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❲♦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♦♦❢✳

π ( x ) = # { p ≤ x s✳t✳ p ✐s ♣r✐♠❡ } ✶ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ❙♦♠❡ ❝♦♥❥❡❝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 , ✳ ✳ ✳ ✳ ✳ ✳ 10 100 + 35737 ❛♥❞ 10 100 + 35739 , ✳ ✳ ✳ ✳ ✳ ✳ 3756801695685 · 2 666669 ± 1 , ✳ ✳ ✳✳ ❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

π ( x ) = # { p ≤ x s✳t✳ p ✐s ♣r✐♠❡ } ✷ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ❙♦♠❡ ❝♦♥❥❡❝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❡

π ( x ) = # { p ≤ x s✳t✳ p ✐s ♣r✐♠❡ } ✸ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ❙♦♠❡ ❝♦♥❥❡❝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 = 2 2 + 1 ✱ 17 = 4 2 + 1 , 37 = 6 2 + 1 ✱ 101 = 10 2 + 1 ✱ ✳ ✳ ✳ 677 = 26 2 + 1 , ✳ ✳ ✳ 10 100 + 420 · 10 50 + 42437 = (10 50 + 206) 2 + 1 ✳ ✳ ✳ ❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

π ( x ) = # { p ≤ x s✳t✳ p ✐s ♣r✐♠❡ } ✹ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ❙♦♠❡ ❝♦♥❥❡❝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❡

π ( x ) = # { p ≤ x s✳t✳ p ✐s ♣r✐♠❡ } ✺ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ❙♦♠❡ ❝♦♥❥❡❝t✉r❡s r❡❣❛r❞✐♥❣ ♣r✐♠❡ ♥✉♠❜❡rs✿ ✺✴✺ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s✳ ζ ( σ + it ) = 0 , σ ∈ (0 , 1) ⇒ σ = 1 2 ❢♦r ❡①❛♠♣❧❡✿ s 1 = 1 2 + 14 . 135 · · · i, s 2 = 1 2 + 21 . 022 · · · i, s 3 = 1 2 + 25 . 011 · · · i, s 4 = 1 2 + 30 . 425 · · · i, s 5 = 1 2 + 32 . 935 · · · i, ✳ ✳ ●❡♦r❣ ❋r✐❡❞r✐❝❤ ❇❡r♥❤❛r❞ ❘✐❡♠❛♥♥ ✳ ❇✐rt❤✿ ✶✼✳✵✾✳✶✽✷✻ ❇r❡s❡❧❡♥③ ✴ ❑ö♥✐❣r❡✐❝❤ ❍❛♥♥♦✈❡r s 126 = 1 2 + 279 . 229 · · · i, ❉❡❛t❤✿ ✷✵✳✵✼✳✶✽✻✻ ❙❡❧❛s❝❛ ✴ ■t❛❧✐❛ s 127 = 1 2 + 282 . 455 · · · i, ✳ ✳ ✳ ❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

π ( x ) = # { p ≤ x s✳t✳ p ✐s ♣r✐♠❡ } ✻ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ❚❤❡ ❡♥✉♠❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ♣r✐♠❡s ☞ Pr♦❜❧❡♠✿ r❛♣✐❞❧② ♣r♦❞✉❝❡ ♣r✐♠❡s p ≈ 10 150 ❀ ☞ ✐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❡

π ( x ) = # { p ≤ x s✳t✳ p ✐s ♣r✐♠❡ } ✼ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ❚❤❡ ❡♥✉♠❡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) = 10 5 · · · π (10 24 ) = 18435599767349200867866 . · · · ❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

π ( x ) = # { p ≤ x s✳t✳ p ✐s ♣r✐♠❡ } ✽ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s x π ( x ) ✶✵✵✵✵ ✶✷✷✾ ❚❤❡ ♣❧♦t ♦❢ π ( x ) ✶✵✵✵✵✵ ✾✺✾✷ ✶✵✵✵✵✵✵ ✼✽✹✾✽ ✶✵✵✵✵✵✵✵ ✻✻✹✺✼✾ ✶✵✵✵✵✵✵✵✵ ✺✼✻✶✹✺✺ ✶✵✵✵✵✵✵✵✵✵ ✺✵✽✹✼✺✸✹ ✶✵✵✵✵✵✵✵✵✵✵ ✹✺✺✵✺✷✺✶✶ ✶✵✵✵✵✵✵✵✵✵✵✵ ✹✶✶✽✵✺✹✽✶✸ ✶✵✵✵✵✵✵✵✵✵✵✵✵ ✸✼✻✵✼✾✶✷✵✶✽ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵ ✸✹✻✵✻✺✺✸✻✽✸✾ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✸✷✵✹✾✹✶✼✺✵✽✵✷ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✾✽✹✹✺✼✵✹✷✷✻✻✾ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✼✾✷✸✽✸✹✶✵✸✸✾✷✺ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✻✷✸✺✺✼✶✺✼✻✺✹✷✸✸ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✹✼✸✾✾✺✹✷✽✼✼✹✵✽✻✵ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✸✹✵✺✼✻✻✼✷✼✻✸✹✹✻✵✼ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✷✷✵✽✶✾✻✵✷✺✻✵✾✶✽✽✹✵ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✶✶✷✼✷✻✾✹✽✻✵✶✽✼✸✶✾✷✽ ✶✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✷✵✶✹✻✼✷✽✻✻✽✾✸✶✺✾✵✻✷✾✵ ❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

π ( x ) = # { p ≤ x s✳t✳ p ✐s ♣r✐♠❡ } ✾ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ❚❤❡ ❙❝❤♦♦❧ ♦❢ ❆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❡

π ( x ) = # { p ≤ x s✳t✳ p ✐s ♣r✐♠❡ } ✶✵ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ❚❤❡ s✐❡✈❡ t♦ ❝♦✉♥t ♣r✐♠❡s ✷✷✵❆❈ ●r❡❡❦s ✭❍❡r❛t❤♦st❡♥❡s ❢r♦♠ ❈✐r❡♥❡✮ ❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

π ( x ) = # { p ≤ x s✳t✳ p ✐s ♣r✐♠❡ } ✶✶ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ▲❡❣❡♥❞r❡✬s ■♥t✉✐t✐♦♥ ❆❞r✐❡♥✲▼❛r✐❡ ▲❡❣❡♥❞r❡ ✶✼✺✷✲✶✽✸✸ x π ( x ) ✐s ❛♣♣r♦①✐♠❛t❡❧② log x log x ✐s t❤❡ ♥❛t✉r❛❧ ❧♦❣❛r✐t❤♠ ❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡

π ( x ) = # { p ≤ x s✳t✳ p ✐s ♣r✐♠❡ } ✶✷ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ❚❤❡ ❢✉♥❝t✐♦♥ x/ log x f ( x ) = x/ log x ❯♥✐✈❡rs✐tà ❘♦♠❛ ❚r❡