;+" x c c 0 2 R (c 26), " 2 R, ">0, O ". : x 2 R, x 2. ), () =L(x) x L(x) 73. ) x : () =x l x +(2 ; )x + O ;p x : L(x) = = = kmx x km= x () ). )

107 264., x 2 R, x 2 x () = 2 l2 x +2l x + c + O ; x ; 2 c. : ). 265., l 2 N c l > 0, x 2 R, x 2 x () l c l (l x) 2l : : l = 264., l 2 N c l , () l+ () l = () = () l (km) l = km x x x km= kmx kx mx (k) l (m) l km = kx (k) l k 2 c l2 (l x) 2l , l 2 N c l > 0, x 2 R, x 2 x () l c l x (l x) 2l ; : 07

108 : l = 263., l 2 N , () l = x () l+ = x () l () = x kmx (k) l (m) l kx km= (k) l m 2x k kmx (m) l : (km) l, , x 2 R, x 2 () =cx 2 + O(x l x) c = 2 (2). x :, 2 N d,. d () = d = d = k: x x dx dx dj x 0 (mod d) P k = (+) k= 2 [x] x () = 2 = x2 2 dx dx h x d ih x + = di 2 d 2 + O x dx : d dx x 2 k x d d 2 + O x d O(x l x) 9. {, 95 ) 32, dx. d = (2) + O 2 x 08 =

109 268. x k () (k ; )! x (l x + k ; )k; : : k. 269.!() 24., x 2 R, x 0 x!() =x l l x + cx + O x l x c 248. : 77, 248,!() = = = x x px = px pj h x pi = px x 0 (mod p) x p ; xo = x l l x + cx + O x l x p : = x px p + O((x)) = 45., 2 N k{,, k. x 2 R, x. N(x k) 2 N, k{ x. : , k 2 N, k 2 N(x k) =c k x + O ; x k c k = (k) ;. 09

110 : 25 77, h x i N(x k) = (d) = (d) = (d) d k x d k j d kp x x 0 (mod d k ) d kp x. :,.,, M(x) = x () :, x 2 R, x jm(x)j x., M(x) (x).,, lim x! M (x) x =0., M(x) =O ; x 2 +", ">0. 0

111 [].,.,, " \,, 987. [2]..,..,, " \,, 985. [3]..,, " \,, 98. [4].,.,, " \,, 976. [5].,.,.,, " \,, 975. [6]..,.,..,, ". \,, 984. [7].,.,.,, " \,, 99. [8].,.,, " 6\,, 999. [9] \America Mathematical Mothly" " \,, 977. [0]..,, " \,, 983. []., p{, p{ { -, " \,, 98. [2].,, " \,, 97. [3].,,.. ". \,, [4].,., I," \,, 973.

112 [5].,., II," \,, 974. [6]..,, " \,, 97. [7].,, " \,, 967. [8].,, " \,, 996. [9].,, " \,, 966. [20]..,..,, " \,, 978. [2].,,, 959. [22].,, " \,, 974. [23]..,..,..,, " \,, 976. [24] T. M. Apostol, Itroductio to aalytic umber theory, Spriger, 976. [25] M. Aiger, G, Ziegler, Proofs from the Book, Sec. ed., Spriger, [26] G. H. Hardy, E. M. Wright, A itroductio to the theory of umbers, Fifth. ed., Oxford Uiv. Press, 979. [27] G. Teebaum, Itroductio to Aalytic ad Probabilistic Number Theory, Cambridge Uiversity Press,

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