2006 4 2 2 4 3 2 4 3 5 4 6 8 7 9 8 26 9 27 0 32 http://www.hyui.com/girl/harmonic.html Hiroshi Yui c 2006, All rights reserved. http://www.hyui.com/
= = 2
3
2 = = + 2 + 3 + = = = n = = = lim n! = = n = = 4
= n n n n n, 2, 3,..., 6 5
= = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 9 = 0 = = 2 = 3 = 4 = 5 = 6 = = + 2 = + 2 + 3 = + 2 + 3 + 4 = + 2 + 3 + 4 + 5 = + 2 + 3 + 4 + 5 + 6 = + 2 + 3 + 4 + 5 + 6 + 7 = + 2 + 3 + 4 + 5 + 6 + 7 + 8 = + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 0 = + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 0 + = + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 0 + + 2 = + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 0 + + 2 + 3 = + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 0 + + 2 + 3 + 4 = + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 0 + + 2 + 3 + 4 + 5 = + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 0 + + 2 + 3 + 4 + 5 + 6 6
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
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 = ( + 2 + + n + n + = n + ) ( + 2 + + ) n 8
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 = + 2 + 3 S 4 = + 2 + 3 + 4 S 5 = + 2 + 3 + 4 + 5.. S n+ S n 9
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 + 2 + 3 + 4 + 5 + 0
. 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)
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
4 ζ() ζ() ζ() ζ 3
5 S 6 = 6 = = + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 0 + + 2 + 3 + 4 + 5 + 6 = ( ) ( + + 2 3 + ) ( + 4 5 + 6 + 7 + ) ( + 8 9 + 0 + + 2 + 3 + 4 + 5 + ) 6 } {{ } } {{ } } {{ } } {{ } 2 4 8 ( ) ( + + 2 4 + ) ( + 4 8 + 8 + 8 + ) ( + 8 6 + 6 + 6 + 6 + 6 + 6 + 6 + ) 6 = ( ) ( ) ( ) ( ) + 2 + 4 2 + 8 4 + 6 8 = + 2 + 2 + 2 + 2 = + 4 2 S 6 S, S 2, S 4, S 8, S 6,... S + 0 2 S 2 > + 2 S 4 > + 2 2 S 8 > + 3 2 S 6 > + 4 2. 4
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
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
a = = = 2 = 2 + 2 2 + + 2 n = 2 n (?) 2 < 2 = 2 2 + 2 2 + + 2 n = 2 n 2 (?) 2 n = = 2 2 = 2 a = = = 2 a = 2 + 2 2 + + 2 n ( = 2 0 + 2 + 2 2 + + ) 2 n 2 0 2 0 = 2 n+ 2 < 2n+ 2 = n = a = n = n 2 0 M < n 2 M n a = 2 2 2. R, N, N a > 0 M R n N M < = a 7
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
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
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
H(x) = x = h(x) = log e (x + ) H(x) = x x, 2, = x y = x 3,... y = x y 2 3 4 5 6 0 2 3 4 5 6 x H(n) =, 2, 3,..., n y = x + n dx + ) x 2
y = x+ dx n 0 x+ y 2 3 4 5 6 0 2 3 4 5 6 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
y = x y = H(x) y = H(x) y = log e (x + ) H(x) log e (x + ) 2.4500 y (0.504) 2.2833 (0.496) 2.0833 (0.4739).8333 (0.4470).5000 (0.404).0000 (0.3069) 0 2 3 4 5 6 x 0.3069, 0.404, 0.4470, 0.4739, 0.496, 0.504,... x =, 2, 3,..., 6 23
x =, 00, 200,..., 600 y 9.0947 9.2770 8.876 8.584 (0.577) (0.577) (0.577) 8.789 (0.5770) (0.5770) 7.4865 (0.5767).0000 (0.3069) 00 200 300 400 500 x 600 0.3069, 0.5767, 0.5770, 0.5770, 0.577, 0.577, 0.577 24
x =, 000, 2000,..., 6000 y.739.3970.5793 0.8862 0.4808 (0.5772) (0.5772) (0.5772) (0.5772) 9.7877 (0.5772) (0.5772).0000 (0.3069) 000 2000 3000 4000 5000 x 6000 00 0.5772 x H(x) log e x + 0.5772 n H n log e n H(x) h(x) x H(x) = = h(x) = log e x 25
8 H = = 2 4 8 2 m 26
9 ( 2 0 + 2 + 2 2 + ) (3 0 + 3 + 3 2 + ) 27
2 0 3 0 + 2 0 3 + 2 3 0 + 2 0 3 2 + 2 3 + 2 2 3 0 + ( 2 0 3 0) + ( 2 0 3 + 2 3 0) + ( 2 0 3 2 + 2 3 + 2 2 3 0) + 2 3 n n=0 =0 n=0 n 0,, 2,... n =0 2 3 n 2 3 2 3 2 3 0 2 Q 2 Q 2 = ( 2 0 + 2 + ) ( 2 2 + 3 0 + 3 + ) 3 2 + 2 ( Q 2 = 2 0 + 2 + ) ( 2 2 + 3 0 + 3 + ) 3 2 + ( ) ( ) = 2 3 Q 2 2 3 n 28
( Q 2 = 2 0 + 2 + ) ( 2 2 + 3 0 + 3 + ) 3 2 + ( ) ( = 2 0 3 } {{ 0 + 2 } 0 3 + ) ( 2 3 } {{ 0 + 2 } 0 3 2 + 2 3 + ) 2 2 3 0 + } {{ } 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 2 3 5 7 0 2 3 2 3 5 7 0 2 3 5 7 M 2 M 29
M p, p 2,..., p,..., p } {{ M p } M 3 p = 2, p 2 = 3, p 3 = 5 Q M ( Q M = 2 0 + 2 + 2 2 + M = = = M = ) ( ( p 0 + p + p 2 + p 3 0 + 3 + 3 2 + ) ) ( 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 = 2 0 + 2 + 2 2 + M = = = n=0 ) ( ( p 0 + p + p 2 + 2 r 3 r 25 r 3 p r MM 3 0 + 3 + 3 2 + ) ) ( p 0 + M p + ) M p 2 + M r + r 2 + + r M = n r, r 2,..., r M n Q M p r n 30
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 = + 2 + 3 + 4 + Q M Q M Q M = M = p Q M = Q M = = M = p = = Quod Erat Demonstrandum. 3
8 0 32
33
6 n Euler-Mascheroni γ [4] H n = log e n + γ + n 2n 2 + 20n 4 ɛ, 0 < ɛ < 256n 6, γ = 0.577256649 s = p p s = PDF URL http://www.hyui.com/girl/harmonic.html Copyright (C) 2006 Hiroshi Yui ( ) All rights reserved. 2006 2006 4 [6] 34
(partial sum) (infinite series) (harmonic number) (harmonic series) a = a + a 2 + a 3 + + a n = a = lim = = = (Riemann s zeta function) ζ(s) = n! = a n = a + a 2 + a 3 + = + 2 + 3 + + n = + 2 + 3 + = = s ζ() = = ζ(s) = p p s For all n in N... n N... There exists n in N such that... n N... 35
[],, ISBN4-254-73-0, 2005. 4 (Nicole d Oresme, 323? 382) ζ(σ) [2],, http://www.gaushuin.ac.jp/~8879/mathboo/ PDF 2 [3], ISBN4-00-0079-5, 994. (p. 32 34) (p. 34 36) [4] Donald E. Knuth, The Art of Computer Programming Volume ISBN4-756-44-X,, 2004..2.7. n H n 7 (p. 75) [5] http://scienceworld.wolfram.com/biography/euler.html (Leonhard Euler, 707 783) 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,,,, ISBN4-8269- 0025-2,, 985. 20 ISBN4-8269-025-9 (2005 ) 20 [7],,, ISBN4-7973-2973-4, 2005. :-) [8],, http://www.hyui.com/girl/ 36
. (2004 ) 2. (2005 ) 3. (2005 ) 4. (2005 ) 5. (2006 ) 6. (2006 ) 37