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tss str s - - PowerPoint PPT Presentation

Sep 20, 2023 337 likes •744 views

rs r r t tr tss str

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SLIDE 1

❲♦r❦s❤♦♣ ✐♥ ◆✉♠❜❡r ❚❤❡♦r② ❱✐❡t♥❛♠ Ð❛ ✳✐ ❍♦ ✳ ❝ ❈ â♥ ❚❤ơ ❙❡♣t❡♠❜❡r ✺✱ ✷✵✶✺

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s✱ ❍✐st♦r② ❛♥❞ ■❞❡❛s

❆ ❝♦♥❥❡❝t✉r❡ ✭❢r♦♠ ❧❛t✐♥ ❝♦♥✐❡❝t✠ ✉r❛✱ ✈❡r❜ ❝♦♥✠ ✙❝❡r❡✱ ♦r ❛❧s♦ ✏✐♥t❡r♣r❡t✱ ✐♥❢❡r✱ ❝♦♥❝❧✉❞❡✑✮ ✐s ❛ st❛t❡♠❡♥t ♦r ❛ ❥✉❞❣♠❡♥t ❜❛s❡❞ ♦♥ ✐♥t✉✐t✐♦♥✱ ❝♦♥s✐❞❡r❡❞ ♣r♦❜❛❜❧② tr✉❡✱ ❜✉t ♥♦t ♣r♦✈❡♥✳ ❆♥ ❤②♣♦t❤❡s✐s ✭❋r♦♠ t❤❡ ❛♥❝✐❡♥t ❣r❡❡❦ ❤②♣♦t❤❡s✐s✱ ✐t ✐s ❝♦♠♣♦s❡❞ ✇✐t❤ ♦❢ ❤②♣♦✱ ✏✉♥❞❡r✑ ❛♥❞ t❤❡s✐s✱ ✏♣♦s✐t✐♦♥✑✱ ♦r ❛ss✉♠♣t✐♦♥✮ ✐s t❤❡ ♣r❡♠✐s❡ ✉♥❞❡r❧②✐♥❣ ❛ r❡❛s♦♥✐♥❣ ♦r ❛ ♣r♦♦❢✳

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SLIDE 2

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✶

❙♦♠❡ ❝♦♥❥❡❝t✉r❡s r❡❣❛r❞✐♥❣ ♣r✐♠❡ ♥✉♠❜❡rs✿ ✶✴✺

❚✇✐♥ ♣r✐♠❡s ❈♦♥❥❡❝t✉r❡✳ ❚❤❡r❡ ❡①✐sts ✐♥✜♥✐t❡❧② ♠❛♥② ♣r✐♠❡s p s✉❝❤ t❤❛t p + 2 ✐s ♣r✐♠❡ ❢♦r ❡①❛♠♣❧❡✿ 3 ❛♥❞ 5✱ 11 ❛♥❞ 13✱ 17 ❛♥❞ 19✱ 101 ❛♥❞ 103, ✳ ✳ ✳ ✳ ✳ ✳ 10100 + 35737 ❛♥❞ 10100 + 35739, ✳ ✳ ✳ ✳ ✳ ✳ 3756801695685 · 2666669 ± 1, ✳ ✳ ✳✳

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-3
SLIDE 3

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✷

❙♦♠❡ ❝♦♥❥❡❝t✉r❡s r❡❣❛r❞✐♥❣ ♣r✐♠❡ ♥✉♠❜❡rs✿ ✷✴✺

  • ♦❧❞❜❛❝❤ ❝♦♥❥❡❝t✉r❡

❊✈❡r② ❡✈❡♥ ♥✉♠❜❡r ✭❡①❝❡♣t ❢♦r 2✮ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s t❤❡ s✉♠ ♦❢ t✇♦ ♣r✐♠❡s ❢♦r ❡①❛♠♣❧❡✿ 42 = 5 + 37✱ 1000 = 71 + 929✱ 888888 = 601 + 888287, ✳ ✳ ✳

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-4
SLIDE 4

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✸

❙♦♠❡ ❝♦♥❥❡❝t✉r❡s r❡❣❛r❞✐♥❣ ♣r✐♠❡ ♥✉♠❜❡rs✿ ✸✴✺

❍❛r❞②✲▲✐tt❧❡✇♦♦❞ ❈♦♥❥❡❝t✉r❡✳ ∃ ✐♥✜♥✐t❡❧② ♠❛♥② ♣r✐♠❡s p s✳t✳ p − 1 ✐s ❛ ♣r❡❢❡❝t sq✉❛r❡ ❢♦r ❡①❛♠♣❧❡✿ 5 = 22 + 1✱ 17 = 42 + 1, 37 = 62 + 1✱ 101 = 102 + 1✱ ✳ ✳ ✳ 677 = 262 + 1, ✳ ✳ ✳ 10100 + 420 · 1050 + 42437 = (1050 + 206)2 + 1 ✳ ✳ ✳

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

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SLIDE 5

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✹

❙♦♠❡ ❝♦♥❥❡❝t✉r❡s r❡❣❛r❞✐♥❣ ♣r✐♠❡ ♥✉♠❜❡rs✿ ✹✴✺

❆rt✐♥ ❈♦♥❥❡❝t✉r❡✳ ❚❤❡ ♣❡r✐♦❞ ♦❢ 1/p ❤❛s ❧❡♥❣t❤ p − 1 ❢♦r ✐♥✜♥✐t❡❧② ♠❛♥② ♣r✐♠❡s p ❢♦r ❡①❛♠♣❧❡✿

1 7 = 0.142857✱ 1 17 = 0, 0588235294117647✱ 1 19 = 0.052631578947368421,

✳ ✳ ✳1

47 =0.0212765957446808510638297872340425531914893617 · · · ♣r✐♠❡s ✇✐t❤ t❤✐s ♣r♦♣❡rt②✿ 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, . . .

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

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SLIDE 6

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✺

❙♦♠❡ ❝♦♥❥❡❝t✉r❡s r❡❣❛r❞✐♥❣ ♣r✐♠❡ ♥✉♠❜❡rs✿ ✺✴✺

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s✳ ζ(σ + it) = 0, σ ∈ (0, 1) ⇒ σ = 1

2

  • ❡♦r❣ ❋r✐❡❞r✐❝❤ ❇❡r♥❤❛r❞ ❘✐❡♠❛♥♥

❇✐rt❤✿ ✶✼✳✵✾✳✶✽✷✻ ❇r❡s❡❧❡♥③ ✴ ❑ö♥✐❣r❡✐❝❤ ❍❛♥♥♦✈❡r ❉❡❛t❤✿ ✷✵✳✵✼✳✶✽✻✻ ❙❡❧❛s❝❛ ✴ ■t❛❧✐❛

❢♦r ❡①❛♠♣❧❡✿ s1 = 1

2 + 14.135 · · · i,

s2 = 1

2 + 21.022 · · · i,

s3 = 1

2 + 25.011 · · · i,

s4 = 1

2 + 30.425 · · · i,

s5 = 1

2 + 32.935 · · · i,

✳ ✳ ✳ s126 = 1

2 + 279.229 · · · i,

s127 = 1

2 + 282.455 · · · i,

✳ ✳ ✳

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-7
SLIDE 7

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✻

❚❤❡ ❡♥✉♠❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ♣r✐♠❡s

☞ Pr♦❜❧❡♠✿ r❛♣✐❞❧② ♣r♦❞✉❝❡ ♣r✐♠❡s p ≈ 10150❀ ☞ ✐t ✐s ❝r✉❝✐❛❧ t♦ ✉♥❞❡rst❛♥❞ ❤♦✇ ♣r✐♠❡s ❛r❡ ❞✐str✐❜✉t❡❞❀ ☞ π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡}❀ ☞ ❚❤❛t ✐s π(x) ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♣r✐♠❡ ♥✉♠❜❡rs ✉♣ t♦ x❀ ☞ ❊①❛♠♣❧❡s✿ π(10) = 4 π(100) = 25 π(1, 000) = 168

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-8
SLIDE 8

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✼

❚❤❡ ❡♥✉♠❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ♣r✐♠❡s

✞ ✝ ☎ ✆

π(x) = #{p ≤ x s✉❝❤ t❤❛t p ✐s ♣r✐♠❡}

❚❤❛t ✐s π(x) ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♣r✐♠❡ ♥✉♠❜❡rs ✉♣ t♦ x ❢♦r ❡①❛♠♣❧❡✿ π(10) = 4✱ π(100) = 25✱ π(1, 000) = 168 · · · π(104729) = 105 · · · π(1024) = 18435599767349200867866. · · ·

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-9
SLIDE 9

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✽ x π(x) ✶✵✵✵✵ ✶✷✷✾ ✶✵✵✵✵✵ ✾✺✾✷ ✶✵✵✵✵✵✵ ✼✽✹✾✽ ✶✵✵✵✵✵✵✵ ✻✻✹✺✼✾ ✶✵✵✵✵✵✵✵✵ ✺✼✻✶✹✺✺ ✶✵✵✵✵✵✵✵✵✵ ✺✵✽✹✼✺✸✹ ✶✵✵✵✵✵✵✵✵✵✵ ✹✺✺✵✺✷✺✶✶ ✶✵✵✵✵✵✵✵✵✵✵✵ ✹✶✶✽✵✺✹✽✶✸ ✶✵✵✵✵✵✵✵✵✵✵✵✵ ✸✼✻✵✼✾✶✷✵✶✽ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵ ✸✹✻✵✻✺✺✸✻✽✸✾ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✸✷✵✹✾✹✶✼✺✵✽✵✷ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✾✽✹✹✺✼✵✹✷✷✻✻✾ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✼✾✷✸✽✸✹✶✵✸✸✾✷✺ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✻✷✸✺✺✼✶✺✼✻✺✹✷✸✸ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✹✼✸✾✾✺✹✷✽✼✼✹✵✽✻✵ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✸✹✵✺✼✻✻✼✷✼✻✸✹✹✻✵✼ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✷✷✵✽✶✾✻✵✷✺✻✵✾✶✽✽✹✵ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✶✶✷✼✷✻✾✹✽✻✵✶✽✼✸✶✾✷✽ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✵✶✹✻✼✷✽✻✻✽✾✸✶✺✾✵✻✷✾✵

❚❤❡ ♣❧♦t ♦❢ π(x)

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-10
SLIDE 10

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✾

❚❤❡ ❙❝❤♦♦❧ ♦❢ ❆t❤❡♥s ✭❘❛✛❛❡❧❧♦ ❙❛♥③✐♦✮ ❊✉❝❧✐❞ ♦❢ ❆❧❡ss❛♥❞r✐❛

❇✐rt❤✿ ✸✷✺ ❇✳❈✳ ✭❛♣♣r♦①✐♠❛t❡❧②✮ ❉❡❛t❤✿ ✷✻✺ ❇✳❈✳ ✭❛♣♣r♦①✐♠❛t❡❧②✮

❚❤❡r❡ ❡①✐sts ✐♥✜♥✐t❡❧② ♠❛♥② ♣r✐♠❡s✿ π(x) → ∞ ✐❢ x → ∞

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-11
SLIDE 11

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✶✵

❚❤❡ s✐❡✈❡ t♦ ❝♦✉♥t ♣r✐♠❡s

✷✷✵❆❈ ●r❡❡❦s ✭❍❡r❛t❤♦st❡♥❡s ❢r♦♠ ❈✐r❡♥❡✮

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-12
SLIDE 12

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✶✶

▲❡❣❡♥❞r❡✬s ■♥t✉✐t✐♦♥

❆❞r✐❡♥✲▼❛r✐❡ ▲❡❣❡♥❞r❡ ✶✼✺✷✲✶✽✸✸

π(x) ✐s ❛♣♣r♦①✐♠❛t❡❧② x log x

log x ✐s t❤❡ ♥❛t✉r❛❧ ❧♦❣❛r✐t❤♠

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-13
SLIDE 13

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✶✷

❚❤❡ ❢✉♥❝t✐♦♥ x/ log x

f(x) = x/ log x

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-14
SLIDE 14

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✶✸

π(x) ✐s ❛♣♣r♦①✐♠❛t❡❧②

x log x

t❤❛t ✐s lim

x→∞

π(x) x/ log x = 1 ❛♥❞ ✐t ✐s ✇r✐tt❡s ❛s π(x) ∼ x log x

x π(x) x log x ✶✵✵✵ ✶✻✽ ✶✹✺ ✶✵✵✵✵ ✶✷✷✾ ✶✵✽✻ ✶✵✵✵✵✵ ✾✺✾✷ ✽✻✽✻ ✶✵✵✵✵✵✵ ✼✽✹✾✽ ✼✷✸✽✷ ✶✵✵✵✵✵✵✵ ✻✻✹✺✼✾ ✻✷✵✹✷✵ ✶✵✵✵✵✵✵✵✵ ✺✼✻✶✹✺✺ ✺✹✷✽✻✽✶ ✶✵✵✵✵✵✵✵✵✵ ✺✵✽✹✼✺✸✹ ✹✽✷✺✹✾✹✷ ✶✵✵✵✵✵✵✵✵✵✵ ✹✺✺✵✺✷✺✶✶ ✹✸✹✷✾✹✹✽✷ ✶✵✵✵✵✵✵✵✵✵✵✵ ✹✶✶✽✵✺✹✽✶✸ ✸✾✹✽✶✸✶✻✺✹ ✶✵✵✵✵✵✵✵✵✵✵✵✵ ✸✼✻✵✼✾✶✷✵✶✽ ✸✻✶✾✶✷✵✻✽✷✺ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵ ✸✹✻✵✻✺✺✸✻✽✸✾ ✸✸✹✵✼✷✻✼✽✸✽✼ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✸✷✵✹✾✹✶✼✺✵✽✵✷ ✸✶✵✷✶✵✸✹✹✷✶✻✻ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✾✽✹✹✺✼✵✹✷✷✻✻✾ ✷✽✾✺✷✾✻✺✹✻✵✷✶✼ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✼✾✷✸✽✸✹✶✵✸✸✾✷✺ ✷✼✶✹✸✹✵✺✶✶✽✾✺✸✷ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✻✷✸✺✺✼✶✺✼✻✺✹✷✸✸ ✷✺✺✹✻✼✸✹✷✷✾✻✵✸✵✺ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✹✼✸✾✾✺✹✷✽✼✼✹✵✽✻✵ ✷✹✶✷✼✹✼✶✷✶✻✽✹✼✸✷✹ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✸✹✵✺✼✻✻✼✷✼✻✸✹✹✻✵✼ ✷✷✽✺✼✻✵✹✸✶✵✻✾✼✹✻✹✻ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✷✷✵✽✶✾✻✵✷✺✻✵✾✶✽✽✹✵ ✷✶✼✶✹✼✷✹✵✾✺✶✻✷✺✾✶✸✽ ❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-15
SLIDE 15

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✶✹

  • ❛✉ÿ ❈♦♥❥❡❝t✉r❡

❏♦❤❛♥♥ ❈❛r❧ ❋r✐❡❞r✐❝❤ ●❛✉ÿ✭✶✼✼✼ ✲ ✶✽✺✺✮

π(x) ∼ x du log u

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-16
SLIDE 16

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✶✺

❚❤❡ ❢✉♥❝t✐♦♥ ▲♦❣❛r✐t❤♠✐❝ ■♥t❡❣r❛❧

❲❡ s❡t li(x) = x

du log u✱ t❤❡ ❢✉♥❝t✐♦♥ ▲♦❣❛r✐t❤♠✐❝ ■♥t❡❣r❛❧✳ ❍❡r❡ ✐s t❤❡ ♣❧♦t✿

li(x)

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-17
SLIDE 17

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✶✻

▼♦r❡ r❡❝❡♥t ♣❤♦t♦ ♦❢ ●❛✉ÿ

❏♦❤❛♥♥ ❈❛r❧ ❋r✐❡❞r✐❝❤ ●❛✉ÿ✭✶✼✼✼ ✲ ✶✽✺✺✮

π(x) ∼ li(x) := x du log u

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-18
SLIDE 18

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✶✼

❚❤❡ ❢✉♥❝t✐♦♥ ✧❧♦❣❛r✐t❤♠✐❝ ✐♥t❡❣r❛❧✧ ♦❢ ●❛✉ÿ

li(x) = x du log u

x π(x) li(x) x log x ✶✵✵✵ ✶✻✽ ✶✼✽ ✶✹✺ ✶✵✵✵✵ ✶✷✷✾ ✶✷✹✻ ✶✵✽✻ ✶✵✵✵✵✵ ✾✺✾✷ ✾✻✸✵ ✽✻✽✻ ✶✵✵✵✵✵✵ ✼✽✹✾✽ ✼✽✻✷✽ ✼✷✸✽✷ ✶✵✵✵✵✵✵✵ ✻✻✹✺✼✾ ✻✻✹✾✶✽ ✻✷✵✹✷✵ ✶✵✵✵✵✵✵✵✵ ✺✼✻✶✹✺✺ ✺✼✻✷✷✵✾ ✺✹✷✽✻✽✶ ✶✵✵✵✵✵✵✵✵✵ ✺✵✽✹✼✺✸✹ ✺✵✽✹✾✷✸✺ ✹✽✷✺✹✾✹✷ ✶✵✵✵✵✵✵✵✵✵✵ ✹✺✺✵✺✷✺✶✶ ✹✺✺✵✺✺✻✶✹ ✹✸✹✷✾✹✹✽✷ ✶✵✵✵✵✵✵✵✵✵✵✵ ✹✶✶✽✵✺✹✽✶✸ ✹✶✶✽✵✻✻✹✵✶ ✸✾✹✽✶✸✶✻✺✹ ✶✵✵✵✵✵✵✵✵✵✵✵✵ ✸✼✻✵✼✾✶✷✵✶✽ ✸✼✻✵✼✾✺✵✷✽✶ ✸✻✶✾✶✷✵✻✽✷✺ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵ ✸✹✻✵✻✺✺✸✻✽✸✾ ✸✹✻✵✻✺✻✹✺✽✶✵ ✸✸✹✵✼✷✻✼✽✸✽✼ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✸✷✵✹✾✹✶✼✺✵✽✵✷ ✸✷✵✹✾✹✷✵✻✺✻✾✷ ✸✶✵✷✶✵✸✹✹✷✶✻✻ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✾✽✹✹✺✼✵✹✷✷✻✻✾ ✷✾✽✹✹✺✼✶✹✼✺✷✽✽ ✷✽✾✺✷✾✻✺✹✻✵✷✶✼ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✼✾✷✸✽✸✹✶✵✸✸✾✷✺ ✷✼✾✷✸✽✸✹✹✷✹✽✺✺✼ ✷✼✶✹✸✹✵✺✶✶✽✾✺✸✷ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✻✷✸✺✺✼✶✺✼✻✺✹✷✸✸ ✷✻✷✸✺✺✼✶✻✺✻✶✵✽✷✷ ✷✺✺✹✻✼✸✹✷✷✾✻✵✸✵✺ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✹✼✸✾✾✺✹✷✽✼✼✹✵✽✻✵ ✷✹✼✸✾✾✺✹✸✵✾✻✾✵✹✶✺ ✷✹✶✷✼✹✼✶✷✶✻✽✹✼✸✷✹ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✸✹✵✺✼✻✻✼✷✼✻✸✹✹✻✵✼ ✷✸✹✵✺✼✻✻✼✸✼✻✷✷✷✸✽✷ ✷✷✽✺✼✻✵✹✸✶✵✻✾✼✹✻✹✻ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✷✷✵✽✶✾✻✵✷✺✻✵✾✶✽✽✹✵ ✷✷✷✵✽✶✾✻✵✷✼✽✸✻✻✸✹✽✹ ✷✶✼✶✹✼✷✹✵✾✺✶✻✷✺✾✶✸✽ ❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-19
SLIDE 19

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✶✽

❚❤❡ ❢✉♥❝t✐♦♥ li(x) ✈s

x log x

✎ ✍ ☞ ✌ li(x) = x log x + x dt log2 t ∼ x log x ✈✐❛ ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-20
SLIDE 20

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✶✾

❈❤❡❜②s❤❡✈ ❈♦♥tr✐❜✉t✐♦♥

P❛❢♥✉t② ▲✈♦✈✐❝❤ ❈❤❡❜②s❤❡✈ ✶✽✷✶ ✲ ✶✽✾✹ ❈❤❡❜②s❤❡✈✬s ❚❤❡♦r❡♠s

  • 7

8 ≤ π(x)

x log x ≤ 9

8

  • lim inf

x→∞

π(x) x/ log x ≤ 1

  • lim sup

x→∞

π(x) x/ log x ≥ 1

  • ∀n✱ ∃p✱ n < p < 2n

✭❇❡rtr❛♥❞ P♦st✉❧❛t❡✮

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-21
SLIDE 21

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✷✵

  • r❡❛t ♣r♦❜❧❡♠ ♦❢ t❤❡ ❡♥❞ ♦❢ ✽✵✵✿

☞ Pr♦✈❡ t❤❡ ▲❡❣❡♥❞r❡ ✕ ●❛✉ÿ ❈♦♥❥❡❝t✉r❡ π(x) ∼ x log x ✐❢ x → ∞ ☞ t❤❛t ✐s✿

  • π(x)

x log x

− 1

  • → 0 ✐❢ x → ∞

☞ t❤❛t ✐s✿

  • π(x) −

x log x

  • ✐s ✏♠✉❝❤ s♠❛❧❧❡r✑ t❤❛♥

x log x ✐❢ x → ∞ ☞ t❤❛t ✐s✿

  • π(x) −

x log x

  • = o
  • x

log x

  • ✐❢ x → ∞

☞ t❤❛t ✐s ✭t♦ s❛② ✐t ❛t t❤❡ ●❛✉ÿ ✇❛②✮✿ |π(x) − li(x)| = o (li(x)) ✐❢ x → ∞ ❚❤✐s st❛t❡♠❡♥t ✐s ❤✐st♦r✐❝❛❧❧② r❡❢❡rr❡❞ t♦ ❛s ❚❤❡ Pr✐♠❡ ◆✉♠❜❡r ❚❤❡♦r❡♠✳

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-22
SLIDE 22

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✷✶

❘✐❡♠❛♥♥✬s ♣❛♣❡r ✶✽✺✾

✭❯❡❜❡r ❞✐❡ ❆♥③❛❤❧ ❞❡r Pr✐♠③❛❤❧❡♥ ✉♥t❡r ❡✐♥❡r ❣❡❣❡❜❡♥❡♥ ●röss❡✳✮ ▼♦♥❛ts❜❡r✐❝❤t❡ ❞❡r ❇❡r❧✐♥❡r ❆❦❛❞❡♠✐❡✱✶✽✺✾

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s✿

|π(x) − li(x)| ≪ √x log x

❘❡✈♦❧✉t✐♦♥❛r② ■❞❡❛✿ ❯s❡ t❤❡ ❢✉♥❝t✐♦♥✿

ζ(s) =

∞

  • n=1

1 ns

❛♥❞ ❝♦♠♣❧❡① ❛♥❛❧②s✐s✳

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-23
SLIDE 23

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✷✷

❙✉♠♠❡r②✿

☞ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ✭✶✽✺✾✮ ✞ ✝ ☎ ✆ |π(x) − li(x)| ≪ √x log x ☞ ❘✐❡♠❛♥♥ ❞♦❡ ♥♦t ❝♦♠♣❧❡t❡ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ Pr✐♠❡ ◆✉♠❜❡r ❚❤❡♦r❡♠ ❜✉t ❤❡ s✉❣❣❡sts t❤❡ r✐❣❤t ♣❛t❤✳ ☞ ❚❤❡ ✐❞❡❛ t♦ ✉s❡ t❤❡ ζ ❢✉♥❝t✐♦♥ ❛s ❛ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡ ❢✉♥❝t✐♦♥ ☞ ❍❛❞❛♠❛r❞ ❛♥❞ ❞❡ ❧❛ ❱❛❧❧é❡ P♦✉ss✐♥ ✭✶✽✾✼✮ ❛❞❞ t❤❡ ♠✐ss✐♥❣ ♣✐❡❝❡ t♦ ❘✐❡♠❛♥♥✬s ♣r♦❣r❛♠ ❛♥❞ ♣r♦✈❡ t❤❡ Pr✐♠❡ ◆✉♠❜❡r ❚❤❡♦r❡♠ ✞ ✝ ☎ ✆ |π(x) − li(x)| ≪ x exp

  • −√log x
  • .

☞ ❚❤❡ ✉s❡ ♦❢ ζ t♦ st✉❞② ♣r✐♠❡s ❤❛❞ ❛❧r❡❛❞② ❜❡❡♥ s✉❣❣❡st❡❞ ❜② ❊✉❧❡r✦✦ ☞ ❙❝❤♦❡♥❢❡❧❞ ✭✶✾✼✻✮✱ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ ✞ ✝ ☎ ✆ |π(x) − li(x)| <

1 8π

√x log(x) ✐❢ x ≥ 2657

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-24
SLIDE 24

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✷✸

Pr✐♠❡ ◆✉♠❜❡r ❚❤❡♦r❡♠ ✜♥❛❧❧② ♣r♦✈❡♥ ✭✶✽✾✻✮

❏❛❝q✉❡s ❙❛❧♦♠♦♥ ❍❛❞❛♠❛r❞ ✶✽✻✺ ✲ ✶✾✻✸ ❈❤❛r❧❡s ❏❡❛♥ ●✉st❛✈❡ ◆✐❝♦❧❛s ❇❛r♦♥ ❞❡ ❧❛ ❱❛❧❧é❡ P♦✉ss✐♥ ✶✽✻✻ ✲ ✶✾✻✷

✞ ✝ ☎ ✆ |π(x) − li(x)| ≪ x exp(−a

  • log x)

∃a > 0

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-25
SLIDE 25

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✷✹

❊✉❧❡r ❈♦♥tr✐❜✉t✐♦♥

▲❡♦♥❤❛r❞ ❊✉❧❡r ✭✶✼✵✼ ✲ ✶✼✽✸✮

ζ(s) =

∞

  • n=1

1 ns ✐s r❡❧❛t❡❞ t♦ ♣r✐♠❡ ♥✉♠❜❡rs

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-26
SLIDE 26

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✷✹

❊✉❧❡r ❈♦♥tr✐❜✉t✐♦♥

▲❡♦♥❤❛r❞ ❊✉❧❡r ✭✶✼✵✼ ✲ ✶✼✽✸✮

ζ(s) =

∞

  • n=1

1 ns =

  • p ♣r✐♠❡
  • 1 − 1

ps −1

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-27
SLIDE 27

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✷✺

❚❤❡ ❜❡❛✉t✐❢✉❧ ❢♦r♠✉❧❛ ♦❢ ❘✐❡♠❛♥♥

✤ ✣ ✜ ✢ ζ(s) =

∞

  • n=1

1 ns = π

s 2

1 s(s − 1) + ∞

1

  • x

s 2 −1 + x− s+1 2

∞

  • n=1

e−n2πx

  • dx

∞ e−uu

s 2 −1 du

u

❊①❡r❝✐s❡

❙❤♦✇ t❤❛t✱ ✐❢ σ, t ∈ R ❛r❡ s✉❝❤ t❤❛t            ∞

1

{x} xσ+1 cos(t log x)dx = σ (σ − 1)2 + t2 ∞

1

{x} xσ+1 sin(t log x)dx = t (σ − 1)2 + t2 ❚❤❡♥ σ = 1

2✳

✭❍❡r❡ {x} ✐s t❤❡ ❢r❛❝t✐♦♥❛❧ ♣❛rt ♦❢ x ∈ R✳✮ ❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-28
SLIDE 28

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✷✻

❊①♣❧✐❝✐t ❞✐str✐❜✉t✐♦♥ ♦❢ ♣r✐♠❡ ♥✉♠❜❡rs

❚❤❡♦r❡♠✳ ✭❘♦ss❡r ✲ ❙❝❤♦❡♥❢❡❧❞✮ ✐❢ x ≥ 67 x log x − 1/2 < π(x) < x log x − 3/2 ❍❡♥❝❡

10100 log(10100)−1/2 < π(10100) < 10100 log(10100)−3/2

43523959267026440185153109567281075805591550920049791753399377550746551916373349269826109730287059.61758148

< π(10100) <

43714220863853254827942128416877119789366015267226917261629640806806895897149988858712131777940942.89031 ❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-29
SLIDE 29

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✷✼

❚❤❡ ✜✈❡ ❝♦♥❥❡❝t✉r❡s t♦❞❛② ✕ ❛♥② ◆❡✇s❄

☞ t✇✐♥ ♣r✐♠❡s✳ ❚❤❡r❡ ❡①✐st❡s ✐♥✜♥✐t❡❧② ♠❛♥② ♣r✐♠❡s p s✉❝❤ t❤❛t p + 2 ✐s ♣r✐♠❡❀ t❤❛t ✐s ✞ ✝ ☎ ✆ lim inf

n→∞ (pn+1 − pn) = 2

p1 = 2, p2 = 3, p3 = 5, · · · , pn ✐s t❤❡ n✕t❤ ♣r✐♠❡ ☞ ❊♥r✐❝♦ ❇♦♠❜✐❡r✐ ❛♥❞ ❍❛r❛❧❞ ❉❛✈❡♥♣♦rt ✐♥ ✶✾✻✻❀

☛ ✡ ✟ ✠

lim inf

n→∞

pn+1 − pn log pn < 0.46 · · · ✐♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ✐♥✜♥✐t❡❧② ♠❛♥② n✱ (pn+1 − pn) < 0, 46 · · · log pn ✌ ❉❛♥✐❡❧ ●♦❧❞st♦♥✱ ❏á♥♦s P✐♥t③ ❛♥❞ ❈❡♠ ❨✐❧❞✐r✐♠ ✐♥ ✷✵✵✺❀

☛ ✡ ✟ ✠

lim inf

n→∞

pn+1 − pn √log pn log log pn = 0 ✌ ❨✐t❛♥❣ ❩❤❛♥❣ ♦♥ ▼❛② ✶✹✱ ✷✵✶✸❀

✞ ✝ ☎ ✆

lim inf

n→∞ (pn+1 − pn) ≤ 7 · 107

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-30
SLIDE 30

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✷✽

❩❤❛♥❣ ❈♦♥tr✐❜✉t✐♦♥

❨✐t❛♥❣ ❩❤❛♥❣ ✭❤tt♣✿✴✴❡♥✳✇✐❦✐♣❡❞✐❛✳♦r❣✴✇✐❦✐✴❨✐t❛♥❣❴❩❤❛♥❣✮

▼❛② ✶✹✱ ✷✵✶✸✿ lim inf

n→∞ (pn+1 − pn) ≤ 70.000.000

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-31
SLIDE 31

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✷✾

❚❤❡ r❛❝❡ ♦❢ t❤❡ s✉♠♠❡r ✷✵✶✸ st❛rt❡❞ ♦♥ ▼❛② ✶✹✱

❤tt♣✿✴✴♠✐❝❤❛❡❧♥✐❡❧s❡♥✳♦r❣✴♣♦❧②♠❛t❤✶✴✐♥❞❡①✳♣❤♣❄t✐t❧❡❂❚✐♠❡❧✐♥❡❴♦❢❴♣r✐♠❡❴❣❛♣❴❜♦✉♥❞s

▼❛② ✶✹✱ ✷✵✶✸✿ lim inf

n→∞ (pn+1 − pn) ≤ 70.000.000

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-32
SLIDE 32

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✸✵

❚❤❡ r❛❝❡ ♦❢ t❤❡ s✉♠♠❡r ✷✵✶✸ st❛rt❡❞ ♦♥ ▼❛② ✶✹✱

❤tt♣✿✴✴♠✐❝❤❛❡❧♥✐❡❧s❡♥✳♦r❣✴♣♦❧②♠❛t❤✶✴✐♥❞❡①✳♣❤♣❄t✐t❧❡❂❚✐♠❡❧✐♥❡❴♦❢❴♣r✐♠❡❴❣❛♣❴❜♦✉♥❞s

❏✉♥❡ ✶✺✱ ✷✵✶✸✿ lim inf

n→∞ (pn+1 − pn) ≤ 60.764

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-33
SLIDE 33

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✸✶

❚❤❡ r❛❝❡ ♦❢ t❤❡ s✉♠♠❡r ✷✵✶✸ st❛rt❡❞ ♦♥ ▼❛② ✶✹✱

❤tt♣✿✴✴♠✐❝❤❛❡❧♥✐❡❧s❡♥✳♦r❣✴♣♦❧②♠❛t❤✶✴✐♥❞❡①✳♣❤♣❄t✐t❧❡❂❚✐♠❡❧✐♥❡❴♦❢❴♣r✐♠❡❴❣❛♣❴❜♦✉♥❞s

❏✉❧② ✷✼✱ ✷✵✶✸✿ lim inf

n→∞ (pn+1 − pn) ≤ 4.680

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-34
SLIDE 34

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✸✷

❚❤❡ r❛❝❡ ♦❢ t❤❡ s✉♠♠❡r ✷✵✶✸ st❛rt❡❞ ♦♥ ▼❛② ✶✹✱

❤tt♣✿✴✴♠✐❝❤❛❡❧♥✐❡❧s❡♥✳♦r❣✴♣♦❧②♠❛t❤✶✴✐♥❞❡①✳♣❤♣❄t✐t❧❡❂❚✐♠❡❧✐♥❡❴♦❢❴♣r✐♠❡❴❣❛♣❴❜♦✉♥❞s

❏❛♥✉❛r② ✻✱ ✷✵✶✹✿ lim inf

n→∞ (pn+1 − pn) ≤ 270

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-35
SLIDE 35

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✸✸

❚❤❡ r❛❝❡ ♦❢ t❤❡ s✉♠♠❡r ✷✵✶✸ st❛rt❡❞ ♦♥ ▼❛② ✶✹✱

❤tt♣✿✴✴♠✐❝❤❛❡❧♥✐❡❧s❡♥✳♦r❣✴♣♦❧②♠❛t❤✶✴✐♥❞❡①✳♣❤♣❄t✐t❧❡❂❇♦✉♥❞❡❞❴❣❛♣s❴❜❡t✇❡❡♥❴♣r✐♠❡s

❘❛❝❡ t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ ❛ ♠♦r❡ ❣❡♥❡r❛❧ ♣r♦❜❧❡♠

Hm = ❧❡❛st ✐♥t❡❣❡r s✳t✳ ❛♠♦♥❣ n, n + 1, · · · , n + Hm t❤❡r❡ ❛r❡ m ❝♦♥s❡❝✉t✐✈❡ ♣r✐♠❡s

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-36
SLIDE 36

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✸✹

P♦❧②♠❛t❤✽ ❛♥❞ ❚❡rr② ❚❛♦

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-37
SLIDE 37

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✸✺

❚❤❡ ✜✈❡ ❝♦♥❥❡❝t✉r❡s t♦❞❛② ✕ ❛♥② ◆❡✇s❄

☞●♦❧❞❜❛❝❤ ❝♦♥❥❡❝t✉r❡ ❊✈❡r② ❡✈❡♥ ♥✉♠❜❡r ✭❡①❝❡♣t ❢♦r 2✮ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s t❤❡ s✉♠ ♦❢ t✇♦ ♣r✐♠❡s ❊q✉✐✈❛❧❡♥t ❢♦r♠✉❧❛t✐♦♥✿ ❡✈❡r② ✐♥t❡❣❡r ❣r❡❛t❡r t❤❛♥ 5 ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s t❤❡ s✉♠ ♦❢ t❤r❡❡ ♣r✐♠❡s

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-38
SLIDE 38

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✸✻

❋r♦♠ ❱✐♥♦❣r❛❞♦✈ t♦ ❍❡❧❢❣♦tt

❍❛r❛❧❞ ❍❡❧❢❣♦tt

  • ✭❱✐♥♦❣r❛❞♦✈ ✕ ✶✾✸✼✮ ❡✈❡r②

♦❞❞ ✐♥t❡❣❡r ❣r❡❛t❡r t❤❛♥ 3315 ✐s t❤❡ s✉♠ ♦❢ t❤r❡❡ ♣r✐♠❡s

  • ✭❍❡❧❢❣♦tt ✕ ✷✵✶✸✮ ❡✈❡r② ♦❞❞

✐♥t❡❣❡r ❣r❡❛t❡r t❤❛♥ ✺ ✐s t❤❡ s✉♠ ♦❢ t❤r❡❡ ♣r✐♠❡s

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

slide-39
SLIDE 39

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s

π(x) = #{p ≤ x s✳t✳ p ✐s ♣r✐♠❡} ✸✼

❍♦♦❧❡②✬s ❈♦♥tr✐❜✉t✐♦♥

❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ✐♠♣❧✐❡s ❆rt✐♥ ❈♦♥❥❡❝t✉r❡✳ ❚❤❡ ♣❡r✐♦❞ ♦❢ 1/p ❤❛s ❧❡♥❣t❤ p − 1 ❢♦r ✐♥✜♥✐t❡❧② ♠❛♥② ♣r✐♠❡s p ❢♦r ❡①❛♠♣❧❡✿

1 7 = 0.142857✱ 1 17 = 0, 0588235294117647✱ 1 19 = 0.052631578947368421,

✳ ✳ ✳1

47 =0.0212765957446808510638297872340425531914893617 · · · Pr✐♠❡s ✇✐t❤ t❤✐s ♣r♦♣❡rt②✿ 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, . . .

❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡