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)

иσ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 

ир 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

ир计 B Θ计 A い┾ǐㄇ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┾ǐㄇ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иσ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 111317192223 ㎝ 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