计计计
硂琌よ猭靡计 (Prime Number) Τ礚и辨硓筁
眖τ眔计计礚
и┮芠代癸禜Τ计计 (Reciprocal) 舱Θ计
1/2 + 1/3 + 1/5 + 1/7 + ...... (P)
иσ納 2 计
1 + 1/2 + 1/3 + 1/4 + 1/5 + ...... (A)
1 + 1/2 + 1/4 + 1/8 + 1/16 + ..... (B)
陪瞷τ计 A 琌祇床 (Divergent) 繦 A 兜计糤ㄤ羆㎝穦镣礚絘计 B 琌Μ滥 (Convergent)Τ絘计ㄏ繦 B 兜计糤ㄤ羆㎝穦镣赣计
眖碭计 (Geometric Series) 瞶阶いи眔璝ㄤそゑ荡癸ぶ 1 玥赣计Μ滥瞷计 B そゑ 1/2 玥ㄤΜ滥Μ滥
1 + 1/2 + 1/4 + 1/8 + 1/16 + ..... 1/(2n-1) = [1 - (1/2)n] / (1 - 1/2) = 2 - 1/(2n-1)
讽 n 禫 B 玥禫亥 2
A
ир A だΘ计琿ㄓ
1 + 1/2 > 1/2 + 1/2 = 1
1/3 + 1/4 > 1/4 + 1/4 = 1/2
1/5 + 1/6 + 1/7 + 1/8 > 1/8 + 1/8 + 1/8 + 1/8 = 1/2
....................
1/(2n-1 + 1) + 1/(2n-1 + 2) + ...... + 1/(2n) > 1/(2n) + 1/(2n) + ...... + 1/(2n) = 1/2
A 兜计糤ゑ挡狦ぃ耞糤 1/2 硂妓и倒﹚ヴ计硂计禬筁硂计讽礛琌祇床
ир计 B Θ计 A い┾ǐㄇ兜璶┾ǐì镑兜硂计獽パ祇床跑ΘΜ滥瞶パ计计舱Θ计 P 砆跌计 A い┾ǐㄇ兜拜肈琌ì镑籔
瞷и安﹚ P Μ滥 S穦祇ネ薄猵
1/2 + 1/3 + 1/5 + ..... 1/pn-1 ぃ S - 1/2
1/2 + 1/3 + 1/5 + ..... 1/pn-1 + 1/pn > S - 1/2
ê或緇兜
1/pn+1 + 1/pn+2 + ...... < 1/2
ぃ礛硂羆㎝獽穦 S
礛иσ納材 k 计ぇрウ患糤抖逼
pk+1 , pk+2 , pk+3 , ......
N(x) ボ┪单 x タ俱计いぃ砆计俱埃计
k = 4 Τ计栋 (Set of Primes){11, 13, 17, 19, ..... }
N(10) = 10 N(15) = 13 N(27) = 20
1 10 い⊿Τタ俱计砆 11 单计俱埃τ 15 玥Τ 11 ㎝ 13 ㄢ计 27 玥Τ 111317192223 ㎝ 26 计砆 11 单计俱埃珿眔ㄒ挡狦
и安﹚ k 计 N(x) 砆絋﹚
ㄓヴ计 y 糶Θ
y = w2v
Αい v 琌礚キよ计 (Squarefree Number) v い⊿Τ滦 w ┪ v 单 1
ê或璶ㄏ y 璸衡 N(x) ず玥ゲ斗才
yぃ x
y ┮Τゲ斗玡
k 计
璶骸ì材兵ン w2v ぃ x w x キよ璶骸ì材兵ン v 琌玡 k 计 v 琌礚キよ (Squarefree) ㄤい玡 k 计计 (Index) 0 ┪ 1 珿 v 计程眔 2k 眔
N(x) ぃ 2k * sqrt(x)
獽呼︽ゅ sqrt(x) 蠢 x キよ
x - N(x) 砆材 k 计ぇ计俱埃计硂计ぃ穦 x ず计计 (Multiple) 计羆㎝讽いΤウそ计眔
N(x) - x ぃ x/pk+1 + x/pk+2 + ......
材 n 计计㎝ (Sum of Reciprocals) 1/2 璝 n = k 玥Τ
N(x) - x < x/2
x/2 < N(x)
侯眔
x/2 < N(x) ぃ 2n * sqrt(x)
讽 x = 22n+2
22n+1 < N(x) ぃ 22n+1
τ玻ネベ珿眔计计舱Θ计 P 琌祇床璝计计Τ赣计羆㎝穦Τ瞷计琌祇床计计琌礚絘ぇ计