2019_Boston_HP

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1

2 (k 1,...,k r )= X 1 m k 1 0<m 1 < <m r 1 mk r r k i 2 N, k r > 1 P 1 \{0, 1, 1}

3 r =1 L. Euler ( ) ζ(2n) = ( 1)n 1 2 B 2n (2n)! (2π)2n (n =1, 2, 3,...) ζ( n) = B n+1 n+1 (n =0, 1, 2,...) B n x 1 e x = n=0 x n B n n!

4 r =2

5

6

7 Q Z k := k 1 + +k r =k d k Q ζ(k 1,...,k r ) d k = d k 2 + d k 3, d 0 =1,d <0 =0 k d k k

8 dim Q Z k apple d k.

9

10 ζ(2) = 0<t 1 <t 2 <1 dt 1 1 t 1 dt 2 t 2

11 = = + + ζ(2) 2 = ( = 0<t 1 <t 2 <s 1 <s 2 <1 0<t 1 <t 2 <s 1 <s 2 <1 0<t 1 <t 2 <1 0<t 1 <t 2 <1 s 0<s 1 <s 2 <1 2 ( <t 1 <s 1 <t 2 <s 2 <1 0<t 1 <s 1 <s 2 <t 2 <1 0<s 1 <t 1 <t 2 <s 2 <1 ) dt1 dt 2 ds 1 ds <s 1 <t 1 <s 2 <t 2 <1 0<s 1 <s 2 <t 1 <t 2 <1 1 t 1 t 2 1 s 1 s 2 dt 1 dt 2 ds 1 ds 2 dt 1 ds 1 dt 2 ds t 1 t 2 1 s 1 s 2 1 t 1 1 s 1 t 2 0<t 1 <s 1 <s 2 <t 2 <1 0<s 1 <t 1 <s 2 <t 2 <1 dt 1 ds 1 ds 2 dt t 1 1 s 1 s 2 t 2 ds 1 dt 1 ds 2 dt s 1 1 t 1 s 2 t 2 0<t 1 <s 1 <t 2 <s 2 <1 0<s 1 <t 1 <t 2 <s 2 <1 0<s 1 <s 2 <t 1 <t 2 <1 = ζ(2, 2) + ζ(1, 3) + ζ(1, 3) + ζ(1, 3) + ζ(1, 3) + ζ(2, 2) =2ζ(2, 2) + 4ζ(1, 3). dt 1 1 t 1 dt 2 t 2 0<s 1 <s 2 <1 dt 1 dt 2 ds 1 ds 2 1 t 1 t 2 1 s 1 ds 1 1 s 1 ds 2 s 2 s 2 ds 1 dt 1 dt 2 ds 2 1 s 1 1 t 1 t 2 s 2 ds 1 ds 2 dt 1 dt 2 1 s 1 s 2 1 t 1 t 2

12 x x x

13 x x

14

15 ζ(k 1,...,k r ) k i

16 p X 0<m 1 < <m r <p 1 m k 1 1 mk r r mod p 2 Z/pZ p A := Y Z/pZ. M Z/pZ p prime p prime Q

17 A (k 1,...,k r )= p (k 1,...,k r )modp p prime 2 A p (k 1,...,k r )= X 0<m 1 < <m r <p 1 m k 1 1 mk r r

18 k 6= 0 A (k) =0. * ) p 1 - k ) p 1 X m=1 1 m k 0 (mod p). A (k 1,k 2 )=( 1) k k1 + k 2 2 Z(k 1 + k 2 ). Z(k) := Bp k k k 1 mod p p 2 A. * ) X 0<m<n<p 1 X m k 1 n k 2 0<m<n<p m p 1 k 1 n k 2 (mod p).

19 Q X Z A,k := Q A (k 1,...,k r ) k 1 + +k r =k d k 3 (= d k d k 2 )

20 ζ(k 1,...,k r ) = ζ(k) k 1 + +kr =k kr 2 k 1 + +kr =k kr 2 ζ A (k 1,...,k r ) = ( 1+( 1) r ( k 1 r 1 Z(k) = Bp k k mod p p ) )Z(k) 2 A!

21 ζ(k) Z(k) ζ(k) Fermat ζ(k (p 1)) = Euler B p k p k

22 k, l A (k) A (l) = A (k l) ζ A (kxl) = ( 1) l ζ A (k, l), Q

23 ζ A (k 1,...,k r ) ζ(k 1,...,k r ) A (k) =0 ζ A (k 1,...,k r )

24 S (k 1,...,k r ):= rx ( 1) k i+1+ +k r (k 1,...,k i ) (k r,...,k i+1 ) i=0 = X, ζ X 1 S (k 1,...,k r )= m 1 mr m i 0 m k 1 1 mk r r ( = ) 3 2 1

25 X S (k 1,...,k r ) S (k 1,...,k r ) mod 2. S (k 1,...,k r ) 2 Z/ 2 Z S (k 1,...,k r ):=S (k 1,...,k r ) mod 2 k r =1 Z Q

26 S (k) =( 1) k (k)+ (k) = ( 2 (k) k: even, 0 k: odd. S (k) =0 S (k 1,k 2 ) ( 0 (mod 2 ) k 1 + k 2 :even, ( 1) k 2 k 1+k 2 (k k k 2 ) (mod 2 ) k 1 + k 2 :odd. S (k 1,k 2 )=( 1) k 2 k1 + k 2 k 1 (k 1 + k 2 )mod 2.

27 k 6= 0 A (k) =0. k1 + k 2 A (k 1,k 2 )=( 1) k 2 Z(k) := Bp k k k 1 Z(k 1 + k 2 ). mod p 2 A. p

28 Q ζ A (k 1,...,k r ) Z/ 2 Z A (k 1,...,k r )! S (k 1,...,k r ) k = d k d k 2 = d k 3.

29 X k 1 + +kr =k kr 2 S (k 1,...,k r )= 1+( 1) r k 1 r 1 (k) mod 2 k 1 + +kr =k kr 2 ζ A ( ( ) (k 1,...,k r ) = 1+( 1) r k 1 )Z(k) r 1 ζ A (k 1,...,k r )

30 k, l S (k) S (l) = S (k l) S (kxl) =( 1) l S (k, l) A (k) A (l) = A (k l) ζ A (kxl) = ( 1) l ζ A (k, l),

31 dim Q Z A,k apple d k 3. ζ A (k 1,...,k r )

32

2

2 CONTENT S 01 02 04 06 08 10 20 25 25 30 38 49 53 58 60 64 74 8 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 6 1 2 2002,000 200 7 Web 20 20 13 13 4 13 13F 13F 20 20 13 13 4 13 13F 13F 13 6 A B C A B 13 20 13 20

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