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  1. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ♦r✐❣✐♥❛❧ ❤②♣♦t❤❡s✐s ❩❡t❛ ❢✉♥❝t✐♦♥s ❢♦r ❣r❛♣❤s ▼♦r❡ ③❡t❛s ◆✉♠❜❡r t❤❡♦r② ❛❣❛✐♥ ❈♦♥❝❧✉s✐♦♥ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s❡s ❆❧❡① ❏✳ ❇❡st ❲▼❙ ❚❛❧❦s ✹✴✷✴✷✵✶✹

  2. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ♦r✐❣✐♥❛❧ ❤②♣♦t❤❡s✐s ❩❡t❛ ❢✉♥❝t✐♦♥s ❢♦r ❣r❛♣❤s ▼♦r❡ ③❡t❛s ◆✉♠❜❡r t❤❡♦r② ❛❣❛✐♥ ❈♦♥❝❧✉s✐♦♥ ■♥ t❤✐s t❛❧❦✿ ■♥tr♦❞✉❝t✐♦♥ ✶ ❚❤❡ ♦r✐❣✐♥❛❧ ❤②♣♦t❤❡s✐s ✷ ❩❡t❛ ❢✉♥❝t✐♦♥s ❢♦r ❣r❛♣❤s ✸ ▼♦r❡ ❛ss♦rt❡❞ ③❡t❛s ✹ ❇❛❝❦ t♦ ♥✉♠❜❡r t❤❡♦r② ✺ ❈♦♥❝❧✉s✐♦♥ ✻

  3. ✷ ❦ ✐♥ ❍❡ ❛❧s♦ ❞✐s❝♦✈❡r❡❞ ♠♦r❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛❡ ❢♦r ✶ ♥ ♥ t❡r♠s ♦❢ t❤❡ ❇❡r♥♦✉❧❧✐ ♥✉♠❜❡rs ❇ ✷ ❦ ❢♦r ❛❧❧ ♥❛t✉r❛❧ ❦ ✳ ■♥ ❢❛❝t✱ ❛ ♥✐❝❡ ❢♦r♠ ❢♦r ✷ ❦ ✶ ♥ ✶ ♥ ✐s st✐❧❧ ✉♥❦♥♦✇♥ t♦❞❛②✳ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ♦r✐❣✐♥❛❧ ❤②♣♦t❤❡s✐s ❩❡t❛ ❢✉♥❝t✐♦♥s ❢♦r ❣r❛♣❤s ▼♦r❡ ③❡t❛s ◆✉♠❜❡r t❤❡♦r② ❛❣❛✐♥ ❈♦♥❝❧✉s✐♦♥ ❚❤❡ ❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥✿ ❊✉❧❡r✬s ✇♦r❦ ❆ ❜r✐❡❢ ❤✐st♦r② ♦❢ ζ ✿ ■♥ ✶✼✸✺ ❊✉❧❡r s♦❧✈❡s t❤❡ ❇❛s❡❧ ♣r♦❜❧❡♠ ❜② ✜♥❞✐♥❣ t❤❛t ∞ ♥ ✷ = π ✷ ✶ � ✻ . ♥ = ✶

  4. ■♥ ❢❛❝t✱ ❛ ♥✐❝❡ ❢♦r♠ ❢♦r ✷ ❦ ✶ ♥ ✶ ♥ ✐s st✐❧❧ ✉♥❦♥♦✇♥ t♦❞❛②✳ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ♦r✐❣✐♥❛❧ ❤②♣♦t❤❡s✐s ❩❡t❛ ❢✉♥❝t✐♦♥s ❢♦r ❣r❛♣❤s ▼♦r❡ ③❡t❛s ◆✉♠❜❡r t❤❡♦r② ❛❣❛✐♥ ❈♦♥❝❧✉s✐♦♥ ❚❤❡ ❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥✿ ❊✉❧❡r✬s ✇♦r❦ ❆ ❜r✐❡❢ ❤✐st♦r② ♦❢ ζ ✿ ■♥ ✶✼✸✺ ❊✉❧❡r s♦❧✈❡s t❤❡ ❇❛s❡❧ ♣r♦❜❧❡♠ ❜② ✜♥❞✐♥❣ t❤❛t ∞ ♥ ✷ = π ✷ ✶ � ✻ . ♥ = ✶ ♥ = ✶ ♥ − ✷ ❦ ✐♥ ❍❡ ❛❧s♦ ❞✐s❝♦✈❡r❡❞ ♠♦r❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛❡ ❢♦r � ∞ t❡r♠s ♦❢ t❤❡ ❇❡r♥♦✉❧❧✐ ♥✉♠❜❡rs ❇ ✷ ❦ ❢♦r ❛❧❧ ♥❛t✉r❛❧ ❦ ✳

  5. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ♦r✐❣✐♥❛❧ ❤②♣♦t❤❡s✐s ❩❡t❛ ❢✉♥❝t✐♦♥s ❢♦r ❣r❛♣❤s ▼♦r❡ ③❡t❛s ◆✉♠❜❡r t❤❡♦r② ❛❣❛✐♥ ❈♦♥❝❧✉s✐♦♥ ❚❤❡ ❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥✿ ❊✉❧❡r✬s ✇♦r❦ ❆ ❜r✐❡❢ ❤✐st♦r② ♦❢ ζ ✿ ■♥ ✶✼✸✺ ❊✉❧❡r s♦❧✈❡s t❤❡ ❇❛s❡❧ ♣r♦❜❧❡♠ ❜② ✜♥❞✐♥❣ t❤❛t ∞ ♥ ✷ = π ✷ ✶ � ✻ . ♥ = ✶ ♥ = ✶ ♥ − ✷ ❦ ✐♥ ❍❡ ❛❧s♦ ❞✐s❝♦✈❡r❡❞ ♠♦r❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛❡ ❢♦r � ∞ t❡r♠s ♦❢ t❤❡ ❇❡r♥♦✉❧❧✐ ♥✉♠❜❡rs ❇ ✷ ❦ ❢♦r ❛❧❧ ♥❛t✉r❛❧ ❦ ✳ ■♥ ❢❛❝t✱ ❛ ♥✐❝❡ ❢♦r♠ ❢♦r ∞ � ♥ − ✷ ❦ − ✶ , ♥ = ✶ ✐s st✐❧❧ ✉♥❦♥♦✇♥ t♦❞❛②✳

  6. ■♥ t❤❡ ♣❛♣❡r ❤❡ ❝♦♥s✐❞❡rs ✶ s ♥ s ♥ ✶ ❍❡r❡ t❤❡ ♥♦t❛t✐♦♥ ❢♦r t❤✐s ❢✉♥❝t✐♦♥ ✐s ✉s❡❞ ❢♦r t❤❡ ✜rst t✐♠❡✳ ❆❧♦♥❣ t❤❡ ✇❛② ❤❡ ✭❡ss❡♥t✐❛❧❧②✮ ♠❛❦❡s ❢♦✉r ❤②♣♦t❤❡s❡s✳ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ♦r✐❣✐♥❛❧ ❤②♣♦t❤❡s✐s ❩❡t❛ ❢✉♥❝t✐♦♥s ❢♦r ❣r❛♣❤s ▼♦r❡ ③❡t❛s ◆✉♠❜❡r t❤❡♦r② ❛❣❛✐♥ ❈♦♥❝❧✉s✐♦♥ ❚❤❡ ❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥✿ ❆❧♦♥❣ ❝♦♠❡s ❘✐❡♠❛♥♥ ❆ ❜r✐❡❢ ❤✐st♦r② ♦❢ ζ ✿ ■♥ ✶✽✺✾ ❇❡r♥❤❛r❞ ❘✐❡♠❛♥♥✱ ❛ ✇❡❧❧ ❦♥♦✇♥ ❛♥❛❧②st✱ ♣✉❜❧✐s❤❡s ❛ ♣❛♣❡r ♦♥ ❤✐s ✇♦r❦ ❝♦✉♥t✐♥❣ t❤❡ ♣r✐♠❡s ✉s✐♥❣ ♠❡t❤♦❞s ❢r♦♠ ❛♥❛❧②s✐s✳

  7. ❍❡r❡ t❤❡ ♥♦t❛t✐♦♥ ❢♦r t❤✐s ❢✉♥❝t✐♦♥ ✐s ✉s❡❞ ❢♦r t❤❡ ✜rst t✐♠❡✳ ❆❧♦♥❣ t❤❡ ✇❛② ❤❡ ✭❡ss❡♥t✐❛❧❧②✮ ♠❛❦❡s ❢♦✉r ❤②♣♦t❤❡s❡s✳ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ♦r✐❣✐♥❛❧ ❤②♣♦t❤❡s✐s ❩❡t❛ ❢✉♥❝t✐♦♥s ❢♦r ❣r❛♣❤s ▼♦r❡ ③❡t❛s ◆✉♠❜❡r t❤❡♦r② ❛❣❛✐♥ ❈♦♥❝❧✉s✐♦♥ ❚❤❡ ❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥✿ ❆❧♦♥❣ ❝♦♠❡s ❘✐❡♠❛♥♥ ❆ ❜r✐❡❢ ❤✐st♦r② ♦❢ ζ ✿ ■♥ ✶✽✺✾ ❇❡r♥❤❛r❞ ❘✐❡♠❛♥♥✱ ❛ ✇❡❧❧ ❦♥♦✇♥ ❛♥❛❧②st✱ ♣✉❜❧✐s❤❡s ❛ ♣❛♣❡r ♦♥ ❤✐s ✇♦r❦ ❝♦✉♥t✐♥❣ t❤❡ ♣r✐♠❡s ✉s✐♥❣ ♠❡t❤♦❞s ❢r♦♠ ❛♥❛❧②s✐s✳ ■♥ t❤❡ ♣❛♣❡r ❤❡ ❝♦♥s✐❞❡rs ∞ ✶ � ζ ( s ) = ♥ s . ♥ = ✶

  8. ❆❧♦♥❣ t❤❡ ✇❛② ❤❡ ✭❡ss❡♥t✐❛❧❧②✮ ♠❛❦❡s ❢♦✉r ❤②♣♦t❤❡s❡s✳ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ♦r✐❣✐♥❛❧ ❤②♣♦t❤❡s✐s ❩❡t❛ ❢✉♥❝t✐♦♥s ❢♦r ❣r❛♣❤s ▼♦r❡ ③❡t❛s ◆✉♠❜❡r t❤❡♦r② ❛❣❛✐♥ ❈♦♥❝❧✉s✐♦♥ ❚❤❡ ❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥✿ ❆❧♦♥❣ ❝♦♠❡s ❘✐❡♠❛♥♥ ❆ ❜r✐❡❢ ❤✐st♦r② ♦❢ ζ ✿ ■♥ ✶✽✺✾ ❇❡r♥❤❛r❞ ❘✐❡♠❛♥♥✱ ❛ ✇❡❧❧ ❦♥♦✇♥ ❛♥❛❧②st✱ ♣✉❜❧✐s❤❡s ❛ ♣❛♣❡r ♦♥ ❤✐s ✇♦r❦ ❝♦✉♥t✐♥❣ t❤❡ ♣r✐♠❡s ✉s✐♥❣ ♠❡t❤♦❞s ❢r♦♠ ❛♥❛❧②s✐s✳ ■♥ t❤❡ ♣❛♣❡r ❤❡ ❝♦♥s✐❞❡rs ∞ ✶ � ζ ( s ) = ♥ s . ♥ = ✶ ❍❡r❡ t❤❡ ♥♦t❛t✐♦♥ ζ ❢♦r t❤✐s ❢✉♥❝t✐♦♥ ✐s ✉s❡❞ ❢♦r t❤❡ ✜rst t✐♠❡✳

  9. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ♦r✐❣✐♥❛❧ ❤②♣♦t❤❡s✐s ❩❡t❛ ❢✉♥❝t✐♦♥s ❢♦r ❣r❛♣❤s ▼♦r❡ ③❡t❛s ◆✉♠❜❡r t❤❡♦r② ❛❣❛✐♥ ❈♦♥❝❧✉s✐♦♥ ❚❤❡ ❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥✿ ❆❧♦♥❣ ❝♦♠❡s ❘✐❡♠❛♥♥ ❆ ❜r✐❡❢ ❤✐st♦r② ♦❢ ζ ✿ ■♥ ✶✽✺✾ ❇❡r♥❤❛r❞ ❘✐❡♠❛♥♥✱ ❛ ✇❡❧❧ ❦♥♦✇♥ ❛♥❛❧②st✱ ♣✉❜❧✐s❤❡s ❛ ♣❛♣❡r ♦♥ ❤✐s ✇♦r❦ ❝♦✉♥t✐♥❣ t❤❡ ♣r✐♠❡s ✉s✐♥❣ ♠❡t❤♦❞s ❢r♦♠ ❛♥❛❧②s✐s✳ ■♥ t❤❡ ♣❛♣❡r ❤❡ ❝♦♥s✐❞❡rs ∞ ✶ � ζ ( s ) = ♥ s . ♥ = ✶ ❍❡r❡ t❤❡ ♥♦t❛t✐♦♥ ζ ❢♦r t❤✐s ❢✉♥❝t✐♦♥ ✐s ✉s❡❞ ❢♦r t❤❡ ✜rst t✐♠❡✳ ❆❧♦♥❣ t❤❡ ✇❛② ❤❡ ✭❡ss❡♥t✐❛❧❧②✮ ♠❛❦❡s ❢♦✉r ❤②♣♦t❤❡s❡s✳

  10. ■♥ ❤✐s ♣❛♣❡r ❘✐❡♠❛♥♥ ✉s❡s t❤✐s t♦ t❛❦❡ t❤❡ ❢✉♥❝t✐♦♥ ✐t ✶ ❛♥❞ ❡①t❡♥❞ ✐t t♦ ❛❧❧ ♦❢ ✳ ❍❡ ❞❡✜♥❡s ❛♥ ❛♥❛❧②t✐❝ ❝♦♥t✐♥✉❛t✐♦♥ ❢r♦♠ ✇❤✐❝❤ ♠❛t❝❤❡s t❤❡ s❡r✐❡s ❞❡✜♥✐t✐♦♥ ❣✐✈❡♥ ❛❜♦✈❡ ✇❤❡♥ t❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s ✭✇❤❡♥ ❘❡ s ✶✮✳ ❘✐❡♠❛♥♥ ❛❧s♦ ❞✐s❝♦✈❡rs ❛ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ③❡t❛ ❢✉♥❝t✐♦♥ ❜② s❤♦✇✐♥❣ t❤❛t ✶ s s s ✷ ✶ s ✷ s ✶ s ✷ ✷ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ♦r✐❣✐♥❛❧ ❤②♣♦t❤❡s✐s ❩❡t❛ ❢✉♥❝t✐♦♥s ❢♦r ❣r❛♣❤s ▼♦r❡ ③❡t❛s ◆✉♠❜❡r t❤❡♦r② ❛❣❛✐♥ ❈♦♥❝❧✉s✐♦♥ ❚❤❡ ❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥✿ ❲❤❛t ❘✐❡♠❛♥♥ ❞✐❞ ■♥ ❤✐s ✇♦r❦ ❊✉❧❡r ❤❛❞ ✭♠♦r❡ ♦r ❧❡ss✮ ❢♦✉♥❞ t❤❛t ∞ ✶ ✶ � � ζ ( s ) = ♥ s = ✶ − ♣ s . ♥ = ✶ ♣ ♣r✐♠❡

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