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[]ǿʥ10κͽۡ른ɥͽۡ1896ǯ˥ޡȥɡ顦Хס󤬤ۤȤƱΩ˾
\pi(x)~\sim~{\Bigint}_{2}^{x}~\frac{dx}{\log{x}}
(x)xʲǿθĿ¨(x)=def#{pǿ|px}Ȥ롣

Ȥΰ̣ϡxΤȤ˺դȱդ椬1˼«ȤȤǤ롣

[]xΤȤ\frac{\pi(x)}{{\Bigint}_{2}^{x}~\frac{dx}{\log{x}}}~\to~1

xΤȤ˺Ǥ礭ʤΤȤ׹ʳιʤ뤤Ȥμ׹ܤȡǿϦ(x)μ׹ΤߤͿƤȤ롣ǿαդʬʬˤäơΤ褦ʬlog(x)μ⤯ʤηŸ[1]

\frac{x}{\log{x}}+\frac{x}{(\log{x})^2}+\frac{2x}{(\log{x})^3}+~\cdots

εμ׹\frac{x}{\log{x}}Ϥʳˤʤ롣äơǿϼΤ褦ɽǤʤĤۤɤɽ褯Ƥˡ

[]άǤǿ
\pi(x)~\sim~\frac{x}{\log{x}}

ɽɤ߼뤳ȤϡľŪx餤礭οǿǤΨ\frac{1}{\log{x}}ȤȤˤʤ롣

Ȥǡx¤ʤ礭Ȥ\frac{x}{\log{x}}ϤǤ礭ʤ롣ʤʤС\log{x}=nȤȤx=e^nǤꡢ

\frac{x}{\log{x}}~=~\frac{e^n}{n}

Ȥʤ롣x¤ʤ礭ʤȤn¤ʤ礭ʤ롣Ĥޤꡢؿؿen޷©Υԡɤn®äơ\frac{e^n}{n}ϤǤ礭ʤ[2]

f(x)=~\frac{x}{\log{x}}Ȥ[4]

ΤȤ

f'(x)
=\frac{\log{x}~-~x~\cdot~\frac{1}{x}}{(\log{x})^2}
=\frac{\log{x}~-~1}{(\log{x})^2}
\approx~\frac{1}{\log{x}}

äơx=10100ΤȤf'(10^{100})~=~0.0043429448190325175~=~4.3429448190325175~\times~10^{-3}Ǥ롣
׻ѤMathematicaΥɤϼ̤ꡣ//N϶ͤȤդ롣NϿ̣ͤ"Numerical"ƬʸȤäΤǡʸǤʤФʤ

1/Log[10^100]//N

äơx=10100նǿƱΤʿѥåפ\frac{1}{4.3429448190325175~\times~10^{-3}}~\approx~230.259~\approx~230ˤʤ롣

ˡx=10100ϤƲåǿĤ뤳ȤԤǤ뤫Ĵ٤롣2ʳϤ٤ƤǿʤΤǡĴ٤Ȥ롣

x=10100նǿƱΤʿѥåפ230ʤΤǡʿѥåפβǡ2ĤǿϤ᤿Ȳꤹȡ10100餫ǿǤʤˡ\frac{230}{4}~=57.5\approx~58Ĵ٤мǿФƤ뤳ȤԤǤ롣

[ͻ]㤨С1,024ӥåȤʤпĤ1Ĥǿ¸ߤ[5]

ȤǿޤäޤޤʤǤդĹ֤ǿˡ

ʤʤС

  • (n+1)!+22dzڤ
  • (n+1)!+33dzڤ
  • (n+1)!+(n+1)n+1dzڤ

nĤδ֤ˤǿ1ĤäƤʤǤ롣

ǿξ

  • ǿϥ꡼ޥ󡦥ؿʦ(s)=0Ȥʤsˤ򸦵椹뤳Ȥ줿

ǿ줳

  • ǿθ̤꡼ޥ󡦥ؿʦ(s)=0ȤʤʣǿsΤȡˤѤɽȤǤ[1]
    • ̵¤ˤ󤢤뤳ȤΤƤ롣
    • ǿθμϦ(s)μξ¤Ĥޤꡢ\rho~=~\sup~\{~Re(s)~|~\zeta(s)~=~0~\}Ȥȡx^{\rho}η򤷤Ƥ롣
      • Ѥ̤ͤǤ롣=1/2Ǥ뤳ȤƤơ꡼ޥͽۤȤ
  • ǿ¬˻Ƥ[1]
    • Сؿμκͤ\rho~=~\sup~\{~Re(s)~+~1~|~\zeta(s)~=~0~\}Ȥȡ¬θx^{\frac{\rho~+~1}{2}}ɽ롣
    • Ѥ̤衣
    • \frac{1}{2}~\le~\rho~\le~1Ωġ
    • \rho~=~\frac{1}{2}Ȥ꡼ޥͽۤϿƤ롣
    • ǿξȰۤʤꡢѡ1/2ΤȤsͭ¸ĤʤȤƤ롣

ʸ

  • [1]ؿإߥʡ1999ǯ3ýʥʤǿǥ pp.18-21
  • [2]شؿȤϤʤpp.180-181
  • [3]ֺ¡ؤȯ6ʤ1ˡֻ䤬οǸȯŸ
  • [4]ذŹ桦桦СɤλȤߤ狼١ʥԥ󡦥ǥ奱7ϡեޡȥ顼7.4.9 pp.160
  • [5]ֵֿáץΡ