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1 Hiroshi Yui c 2006, All rights reserved.

2 = = 2

3 3

4 2 = = = = = n = = = lim n! = = n = = 4

5 = n n n n n, 2, 3,..., 6 5

6 = = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 9 = 0 = = 2 = 3 = 4 = 5 = 6 = = + 2 = = = = = = = = = = = = = =

7 n n = S n S n = = (S n ) n n 5 n 000 = n n = n S n S sum sum sum up Sn n n = s t n n n S n n = S n = S = n S n > 0 n n S n n n S n 7

8 n S n < S n+ n S n S n < S n+ be be Reality of Real Number n S n+ S n = n (?) S n S n = = S n+ S n = n n S n+ S n = n + S n+ S n = = n+ = n = ( n + n + = n + ) ( ) n 8

9 S n+ S n n n S n+ S n n S n+ S n > S n+2 S n+ 2, 3, 4,... n N n N S n+ S n > S n+2 S n+ n N... For all n in N... n All A S = S 2 = + 2 S 3 = S 4 = S 5 = S n+ S n 9

10 S 2 S = 2 S 3 S 2 = 3 S 4 S 3 = 4 S 5 S 4 = 5 S 6 S 5 = 6. S n+ S n S, S 2, S 3, S 4, S 5,... n S n+ S n > S n+2 S n+ n N S n+ S n > S n+2 S n+ S n = n = n S n S n n S n n

11 . M M < = n M M M M n S n M n S n a. R N M R n N M < Exists E = For all M in R n exists in N... n exists there exists n such that For all M in R there exists n in N such that M < n =. () (2) M R n N M < n N M R M < = = () (2)

12 For all M in R there exists n in N such that M < There exists n in N such that for all M in R M < = =. (). (2) () M n n M (2) n n M () M n M M n (2) n n M n (2) M n a () 3 2

13 4 ζ() ζ() ζ() ζ 3

14 5 S 6 = 6 = = = ( ) ( ) ( ) ( ) 6 } {{ } } {{ } } {{ } } {{ } ( ) ( ) ( ) ( ) 6 = ( ) ( ) ( ) ( ) = = S 6 S, S 2, S 4, S 8, S 6,... S S 2 > + 2 S 4 > S 8 > S 6 >

15 m 0 S 2 m + m 2 S 2 m S 2 m S 2 m m m + m 2 S 2 m m M M < n = n M m M < + m 2 m m = 2M m n = 2 m m n n n M < + m 2 S 2 m = S n =. M M < n a. R N = M R n N M < + m 2 S 2 m n = 0 = = 5

16 0 a 0 ( =, 2, 3,...) * n = a M R n N M < = a n = a n a = = * 2 a = = 2. R, N, N a > 0 M R n N M < 2 a n n M = a M a a = 2 n a > 0 = a = a * a = 0 ( : M R m N n = m M < P n = a ) * 2 (=) 6

17 a = = = 2 = n = 2 n (?) 2 < 2 = n = 2 n 2 (?) 2 n = = 2 2 = 2 a = = = 2 a = n ( = ) 2 n = 2 n+ 2 < 2n+ 2 = n = a = n = n 2 0 M < n 2 M n a = R, N, N a > 0 M R n N M < = a 7

18 n 6 ζ() ζ ζ() ζ(s) (Riemann s zeta function) ζ(s) = = s ( ) ζ(s) s = (harmonic series) Harmonic H H H = = ( ) s = H = ζ() H n (harmonic number) H n = = ( ) n H n H 8

19 H = lim n! H n H n n lim H n = n! H = ζ() ζ() = m 0 H 2 m + m 2 4 L H n = H(n) H(x) H(x) h(x) h(x) H(x) h(x) 7 H n H(x) h(x) 9

20 H(x) = x = x h(x) =? f(x) f(x) = f(x + ) f(x + 0) (x + ) (x + 0) f(x) D Df(x) = lim h!0 f(x + h) f(x + 0) (x + h) (x + 0) f(x + ) f(x + 0) f(x + h) f(x + 0) f(x) = Df(x) = lim (x + ) (x + 0) h!0 (x + h) (x + 0) H n H n H(n) H(n + ) H(n + 0) H(n) = (n + ) (n + 0) = H(n + ) H(n) H(n + ) H(n) H(n) = H(n + ) H(n) = n + H(x) x+ h(x) x+ x + dx = log e (x + ) + C C h(x) = log e (x + ) 20

21 H(x) = x = h(x) = log e (x + ) H(x) = x x, 2, = x y = x 3,... y = x y x H(n) =, 2, 3,..., n y = x + n dx + ) x 2

22 y = x+ dx n 0 x+ y x n 0 n x + dx < H n < + x dx n n 0 x+ dx = log e(n + ) log e (0 + ) = log e (n + ) + n x dx = + log e n log e = + log e n log e (n + ) < H n < + log e n n log e (n + ) log e n log e (n + ) + log e n 22

23 y = x y = H(x) y = H(x) y = log e (x + ) H(x) log e (x + ) y (0.504) (0.496) (0.4739).8333 (0.4470).5000 (0.404).0000 (0.3069) x , 0.404, , , 0.496, 0.504,... x =, 2, 3,..., 6 23

24 x =, 00, 200,..., 600 y (0.577) (0.577) (0.577) (0.5770) (0.5770) (0.5767).0000 (0.3069) x , , , , 0.577, 0.577,

25 x =, 000, 2000,..., 6000 y (0.5772) (0.5772) (0.5772) (0.5772) (0.5772) (0.5772).0000 (0.3069) x x H(x) log e x n H n log e n H(x) h(x) x H(x) = = h(x) = log e x 25

26 8 H = = m 26

27 9 ( ) ( ) 27

28 ( ) + ( ) + ( ) n n=0 =0 n=0 n 0,, 2,... n =0 2 3 n Q 2 Q 2 = ( ) ( ) ( Q 2 = ) ( ) ( ) ( ) = 2 3 Q n 28

29 ( Q 2 = ) ( ) ( ) ( = } {{ } ) ( 2 3 } {{ } ) } {{ } n=0 n= n=2 = n=0 =0 2 3 n Q 2 ( 2 ) ( 3 ) = n=0 =0 2 3 n 2 3 n n=0 =0 2 3 n 2 3 n M 2 M 29

30 M p, p 2,..., p,..., p } {{ M p } M 3 p = 2, p 2 = 3, p 3 = 5 Q M ( Q M = M = = = M = ) ( ( p 0 + p + p 2 + p ) ) ( p 0 + M p + ) M p 2 + M Q 2 2 M M Q M p 0 + p + p 2 p 2 + p 2 3 M 2 M ( Q M = M = = = n=0 ) ( ( p 0 + p + p r 3 r 25 r 3 p r MM ) ) ( p 0 + M p + ) M p 2 + M r + r r M = n r, r 2,..., r M n Q M p r n 30

31 2 r 3 r 2 5 r3 p r M M M p r p r 2 2 p r 3 3 p r M M Q M 2 3 Q M = Q M Q M Q M = M = p Q M = Q M = = M = p = = Quod Erat Demonstrandum. 3

32 8 0 32

33 33

34 6 n Euler-Mascheroni γ [4] H n = log e n + γ + n 2n n 4 ɛ, 0 < ɛ < 256n 6, γ = s = p p s = PDF URL Copyright (C) 2006 Hiroshi Yui ( ) All rights reserved [6] 34

35 (partial sum) (infinite series) (harmonic number) (harmonic series) a = a + a 2 + a a n = a = lim = = = (Riemann s zeta function) ζ(s) = n! = a n = a + a 2 + a 3 + = n = = = s ζ() = = ζ(s) = p p s For all n in N... n N... There exists n in N such that... n N... 35

36 [],, ISBN , (Nicole d Oresme, 323? 382) ζ(σ) [2],, PDF 2 [3], ISBN , 994. (p ) (p ) [4] Donald E. Knuth, The Art of Computer Programming Volume ISBN X,, n H n 7 (p. 75) [5] (Leonhard Euler, ) He calculated just as men breathe, as eagles sustain themselves in the air (by François Arago) Read Euler, read Euler, he is our master in everything (by Pierre Laplace) [6] R,,,, ISBN ,, ISBN (2005 ) 20 [7],,, ISBN , :-) [8],, 36

37 . (2004 ) 2. (2005 ) 3. (2005 ) 4. (2005 ) 5. (2006 ) 6. (2006 ) 37

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