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P❛✉❧ ❊r❞➤s ❛♥❞ t❤❡ r✐s❡ ♦❢ st❛t✐st✐❝❛❧ t❤✐♥❦✐♥❣ ✐♥ ❡❧❡♠❡♥t❛r② ♥✉♠❜❡r t❤❡♦r②

❈❛r❧ P♦♠❡r❛♥❝❡✱ ❉❛rt♠♦✉t❤ ❈♦❧❧❡❣❡ ❜❛s❡❞ ♦♥ t❤❡ ❥♦✐♥t s✉r✈❡② ✇✐t❤ P❛✉❧ P♦❧❧❛❝❦✱ ❯♥✐✈❡rs✐t② ♦❢ ●❡♦r❣✐❛

✶

▲❡t ✉s ❜❡❣✐♥ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣✿

✷

P②t❤❛❣♦r❛s

✸

❙✉♠ ♦❢ ♣r♦♣❡r ❞✐✈✐s♦rs ▲❡t s(n) ❜❡ t❤❡ s✉♠ ♦❢ t❤❡ ♣r♦♣❡r ❞✐✈✐s♦rs ♦❢ n✿ ❋♦r ❡①❛♠♣❧❡✿

s(10) = 1 + 2 + 5 = 8, s(11) = 1, s(12) = 1 + 2 + 3 + 4 + 6 = 16.

✹

■♥ ♠♦❞❡r♥ ♥♦t❛t✐♦♥✿ s(n) = σ(n) − n✱ ✇❤❡r❡ σ(n) ✐s t❤❡ s✉♠ ♦❢ ❛❧❧ ♦❢ n✬s ♥❛t✉r❛❧ ❞✐✈✐s♦rs✳ ❚❤❡ ❢✉♥❝t✐♦♥ s(n) ✇❛s ❝♦♥s✐❞❡r❡❞ ❜② P②t❤❛❣♦r❛s✱ ❛❜♦✉t ✷✺✵✵ ②❡❛rs ❛❣♦✳

✺

P②t❤❛❣♦r❛s✿ ♥♦t✐❝❡❞ t❤❛t s(6) = 1 + 2 + 3 = 6 ✭■❢ s(n) = n✱ ✇❡ s❛② n ✐s ♣❡r❢❡❝t✳✮ ❆♥❞ ❤❡ ♥♦t✐❝❡❞ t❤❛t

s(220) = 284, s(284) = 220.

✻

■❢ s(n) = m✱ s(m) = n✱ ❛♥❞ m = n✱ ✇❡ s❛②

n, m ❛r❡ ❛♥ ❛♠✐❝❛❜❧❡ ♣❛✐r ❛♥❞ t❤❛t t❤❡②

❛r❡ ❛♠✐❝❛❜❧❡ ♥✉♠❜❡rs✳ ❙♦ ✷✷✵ ❛♥❞ ✷✽✹ ❛r❡ ❛♠✐❝❛❜❧❡ ♥✉♠❜❡rs✳

✼

■♥ ✶✾✼✻✱ ❊♥r✐❝♦ ❇♦♠❜✐❡r✐ ✇r♦t❡✿

✽

✏❚❤❡r❡ ❛r❡ ✈❡r② ♠❛♥② ♦❧❞ ♣r♦❜❧❡♠s ✐♥ ❛r✐t❤♠❡t✐❝ ✇❤♦s❡ ✐♥t❡r❡st ✐s ♣r❛❝t✐❝❛❧❧② ♥✐❧✱ ❡✳❣✳✱ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♦❞❞ ♣❡r❢❡❝t ♥✉♠❜❡rs✱ ♣r♦❜❧❡♠s ❛❜♦✉t t❤❡ ✐t❡r❛t✐♦♥ ♦❢ ♥✉♠❡r✐❝❛❧ ❢✉♥❝t✐♦♥s✱ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ✐♥✜♥✐t❡❧② ♠❛♥② ❋❡r♠❛t ♣r✐♠❡s 22n + 1✱ ❡t❝✳✑

✾

❙✐r ❋r❡❞ ❍♦②❧❡ ✇r♦t❡ ✐♥ ✶✾✻✷ t❤❛t t❤❡r❡ ✇❡r❡ t✇♦ ❞✐✣❝✉❧t ❛str♦♥♦♠✐❝❛❧ ♣r♦❜❧❡♠s ❢❛❝❡❞ ❜② t❤❡ ❛♥❝✐❡♥ts✳ ❖♥❡ ✇❛s ❛ ❣♦♦❞ ♣r♦❜❧❡♠✱ t❤❡ ♦t❤❡r ✇❛s ♥♦t s♦ ❣♦♦❞✳

✶✵

❚❤❡ ❣♦♦❞ ♣r♦❜❧❡♠✿ ❲❤② ❞♦ t❤❡ ♣❧❛♥❡ts ✇❛♥❞❡r t❤r♦✉❣❤ t❤❡ ❝♦♥st❡❧❧❛t✐♦♥s ✐♥ t❤❡ ♥✐❣❤t s❦②❄ ❚❤❡ ♥♦t✲s♦✲❣♦♦❞ ♣r♦❜❧❡♠✿ ❲❤② ✐s ✐t t❤❛t t❤❡ s✉♥ ❛♥❞ t❤❡ ♠♦♦♥ ❛r❡ t❤❡ s❛♠❡ ❛♣♣❛r❡♥t s✐③❡❄

✶✶

P❡r❢❡❝t ♥✉♠❜❡rs✱ ❛♠✐❝❛❜❧❡ ♥✉♠❜❡rs✱ ❛♥❞ s✐♠✐❧❛r t♦♣✐❝s ✇❡r❡ ✐♠♣♦rt❛♥t t♦ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❡❧❡♠❡♥t❛r② ♥✉♠❜❡r t❤❡♦r②✳ Pr♦❜❛❜✐❧✐st✐❝ ♥✉♠❜❡r t❤❡♦r② ❛❧s♦ ♦✇❡s ✐ts ✐♥s♣✐r❛t✐♦♥ t♦ s♦♠❡ ♦❢ t❤❡s❡ ❛♥❝✐❡♥t ♣r♦❜❧❡♠s✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝❤❛❧❧❡♥❣❡s ♣♦s❡❞ ❜② ♣❡r❢❡❝t ♥✉♠❜❡rs ✐♥ ♣❛rt✐❝✉❧❛r ❤❛✈❡ ❧❡❞ t♦ t❤❡ ❜❧♦ss♦♠✐♥❣ ♦❢ ♣r✐♠❛❧✐t② t❡st✐♥❣ ❛♥❞

✶✷

♠♦r❡ ❣❡♥❡r❛❧❧② ❛❧❣♦r✐t❤♠✐❝ ♥✉♠❜❡r t❤❡♦r②✳ ❆♥❞✱ ❡①♣♦♥❡♥t✐❛❧ ❞✐♦♣❤❛♥t✐♥❡ ❡q✉❛t✐♦♥s✱ s✉❝❤ ❛s t❤❡ ❈❛t❛❧❛♥ ❡q✉❛t✐♦♥ xn + 1 = yk✱ ❤❛✈❡ ❛ ❧✐♥❦ t♦ t❤❡ q✉❡st✐♦♥ ♦❢ ✇❤❡t❤❡r ♦❞❞ ♣❡r❢❡❝t ♥✉♠❜❡rs ❡①✐st✳ ❙♦✱ ♣❡r❤❛♣s ✐t ❝♦✉❧❞ ❜❡ ❛r❣✉❡❞ t❤❛t t❤❡② ❛r❡ ✏❣♦♦❞✑ ♣r♦❜❧❡♠s✱ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❍♦②❧❡✳

✶✸

❚❤♦✉❣❤ ❇♦♠❜✐❡r✐✬s ♣♦✐♥t ♦❢ ✈✐❡✇ ✐s ♥❡✈❡rt❤❡❧❡ss ✉♥❞❡rst❛♥❞❛❜❧❡✱ t❤❡s❡ ❛♥❝✐❡♥t ♣r♦❜❧❡♠s ❝♦♥t✐♥✉❡ t♦ ❢❛s❝✐♥❛t❡✳ ❆♥❞ t❤❡② ❛r❡ ❢❛s❝✐♥❛t✐♥❣ t♦ ♠♦r❡ t❤❛♥ ❥✉st ♥✉♠❜❡r t❤❡♦r✐sts✿

✶✹

❙t✳ ❆✉❣✉st✐♥❡ ✇r♦t❡ ❛❜♦✉t ♣❡r❢❡❝t ♥✉♠❜❡rs ✐♥ t❤❡ ❜✐❜❧❡✿ ✏❙✐① ✐s ❛ ♣❡r❢❡❝t ♥✉♠❜❡r ✐♥ ✐ts❡❧❢✱ ❛♥❞ ♥♦t ❜❡❝❛✉s❡ ●♦❞ ❝r❡❛t❡❞ ❛❧❧ t❤✐♥❣s ✐♥ s✐① ❞❛②s❀ r❛t❤❡r t❤❡ ❝♦♥✈❡rs❡ ✐s tr✉❡ ✖ ●♦❞ ❝r❡❛t❡❞ ❛❧❧ t❤✐♥❣s ✐♥ s✐① ❞❛②s ❜❡❝❛✉s❡ t❤❡ ♥✉♠❜❡r ✐s ♣❡r❢❡❝t✳✑

✶✺

■♥ ●❡♥❡s✐s ✐t ✐s r❡❧❛t❡❞ t❤❛t ❏❛❝♦❜ ❣❛✈❡ ❤✐s ❜r♦t❤❡r ❊s❛✉ ❛ ❧❛✈✐s❤ ❣✐❢t s♦ ❛s t♦ ✇✐♥ ❤✐s ❢r✐❡♥❞s❤✐♣✳ ❚❤❡ ❣✐❢t ✐♥❝❧✉❞❡❞ ✷✷✵ ❣♦❛ts ❛♥❞ ✷✷✵ s❤❡❡♣✳ ❆❜r❛❤❛♠ ❆③✉❧❛✐✱ ❝❛✳ ✺✵✵ ②❡❛rs ❛❣♦✿

✶✻

✏❖✉r ❛♥❝❡st♦r ❏❛❝♦❜ ♣r❡♣❛r❡❞ ❤✐s ♣r❡s❡♥t ✐♥ ❛ ✇✐s❡ ✇❛②✳ ❚❤✐s ♥✉♠❜❡r ✷✷✵ ✐s ❛ ❤✐❞❞❡♥ s❡❝r❡t✱ ❜❡✐♥❣ ♦♥❡ ♦❢ ❛ ♣❛✐r ♦❢ ♥✉♠❜❡rs s✉❝❤ t❤❛t t❤❡ ♣❛rts ♦❢ ✐t ❛r❡ ❡q✉❛❧ t♦ t❤❡ ♦t❤❡r ♦♥❡ ✷✽✹✱ ❛♥❞ ❝♦♥✈❡rs❡❧②✳ ❆♥❞ ❏❛❝♦❜ ❤❛❞ t❤✐s ✐♥ ♠✐♥❞❀ t❤✐s ❤❛s ❜❡❡♥ tr✐❡❞ ❜② t❤❡ ❛♥❝✐❡♥ts ✐♥ s❡❝✉r✐♥❣ t❤❡ ❧♦✈❡ ♦❢ ❦✐♥❣s ❛♥❞ ❞✐❣♥✐t❛r✐❡s✳✑

✶✼

■❜♥ ❑❤❛❧❞✉♥✱ ❝❛✳ ✻✵✵ ②❡❛rs ❛❣♦ ✐♥ ✏▼✉q❛❞❞✐♠❛❤✑✿ ✏P❡rs♦♥s ✇❤♦ ❤❛✈❡ ❝♦♥❝❡r♥❡❞ t❤❡♠s❡❧✈❡s ✇✐t❤ t❛❧✐s♠❛♥s ❛✣r♠ t❤❛t t❤❡ ❛♠✐❝❛❜❧❡ ♥✉♠❜❡rs ✷✷✵ ❛♥❞ ✷✽✹ ❤❛✈❡ ❛♥ ✐♥✢✉❡♥❝❡ t♦ ❡st❛❜❧✐s❤ ❛ ✉♥✐♦♥ ♦r ❝❧♦s❡ ❢r✐❡♥❞s❤✐♣ ❜❡t✇❡❡♥ t✇♦ ✐♥❞✐✈✐❞✉❛❧s✳✑

✶✽

■♥ ✏❆✐♠ ♦❢ t❤❡ ❲✐s❡✑✱ ❛ttr✐❜✉t❡❞ t♦ ❆❧✲▼❛❥r✐t✐✱ ❝❛✳ ✶✵✺✵ ②❡❛rs ❛❣♦✱ ✐t ✐s r❡♣♦rt❡❞ t❤❛t t❤❡ ❡r♦t✐❝ ❡✛❡❝t ♦❢ ❛♠✐❝❛❜❧❡ ♥✉♠❜❡rs ❤❛❞ ❜❡❡♥ ♣✉t t♦ t❤❡ t❡st ❜②✿ ✏❣✐✈✐♥❣ ❛♥② ♦♥❡ t❤❡ s♠❛❧❧❡r ♥✉♠❜❡r ✷✷✵ t♦ ❡❛t✱ ❛♥❞ ❤✐♠s❡❧❢ ❡❛t✐♥❣ t❤❡ ❧❛r❣❡r ♥✉♠❜❡r ✷✽✹✳✑

✶✾

✭❚❤✐s ✇❛s ❛ ✈❡r② ❡❛r❧② ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ♥✉♠❜❡r t❤❡♦r②✱ ❢❛r ♣r❡❞❛t✐♥❣ ♣✉❜❧✐❝✲❦❡② ❝r②♣t♦❣r❛♣❤② . . . ✮

✷✵

◆✐❝♦♠❛❝❤✉s

✷✶

◆✐❝♦♠❛❝❤✉s✱ ❝❛✳ ✶✾✵✵ ②❡❛rs ❛❣♦✿ ❆ ♥❛t✉r❛❧ ♥✉♠❜❡r n ✐s ❛❜✉♥❞❛♥t ✐❢

s(n) > n ❛♥❞ ✐s ❞❡✜❝✐❡♥t ✐❢ s(n) < n✳ ❚❤❡s❡

❤❡ ❞❡✜♥❡❞ ✐♥ ✏■♥tr♦❞✉❝t✐♦ ❆r✐t❤♠❡t✐❝❛✑ ❛♥❞ ✇❡♥t ♦♥ t♦ ❣✐✈❡ ✇❤❛t ■ ❝❛❧❧ ❤✐s ❵●♦❧❞✐❧♦❝❦s ❚❤❡♦r②✬✿ ✏ ■♥ t❤❡ ❝❛s❡ ♦❢ t♦♦ ♠✉❝❤✱ ✐s ♣r♦❞✉❝❡❞ ❡①❝❡ss✱ s✉♣❡r✢✉✐t②✱ ❡①❛❣❣❡r❛t✐♦♥s ❛♥❞ ❛❜✉s❡❀ ✐♥ t❤❡ ❝❛s❡ ♦❢ t♦♦ ❧✐tt❧❡✱ ✐s ♣r♦❞✉❝❡❞ ✇❛♥t✐♥❣✱ ❞❡❢❛✉❧ts✱ ♣r✐✈❛t✐♦♥s

✷✷

❛♥❞ ✐♥s✉✣❝✐❡♥❝✐❡s✳ ❆♥❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤♦s❡ t❤❛t ❛r❡ ❢♦✉♥❞ ❜❡t✇❡❡♥ t❤❡ t♦♦ ♠✉❝❤ ❛♥❞ t❤❡ t♦♦ ❧✐tt❧❡✱ t❤❛t ✐s ✐♥ ❡q✉❛❧✐t②✱ ✐s ♣r♦❞✉❝❡❞ ✈✐rt✉❡✱ ❥✉st ♠❡❛s✉r❡✱ ♣r♦♣r✐❡t②✱ ❜❡❛✉t② ❛♥❞ t❤✐♥❣s ♦❢ t❤❛t s♦rt ✖ ♦❢ ✇❤✐❝❤ t❤❡ ♠♦st ❡①❡♠♣❧❛r② ❢♦r♠ ✐s t❤❛t t②♣❡ ♦❢ ♥✉♠❜❡r ✇❤✐❝❤ ✐s ❝❛❧❧❡❞ ♣❡r❢❡❝t✳✑

✷✸

❙♦✱ ✇❤❛t ✐s ❛ ♠♦❞❡r♥ ♥✉♠❜❡r t❤❡♦r✐st t♦ ♠❛❦❡ ♦❢ ❛❧❧ t❤✐s❄ ❆♥s✇❡r✿ ❚❤✐♥❦ st❛t✐st✐❝❛❧❧②✳ ❊r✐❝❤ ❇❡ss❡❧✲❍❛❣❡♥✱ ✐♥ ❛ ✶✾✷✾ s✉r✈❡② ❛rt✐❝❧❡✱ ❛s❦❡❞ ✐❢ t❤❡ ❛s②♠♣t♦t✐❝ ❞❡♥s✐t② ♦❢ t❤❡ ❛❜✉♥❞❛♥t ♥✉♠❜❡rs ❡①✐st✳

✷✹

■♥ ❤✐s ✶✾✸✸ ❇❡r❧✐♥ ❞♦❝t♦r❛❧ t❤❡s✐s✱ ❋❡❧✐① ❇❡❤r❡♥❞ ♣r♦✈❡❞ t❤❛t ✐❢ t❤❡ ❞❡♥s✐t② ❡①✐sts✱ ✐t ❧✐❡s ❜❡t✇❡❡♥ ✵✳✷✹✶ ❛♥❞ ✵✳✸✶✹✳ ❆♥❞ ❧❛t❡r ✐♥ ✶✾✸✸✱ ❍❛r♦❧❞ ❉❛✈❡♥♣♦rt s❤♦✇❡❞ t❤❡ ❞❡♥s✐t② ❡①✐sts✳ ■♥ ❢❛❝t✱ t❤❡ ❞❡♥s✐t② Dσ(u) ♦❢ t❤♦s❡ n ✇✐t❤

σ(n)/n ≤ u ❡①✐sts✱ ❛♥❞ Dσ(u) ✐s ❝♦♥t✐♥✉♦✉s✳

✷✺

◆♦t❡✿ ❚❤❡ ❛❜✉♥❞❛♥t ♥✉♠❜❡rs ❤❛✈❡ ❞❡♥s✐t② 1 − Dσ(2)✳ ❆ ♥✉♠❜❡r ♦❢ ♣❡♦♣❧❡ ❤❛✈❡ ❡st✐♠❛t❡❞ t❤✐s ❞❡♥s✐t②✱ r❡❝❡♥t❧② ✇❡ ❧❡❛r♥❡❞ ✐t t♦ ✹ ❞❡❝✐♠❛❧ ♣❧❛❝❡s✿ ✵✳✷✹✼✻ . . . ✭▼✐ts✉♦ ❑♦❜❛②❛s❤✐✱ ✷✵✶✵✮✳

✷✻

❉❛✈❡♥♣♦rt str♦♥❣❧② ✉s❡❞ ❛ t❡❝❤♥✐❝❛❧ ❝r✐t❡r✐♦♥ ♦❢ ■✳ ❏✳ ❙❝❤♦❡♥❜❡r❣✱ ✇❤♦ ✐♥ ✶✾✷✽ ♣r♦✈❡❞ ❛♥ ❛♥❛❧♦❣♦✉s r❡s✉❧t ❢♦r t❤❡ ❞❡♥s✐t② ♦❢ ♥✉♠❜❡rs n ✇✐t❤ n/ϕ(n) ≤ u✳ ❍❡r❡ ϕ ✐s ❊✉❧❡r✬s ❢✉♥❝t✐♦♥✳ ❇❡❣✐♥♥✐♥❣ ❛r♦✉♥❞ ✶✾✸✺✱ P❛✉❧ ❊r❞➤s ❜❡❣❛♥ st✉❞②✐♥❣ t❤✐s s✉❜❥❡❝t✱ ❧♦♦❦✐♥❣ ❢♦r t❤❡ ❣r❡❛t t❤❡♦r❡♠ t❤❛t ✇♦✉❧❞ ✉♥✐t❡ t❤❡s❡ t❤r❡❛❞s✳

✷✼

■♥ ❛❞❞✐t✐♦♥ ❊r❞➤s ❜❡❣❛♥ ❤✐s q✉❡st ❢♦r ❛♥ ❡❧❡♠❡♥t❛r② ♠❡t❤♦❞✳ ❚❤✐s ❝✉❧♠✐♥❛t❡❞ ✐♥ t❤❡ ❊r❞➤s✕❲✐♥t♥❡r t❤❡♦r❡♠ ✐♥ ✶✾✸✾✳

✷✽

❚❤❡ ❊r❞➤s✕❲✐♥t♥❡r t❤❡♦r❡♠✿ ❋♦r ❛ ♣♦s✐t✐✈❡✲✈❛❧✉❡❞ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❛r✐t❤♠❡t✐❝ ❢✉♥❝t✐♦♥ f✱ ❧❡t g(n) = log f(n)✳ ❋♦r f t♦ ❤❛✈❡ ❛ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ✐t ✐s ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t t❤❛t

• |g(p)|>1

1 p,

• |g(p)|≤1

g(p)2 p ,

• |g(p)|≤1

g(p) p

❛❧❧ ❝♦♥✈❡r❣❡✳ ❋✉rt❤❡r✱ ✐❢

g(p)=0 1/p

❞✐✈❡r❣❡s✱ t❤❡ ❞✐str✐❜✉t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s✳

✷✾

❊①❛♠♣❧❡✿ f(n) = σ(n)/n✱ s♦ t❤❛t

g(p) = log(1 + 1

p) ≈ 1 p✳

✸✵

❙✉r❡❧② t❤✐s ❜❡❛✉t✐❢✉❧ t❤❡♦r❡♠ ❝❛♥ ❥✉st✐❢② t❤❡ ❧♦✇ ♦r✐❣✐♥s ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛❜✉♥❞❛♥t ♥✉♠❜❡rs✦ ❇✉t ✇❤❛t ♦❢ ♦t❤❡r ❢❛♠✐❧✐❛r ❛r✐t❤♠❡t✐❝ ❢✉♥❝t✐♦♥s s✉❝❤ ❛s ω(n)✱ ✇❤✐❝❤ ❝♦✉♥ts t❤❡ ♥✉♠❜❡r ♦❢ ❞✐st✐♥❝t ♣r✐♠❡s t❤❛t ❞✐✈✐❞❡ n❄ ❚❤✐s ❢✉♥❝t✐♦♥ ✐s ❛❞❞✐t✐✈❡✱ s♦ ✐t ✐s ❛❧r❡❛❞② ♣❧❛②✐♥❣ t❤❡ r♦❧❡ ♦❢ g(n)✳

✸✶

❍♦✇❡✈❡r✱ ω(p) = 1 ❢♦r ❛❧❧ ♣r✐♠❡s p✱ s♦ t❤❡ ✷♥❞ ❛♥❞ ✸r❞ s❡r✐❡s ❞✐✈❡r❣❡✳ ■t✬s ✐♥ ❤♦✇ ②♦✉ ♠❡❛s✉r❡✳ ❍❛r❞② ❛♥❞ ❘❛♠❛♥✉❥❛♥ ❤❛❞ s❤♦✇♥ t❤❛t

ω(n)/ log log n → 1 ❛s n → ∞ t❤r♦✉❣❤ ❛ s❡t

♦❢ ❛s②♠♣t♦t✐❝ ❞❡♥s✐t② ✶✳ ❙♦ t❤❡r❡ ✐s ❛ t❤r❡s❤♦❧❞ ❢✉♥❝t✐♦♥✿ ✇❡ s❤♦✉❧❞ ❜❡ st✉❞②✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡ ω(n) − log log n✳

✸✷

❚❤❡ ❊r❞➤s✕❑❛❝ t❤❡♦r❡♠ ✭✶✾✸✾✮✿ ❋♦r ❡❛❝❤ r❡❛❧ ♥✉♠❜❡r u✱ t❤❡ ❛s②♠♣t♦t✐❝ ❞❡♥s✐t② ♦❢ t❤❡ s❡t

• n : ω(n) − log log n ≤ u
• log log n
• ✐s

1 √ 2π u

−∞

e−t2/2 dt.

❚❤✐s ✐s t❤❡ ●❛✉ss✐❛♥ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✱ t❤❡ ❇❡❧❧ ❝✉r✈❡✦

✸✸

❊✐♥st❡✐♥✿ ✏●♦❞ ❞♦❡s ♥♦t ♣❧❛② ❞✐❝❡ ✇✐t❤ t❤❡ ✉♥✐✈❡rs❡✳✑

✸✹

❊✐♥st❡✐♥✿ ✏●♦❞ ❞♦❡s ♥♦t ♣❧❛② ❞✐❝❡ ✇✐t❤ t❤❡ ✉♥✐✈❡rs❡✳✑ ❊r❞➤s ✫ ❑❛❝✿ ▼❛②❜❡ s♦ ❜✉t s♦♠❡t❤✐♥❣✬s ❣♦✐♥❣ ♦♥ ✇✐t❤ t❤❡ ♣r✐♠❡s✳

✸✺

❊✐♥st❡✐♥✿ ✏●♦❞ ❞♦❡s ♥♦t ♣❧❛② ❞✐❝❡ ✇✐t❤ t❤❡ ✉♥✐✈❡rs❡✳✑ ❊r❞➤s ✫ ❑❛❝✿ ▼❛②❜❡ s♦ ❜✉t s♦♠❡t❤✐♥❣✬s ❣♦✐♥❣ ♦♥ ✇✐t❤ t❤❡ ♣r✐♠❡s✳ ✭◆♦t❡✿ ■ ♠❛❞❡ t❤✐s ✉♣✱ ✐t ✇❛s ❛ ❥♦❦❡ . . . ✮

✸✻

Pr✐♠❡ ♥✉♠❜❡rs✱ t❤❡ ♠♦st ♠②st❡r✐♦✉s ✜❣✉r❡s ✐♥ ♠❛t❤✱ ❉✳ ❲❡❧❧s

✸✼

✸✽

❙♦♠❡ ❜❛❝❦❣r♦✉♥❞ ♦♥ t❤❡ ❊r❞➤s✕❑❛❝ t❤❡♦r❡♠ ✇♦✉❧❞ ❜❡ ❤❡❧♣❢✉❧✱ ✐t ❞✐❞♥✬t ❛r✐s❡ s♣♦♥t❛♥❡♦✉s❧②✳ ■♥ ✶✾✸✹✱ P❛✉❧ ❚✉rá♥ ❣❛✈❡ ❛ s✐♠♣❧✐✜❡❞ ♣r♦♦❢ ♦❢ t❤❡ ❍❛r❞②✕❘❛♠❛♥✉❥❛♥ t❤❡♦r❡♠✳ ◗✉♦t❡❞ ✐♥ ❊❧❧✐♦tt✬s Pr♦❜❛❜✐❧✐st✐❝ ◆✉♠❜❡r ❚❤❡♦r②✱ ❚✉rá♥ s❛✐❞ ✐♥ ✶✾✼✻✿ ✏. . . ■ ❞✐❞ ♥♦t ❦♥♦✇ ✇❤❛t ❈❤❡❜②s❤❡✈✬s ✐♥❡q✉❛❧✐t② ✇❛s ❛♥❞ ❛ ❢♦rt✐♦r✐ t❤❡ ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✳ ❊r❞➤s✱ t♦ ♠② ❜❡st ❦♥♦✇❧❡❞❣❡✱ ✇❛s ❛t t❤❛t t✐♠❡ ♥♦t ❛✇❛r❡ t♦♦✳ ■t ✇❛s

✸✾

▼❛r❦ ❑❛❝ ✇❤♦ ✇r♦t❡ t♦ ♠❡ ❛ ❢❡✇ ②❡❛rs ❧❛t❡r t❤❛t ❤❡ ❞✐s❝♦✈❡r❡❞ ✇❤❡♥ r❡❛❞✐♥❣ ♠② ♣r♦♦❢ ✐♥ ❏✳▲✳▼✳❙✳ t❤❛t t❤✐s ✐s ❜❛s✐❝❛❧❧② ♣r♦❜❛❜✐❧✐t② ❛♥❞ s♦ ✇❛s ❤✐s ✐♥t❡r❡st t✉r♥❡❞ t♦ t❤✐s s✉❜❥❡❝t✳✑

✹✵

❊❧❧✐♦tt ❛❧s♦ q✉♦t❡s ▼❛r❦ ❑❛❝✿ ✏■❢ ■ r❡♠❡♠❜❡r ❝♦rr❡❝t❧② ■ ✜rst st❛t❡❞ ✭❛s ❛ ❝♦♥❥❡❝t✉r❡✮ t❤❡ t❤❡♦r❡♠ ♦♥ t❤❡ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣r✐♠❡ ❞✐✈✐s♦rs ❞✉r✐♥❣ ❛ ❧❡❝t✉r❡ ✐♥ Pr✐♥❝❡t♦♥ ✐♥ ▼❛r❝❤ ✶✾✸✾✳ ❋♦rt✉♥❛t❡❧② ❢♦r ♠❡ ❛♥❞ ♣♦ss✐❜❧② ❢♦r ▼❛t❤❡♠❛t✐❝s✱ ❊r❞➤s ✇❛s ✐♥ t❤❡ ❛✉❞✐❡♥❝❡✱ ❛♥❞ ❤❡ ✐♠♠❡❞✐❛t❡❧② ♣❡r❦❡❞ ✉♣✳ ❇❡❢♦r❡ t❤❡ ❧❡❝t✉r❡ ✇❛s ♦✈❡r ❤❡ ❤❛❞ ❝♦♠♣❧❡t❡❞ t❤❡ ♣r♦♦❢✱ ✇❤✐❝❤ ■ ❝♦✉❧❞ ♥♦t ❤❛✈❡ ❞♦♥❡ ♥♦t ❤❛✈✐♥❣ ❜❡❡♥ ✈❡rs❡❞ ✐♥ t❤❡ ♥✉♠❜❡r

✹✶

t❤❡♦r❡t✐❝ ♠❡t❤♦❞s✱ ❡s♣❡❝✐❛❧❧② t❤♦s❡ r❡❧❛t❡❞ t♦ t❤❡ s✐❡✈❡✳✑

✹✷

▲❡t ✉s r❡t✉r♥ t♦ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❛♠✐❝❛❜❧❡ ♥✉♠❜❡rs ✐♥tr♦❞✉❝❡❞ ❜② P②t❤❛❣♦r❛s ✷✺✵✵ ②❡❛rs ❛❣♦✳ ❘❡❝❛❧❧✿ ❚✇♦ ♥✉♠❜❡rs ❛r❡ ❛♠✐❝❛❜❧❡ ✐❢ t❤❡ s✉♠ ♦❢ t❤❡ ♣r♦♣❡r ❞✐✈✐s♦rs ♦❢ ♦♥❡ ✐s t❤❡ ♦t❤❡r ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❚❤❡ P②t❤❛❣♦r❡❛♥ ❡①❛♠♣❧❡✿ ✷✷✵ ❛♥❞ ✷✽✹✳

✹✸

❲❡ ❤❛✈❡ s❡❡♥ t❤❛t ❛♠✐❝❛❜❧❡ ♥✉♠❜❡rs ❤❛✈❡ ❢❛s❝✐♥❛t❡❞ ♣❡♦♣❧❡ t❤r♦✉❣❤ t❤❡ ✐♥t❡r✈❡♥✐♥❣ ❝❡♥t✉r✐❡s✳ ❚❤❛❜✐t ✐❜♥ ❑✉rr❛❤ ❢♦✉♥❞ ❛ ❢♦r♠✉❧❛ t❤❛t ❣❛✈❡ ❛ ❢❡✇ ❡①❛♠♣❧❡s✳ ❊✉❧❡r ❢♦✉♥❞ ❛ ❢❡✇✳ ❙♦ ❢❛r ✇❡ ❦♥♦✇ ❛❜♦✉t t✇❡❧✈❡ ♠✐❧❧✐♦♥ ♣❛✐rs✱ ❛♥❞ ♣r♦❜❛❜❧② t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥②✱ ❜✉t ✇❡ ❤❛✈❡ ♥♦ ♣r♦♦❢✳ ❍♦✇ ✇♦✉❧❞ ❊r❞➤s ❛♣♣r♦❛❝❤ t❤✐s ♣r♦❜❧❡♠❄

✹✹

❲❤② ❝♦✉♥t ♦❢ ❝♦✉rs❡✦ ▲❡t A(x) ❞❡♥♦t❡ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥t❡❣❡rs ✐♥

[1, x] t❤❛t ❜❡❧♦♥❣ t♦ ❛♥ ❛♠✐❝❛❜❧❡ ♣❛✐r✳ ❲❡

❤❛✈❡ ♥♦ ❣♦♦❞ ❧♦✇❡r ❜♦✉♥❞s ❢♦r A(x) ❛s

x → ∞✱ ❜✉t ✇❤❛t ❛❜♦✉t ❛♥ ✉♣♣❡r ❜♦✉♥❞❄

✹✺

❋♦r ♣❡r❢❡❝t ♥✉♠❜❡rs✱ ✇❤✐❝❤ ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞ t♦ t❤❡ ❛♠✐❝❛❜❧❡s✱ ✇❡ ❦♥♦✇ ❛ ❢❛✐r ❛♠♦✉♥t ❛❜♦✉t ✉♣♣❡r ❜♦✉♥❞s✳ ❋✐rst✱ ❢r♦♠ ❉❛✈❡♥♣♦rt✬s t❤❡♦r❡♠ ♦♥ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ σ(n)/n ✐t ✐s ✐♠♠❡❞✐❛t❡ t❤❛t t❤❡ ♣❡r❢❡❝t ♥✉♠❜❡rs ❤❛✈❡ ❛s②♠♣t♦t✐❝ ❞❡♥s✐t② ✵✳

✹✻

❚❤❡r❡ ❛r❡ ♠✉❝❤ ❜❡tt❡r ✉♣♣❡r ❜♦✉♥❞s ❢♦r t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ♣❡r❢❡❝t ♥✉♠❜❡rs✳ ❊r❞➤s ♠❛❞❡ ❛ ❢✉♥❞❛♠❡♥t❛❧ ❝♦♥tr✐❜✉t✐♦♥ ❤❡r❡✱ ❜✉t t❤❡ ❝❤❛♠♣✐♦♥ r❡s✉❧t ✐s ❞✉❡ t♦ ❍♦r♥❢❡❝❦ ❛♥❞ ❲✐rs✐♥❣✿ t❤❡ ♥✉♠❜❡r ♦❢ ♣❡r❢❡❝t ♥✉♠❜❡rs ✐♥ [1, x] ✐s ❛t ♠♦st xo(1)✳ ❇✉t ❛♠✐❝❛❜❧❡s ♣r❡s✉♠❛❜❧② ❢♦r♠ ❛ ❧❛r❣❡r s❡t✱ ♠❛②❜❡ ♠✉❝❤ ❧❛r❣❡r✳

✹✼

❊r❞➤s ✭✶✾✺✺✮ ✇❛s t❤❡ ✜rst t♦ s❤♦✇ t❤❛t

A(x) = o(x)✱ t❤❛t ✐s✱ t❤❡ ❛♠✐❝❛❜❧❡ ♥✉♠❜❡rs

❤❛✈❡ ❛s②♠♣t♦t✐❝ ❞❡♥s✐t② ✵✳ ❍✐s ✐♥s✐❣❤t✿ t❤❡ s♠❛❧❧❡r ♠❡♠❜❡r ♦❢ ❛♥ ❛♠✐❝❛❜❧❡ ♣❛✐r ✐s ❛❜✉♥❞❛♥t✱ t❤❡ ❧❛r❣❡r ✐s ❞❡✜❝✐❡♥t✳ ❚❤✉s✱ ✇❡ ❤❛✈❡ ❛♥ ❛❜✉♥❞❛♥t ♥✉♠❜❡r ✇✐t❤ t❤❡ s✉♠ ♦❢ ✐ts ♣r♦♣❡r ❞✐✈✐s♦rs ❜❡✐♥❣ ❞❡✜❝✐❡♥t✳

✹✽

❚❤✐s ♣r♦♣❡rt② ❛❧♦♥❡ ✐s ❡♥♦✉❣❤ t♦ ♣r♦✈❡ ❞❡♥s✐t② ✵✳ ▲❡t ✉s ❧♦♦❦ ❛t ❛ s❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢✳ ▲❡t

h(n) = σ(n)/n✳ ❲❡ ❛r❡ ❝♦✉♥t✐♥❣ ♥✉♠❜❡rs n ≤ x ❢♦r ✇❤✐❝❤ h(n) > 2

❛♥❞ h(s(n)) < 2.

✹✾

❊r❞➤s ✜rst ✉s❡❞ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ❢♦r h s♦ t❤❛t ✐♥st❡❛❞ ♦❢ ♠❡r❡❧② ❛ss✉♠✐♥❣ h(n) > 2✱ ✇❡ ❤❛✈❡ t❤❡ str♦♥❣❡r ❛ss✉♠♣t✐♦♥ h(n) > 2 + δ ✭❢♦r s♦♠❡ ✜①❡❞ t✐♥② δ > 0✮✳ ◆♦t❡ t❤❛t h(n) ✐s t❤❡ s✉♠ ♦❢ 1/d ❢♦r d | n ❛♥❞ 1 ≤ d✳ ❋♦r ❛ ♣❛r❛♠❡t❡r y✱ ❧❡t hy(n) ❜❡ t❤❡ s✉♠ ♦❢ 1/d ❢♦r d | n ❛♥❞ 1 ≤ d ≤ y✳

✺✵

❊r❞➤s ♥❡①t ❛r❣✉❡❞ ❜② ❛♥ ❛✈❡r❛❣✐♥❣ ❛r❣✉♠❡♥t t❤❛t✱ ❢♦r y ❧❛r❣❡✱ ✇❡ ✉s✉❛❧❧② ❤❛✈❡ hy(n) ≈ h(n)✱ s♦ t❤❛t ✇❡ ♠❛② ❛ss✉♠❡ t❤❛t

hy(n) =

• d|n, d≤y

1 d > 2.

◆♦✇ t❤❡ ❦❡② st❡♣✿ ▲❡t M ❜❡ t❤❡ ❧❝♠ ♦❢

{1, 2, . . . , ⌊y⌋}✳ ❆❧♠♦st ❛❧❧ ♥✉♠❜❡rs n ❛r❡

❞✐✈✐s✐❜❧❡ ❜② ❛ ♣r✐♠❡ p t♦ t❤❡ ✜rst ♣♦✇❡r ✇✐t❤ p + 1 ≡ 0 (mod M)✳ ✭❍✐♥t✿ ✉s❡ ❉✐r✐❝❤❧❡t✳✮

✺✶

❚❤✉s✱ ❢♦r ❛❧♠♦st ❛❧❧ ♥✉♠❜❡rs n✱ ✇❡ ❤❛✈❡

M | σ(n)✱ ❛♥❞ s♦ s(n) = σ(n) − n ❤❛s ❡①❛❝t❧②

t❤❡ s❛♠❡ ❞✐✈✐s♦rs ✉♣ t♦ y ❛s n ❞♦❡s✳ ❆ss✉♠✐♥❣ t❤✐s✱

h(s(n)) ≥ hy(s(n)) = hy(n) > 2,

❝♦♥tr❛❞✐❝t✐♥❣ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t

h(s(n)) < 2✳

◗❊❉

✺✷

■ ❧❛t❡r ❣❛✈❡ ❛ s✐♠♣❧✐✜❡❞ ♣r♦♦❢ ✉s✐♥❣ ❛♥♦t❤❡r ❊r❞➤s ✐♥s✐❣❤t✿ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ♣r✐♠✐t✐✈❡ ❛❜✉♥❞❛♥t ♥✉♠❜❡rs ✭✶✾✸✺✮✳ ✭❆♥❞ t❤❡♥ ✐♥ ❛♥♦t❤❡r ♣❛♣❡r✱ ■ s❤♦✇❡❞ t❤❡ r❡❝✐♣r♦❝❛❧ s✉♠ ♦❢ t❤❡ ❛♠✐❝❛❜❧❡ ♥✉♠❜❡rs ✐s ✜♥✐t❡✳✮

✺✸

❚❤♦✉❣❤ ❊r❞➤s ❞✐❞ ♥♦t ❝♦♥tr✐❜✉t❡ ❞✐r❡❝t❧② t♦ ❝♦♠♣✉t❛t✐♦♥❛❧ ♥✉♠❜❡r t❤❡♦r②✱ ❤✐s st❛t✐st✐❝❛❧ ✈✐❡✇♣♦✐♥t ✐s ♣❛rt ♦❢ t❤❡ ❧❛♥❞s❝❛♣❡ ❤❡r❡ t♦♦✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♥ ❈❛♥✜❡❧❞✱ ❊r❞➤s✱ P ✭✶✾✽✸✮✱ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ✏s♠♦♦t❤✑ ✭♦r ✏❢r✐❛❜❧❡✑✮ ♥✉♠❜❡rs ✇❛s ✇♦r❦❡❞ ♦✉t t♦ ❡♥♦✉❣❤ ❞❡t❛✐❧ t♦ ❣✐✈❡ ❛❝❝✉r❛t❡ ❣✉✐❞❛♥❝❡ t♦ t❤❡ ❝♦♥str✉❝t✐♦♥ ❛♥❞ ❛♥❛❧②s✐s ♦❢ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠s✳

✺✹

❚❤✐s ♣❛♣❡r ✐s ❤✐s ✶✸t❤ ♠♦st✲❝✐t❡❞ ♦♥ ♠❛t❤s❝✐♥❡t ✭❛♥❞ ❈❛♥✜❡❧❞✬s ❛♥❞ ♠② ★✶✮✳

✺✺

❋❡r♠❛t ♣r♦✈❡❞ t❤❛t ✐❢ p ✐s ❛ ♣r✐♠❡ t❤❡♥

ap ≡ a (mod p) ❢♦r ❡✈❡r② ✐♥t❡❣❡r a✳ ■t ✐s ❛♥

❡❛s② ❝♦♥❣r✉❡♥❝❡ t♦ ❝❤❡❝❦✳ ❈❛♥ ♦♥❡ r❡❛s♦♥ ❢r♦♠ t❤❡ ❝♦♥✈❡rs❡❄❄ ❙❛② ❛ ❝♦♠♣♦s✐t❡ ♥✉♠❜❡r n ✐s ❛ ❜❛s❡✲a ♣s❡✉❞♦♣r✐♠❡ ✐❢ an ≡ a (mod n)✳

✺✻

Ps❡✉❞♦♣r✐♠❡s ❡①✐st✳ ❋♦r ❡①❛♠♣❧❡✱

2341 ≡ 2 (mod 341)✳

❊r❞➤s ✭✶✾✹✾✱ ✶✾✺✵✮ ✇❛s t❤❡ ✜rst t♦ s❤♦✇ t❤❛t ❢♦r ❡❛❝❤ ✜①❡❞ ❜❛s❡ a > 1✱ t❤❡ ♣s❡✉❞♦♣r✐♠❡s ❛r❡ ✈❡r② r❛r❡ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ ♣r✐♠❡s✳ ❉✉❡ t♦ t❤✐s✱ ✐❢ ♦♥❡ ❤❛s ❛ ❧❛r❣❡ r❛♥❞♦♠ ♥✉♠❜❡r ❛♥❞ t❡sts ♠❡r❡❧② ✐❢

2n ≡ 2 (mod n)✱ ❛❝❝❡♣t✐♥❣ n ❛s ♣r✐♠❡ ✐❢ t❤❡

❝♦♥❣r✉❡♥❝❡ ❤♦❧❞s✱ ♦♥❡ ✇♦✉❧❞ ❛❧♠♦st s✉r❡❧② ❜❡ r✐❣❤t✦

✺✼

❚❤♦✉❣❤ t❤❡r❡ ✐s ♦❢ ❝♦✉rs❡ s♦♠❡ ❝❤❛♥❝❡ ❢♦r ❡rr♦r ❤❡r❡✱ ✐t ✐s ❛❝t✉❛❧❧② ❛ ♣r❛❝t✐❝❛❧ ✇❛② t♦ r❡❝♦❣♥✐③❡ ♣r✐♠❡s✱ ✐t ✐s ❢❛st✱ ❛♥❞ ✐t ✐s ❡①tr❛♦r❞✐♥❛r✐❧② s✐♠♣❧❡✳

✺✽

❊r❞➤s ✇❛s ✈❡r② ✐♥t❡r❡st❡❞ ✐♥ ❈❛r♠✐❝❤❛❡❧ ♥✉♠❜❡rs✳ ❚❤❡s❡ ❛r❡ ♥✉♠❜❡rs✱ ❧✐❦❡ ✺✻✶✱ ✇❤✐❝❤ ❛r❡ ♣s❡✉❞♦♣r✐♠❡s t♦ ❡✈❡r② ❜❛s❡✳ ■♥ ✶✾✺✻ ❤❡ ❣♦t t❤❡ ❡ss❡♥t✐❛❧❧② ❜❡st✲❦♥♦✇♥ ✉♣♣❡r ❜♦✉♥❞ ❢♦r C(x)✱ t❤❡ ♥✉♠❜❡r ♦❢ ❈❛r♠✐❝❤❛❡❧ ♥✉♠❜❡rs ✐♥ [1, x]✿

C(x) ≤ x1−c log log log x/ log log x.

❍❡ ❛❧s♦ ❣❛✈❡ ❛ ❤❡✉r✐st✐❝ ❛r❣✉♠❡♥t ✭❜❛s❡❞ ♦♥ ❛ s❡♠✐♥❛❧ ♣❛♣❡r ♦❢ ❤✐s ❢r♦♠ ✶✾✸✺✮ t❤❛t t❤✐s ✇❛s ❡ss❡♥t✐❛❧❧② ❜❡st ♣♦ss✐❜❧❡✳

✺✾

❚❤❡ ❊r❞➤s ❝♦♥❥❡❝t✉r❡ ♦♥ ❈❛r♠✐❝❤❛❡❧ ♥✉♠❜❡rs✿ C(x) ≥ x1−ǫ✳ ■♥ ✶✾✾✸✱ ❆❧❢♦r❞✱ ●r❛♥✈✐❧❧❡✱ P ❣❛✈❡ ❛ r✐❣♦r♦✉s ♣r♦♦❢✱ ❜❛s❡❞ ♦♥ t❤❡ ❊r❞➤s ❤❡✉r✐st✐❝✱ t❤❛t C(x) > x2/7 ❢♦r ❛❧❧ ❧❛r❣❡ x ❛♥❞ t❤❛t C(x) > x1−ǫ ❛ss✉♠✐♥❣ t❤❡ ❊❧❧✐♦tt✕❍❛❧❜❡rst❛♠ ❝♦♥❥❡❝t✉r❡ ♦♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ♣r✐♠❡s ✐♥ r❡s✐❞✉❡ ❝❧❛ss❡s✳

✻✵

❆♥❞ ✇❤❛t ✇❛s t❤✐s s❡♠✐♥❛❧ ♣❛♣❡r ❢r♦♠ ✶✾✸✺ ❥✉st ♠❡♥t✐♦♥❡❞❄ ■t ✇❛s ✐♥✿ ◗✉❛rt❡r❧② ❏✳ ▼❛t❤✳ ❖①❢♦r❞ ❙❡r✳ ✻ ✭✶✾✸✺✮✱ ✷✵✺✕✷✶✸✳

✻✶

✻✷

❆s ♠❡♥t✐♦♥❡❞✱ ❍❛r❞② ✫ ❘❛♠❛♥✉❥❛♥ s❤♦✇❡❞ t❤❛t n ♥♦r♠❛❧❧② ❤❛s ❛❜♦✉t log log n ♣r✐♠❡ ❢❛❝t♦rs✳ ❈❧❡❛r❧② t❤❡♥✱ ♣r✐♠❡s ❛r❡ ♥♦t ♥♦r♠❛❧✦ ❇✉t ❛r❡ ♥✉♠❜❡rs p − 1 ♥♦r♠❛❧❄ ■♥ t❤✐s ♣❛♣❡r✱ s✉❜♠✐tt❡❞ ❢♦r ♣✉❜❧✐❝❛t✐♦♥ ❛t t❤❡ ❛❣❡ ♦❢ ✷✶✱ ❊r❞➤s s❤♦✇❡❞ t❤❛t ②❡s✱

p − 1 ✐s ✐♥❞❡❡❞ ♥♦r♠❛❧ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡

♥✉♠❜❡r ♦❢ ✐ts ♣r✐♠❡ ❢❛❝t♦rs✳

✻✸

◆♦t ♦♥❧② ✐s t❤✐s ✐♥t❡r❡st✐♥❣ ♦♥ ✐ts ♦✇♥✱ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♥♦r♠❛❧✐t② ♦❢ p − 1 ✇❛s ♦♥❡ ♦❢ t❤❡ ❡❛r❧② ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❇r✉♥✬s s✐❡✈❡ ♠❡t❤♦❞✱ ♦❢ ✇❤✐❝❤ ❊r❞➤s ✇❛s s♦ ❢❛♠♦✉s✳ ❆♥❞ t❤❡ r❡s✉❧t ✇❛s ❛♥ ❡ss❡♥t✐❛❧ t♦♦❧ ✐♥ s♦❧✈✐♥❣ ❛ ♣r♦❜❧❡♠ ♦❢ P✐❧❧❛✐✿ ❤♦✇ ♠❛♥② ♥✉♠❜❡rs ✐♥ [1, x] ❛r❡ ✈❛❧✉❡s ♦❢ ϕ ✭❊✉❧❡r✬s ❢✉♥❝t✐♦♥✮❄

✻✹

❊r❞➤s s❤♦✇❡❞ t❤❛t t❤✐s ❝♦✉♥t ♦❢ ϕ✲✈❛❧✉❡s ✐♥ [1, x] ✐s ♦❢ t❤❡ s❤❛♣❡ x/(log x)1+o(1)✳ ❆♥❞ ✇❤✐❧❡ ❤❡ ✇❛s ♦♥ t❤❡ t♦♣✐❝✱ ❤❡ ♣r♦✈❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛st♦✉♥❞✐♥❣ r❡s✉❧t✿ ❚❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t c s✉❝❤ t❤❛t ❢♦r ✐♥✜♥✐t❡❧② ♠❛♥② ♥✉♠❜❡rs N✱ t❤❡r❡ ❛r❡ ♠♦r❡ t❤❛♥ N c s♦❧✉t✐♦♥s t♦ ϕ(n) = N✳

✻✺

❍❡ ❣❛✈❡ ❛ ❤❡✉r✐st✐❝ t❤❛t ✏c✑ ❤❡r❡ ❝❛♥ ❜❡ t❛❦❡♥ ❛s ❛♥② ♥✉♠❜❡r s♠❛❧❧❡r t❤❛♥ ✶✳ ■t ✇❛s t❤✐s ❝♦♥str✉❝t✐♦♥ t❤❛t ✇❛s s♦ ✐♠♣♦rt❛♥t ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ✐♥✜♥✐t✉❞❡ ♦❢ ❈❛r♠✐❝❤❛❡❧ ♥✉♠❜❡rs✳ ❚❤❡ ✈❛❧✉❡ ♦❢ c ✐♥ t❤❡ t❤❡♦r❡♠ ❤❛s s❧♦✇❧② ❝❧✐♠❜❡❞ ♦✈❡r t❤❡ ✐♥t❡r✈❡♥✐♥❣ ②❡❛rs✱ ✇✐t❤ ♠❛♥② ♣❧❛②❡rs✳ ❈✉rr❡♥t❧② ✐t ✐s ❛❜♦✉t ✵✳✼✱ ❛ r❡s✉❧t ♦❢ ❇❛❦❡r ❛♥❞ ❍❛r♠❛♥✳

✻✻

❚❤❡ ❝♦✉♥t ♦❢ x/(log x)1+o(1) ❢♦r ϕ✲✈❛❧✉❡s ✐♥

[1, x] ❤❛s ❜❡❡♥ r❡✜♥❡❞ ❛s ✇❡❧❧✱ ✇✐t❤ t❤❡

❝✉rr❡♥t ❜❡st r❡s✉❧t ❞✉❡ t♦ ❋♦r❞✿ ✐t ✐s ♦❢ ♠❛❣♥✐t✉❞❡

x log x exp

• c1(log3 x− log4 x)2 + c2 log3 x + c3 log4 x
• ❢♦r ❝❡rt❛✐♥ ❡①♣❧✐❝✐t ❝♦♥st❛♥ts c1, c2, c3✳

❚❤❡ s❛♠❡ ✐s tr✉❡ ❢♦r t❤❡ s❡t ♦❢ ✈❛❧✉❡s ♦❢

σ(n) ✐♥ [1, x]✳

✻✼

■♥ ♠❛♥② ✇❛②s✱ σ ❛♥❞ ϕ ❛r❡ t✇✐♥s✳ ❊r❞➤s ❛s❦❡❞ ✐♥ ✶✾✺✾ ✐❢ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② s♦❧✉t✐♦♥s t♦ ϕ(m) = σ(n)✳ ❨❡s✱ ✐❢ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② t✇✐♥ ♣r✐♠❡s✿ ■❢ p✱ p + 2 ❛r❡ ❜♦t❤ ♣r✐♠❡✱ t❤❡♥

ϕ(p + 2) = p + 1 = σ(p)✳

✻✽

■♥✜♥✐t❡❧② ♠❛♥② s♦❧✉t✐♦♥s t♦ ϕ(m) = σ(n)❄ ❨❡s✱ ✐❢ t❤❡r❡ ❛r❡ ✐♥✜♥t❡❧② ♠❛♥② ▼❡rs❡♥♥❡ ♣r✐♠❡s✿ ■❢ 2p − 1 ✐s ♣r✐♠❡✱ t❤❡♥

ϕ(2p+1) = 2p = σ(2p − 1)✳

❨❡s✱ ✐❢ t❤❡ ❊①t❡♥❞❡❞ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ❤♦❧❞s✳

✻✾

■♥✜♥✐t❡❧② ♠❛♥② s♦❧✉t✐♦♥s t♦ ϕ(m) = σ(n)❄ ❨❡s✱ ✐❢ t❤❡r❡ ❛r❡ ✐♥✜♥t❡❧② ♠❛♥② ▼❡rs❡♥♥❡ ♣r✐♠❡s✿ ■❢ 2p − 1 ✐s ♣r✐♠❡✱ t❤❡♥

ϕ(2p+1) = 2p = σ(2p − 1)✳

❨❡s✱ ✐❢ t❤❡ ❊①t❡♥❞❡❞ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ❤♦❧❞s✳ ❋♦r❞✱ ▲✉❝❛✱ ✫ P ✭✷✵✶✵✮✿ ❨❡s✳

✼✵

■ ✇♦✉❧❞ ❧✐❦❡ t♦ ❝❧♦s❡ ✇✐t❤ ♦♥❡ ❧❛st ❛♥❝✐❡♥t ♣r♦❜❧❡♠✿ ♣r✐♠❡ ♥✉♠❜❡rs✳ ✷✸✵✵ ②❡❛rs ❛❣♦✱ ❊✉❝❧✐❞ ✇❛s t❤❡ ✜rst t♦ ❝♦♥s✐❞❡r ❝♦✉♥t✐♥❣ ♣r✐♠❡s✿ ❤❡ ♣r♦✈❡❞ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥②✳ ❖♥❡ ♠✐❣❤t ❛r❣✉❡ t❤❡♥ t❤❛t ✐t ✐s ❊✉❝❧✐❞ ✇❤♦ ✜rst ♦✛❡r❡❞ t❤❡ st❛t✐st✐❝❛❧ ✈✐❡✇♣♦✐♥t✳

✼✶

✼✷

❉❡t❛✐❧ ❢r♦♠ ❘❛♣❤❛❡❧✬s ♠✉r❛❧ ❚❤❡ ❙❝❤♦♦❧ ♦❢ ❆t❤❡♥s✱ ❝❛✳ ✶✺✶✵

✼✸

❋✉rt❤❡r ♣r♦❣r❡ss ✇❛s ♠❛❞❡ ✷✵✵✵ ②❡❛rs ❧❛t❡r ❜② ❊✉❧❡r✿

• p≤x

1 p ∼ log log x.

❋✐❢t② ②❡❛rs ❧❛t❡r✿ ●❛✉ss ❛♥❞ ▲❡❣❡♥❞r❡ ❝♦♥❥❡❝t✉r❡❞ t❤❛t

π(x) :=

• p≤x

1 ∼ x log x.

✼✹

❋✐❢t② ②❡❛rs ❧❛t❡r✿ ❈❤❡❜②s❤❡✈ ♣r♦✈❡❞ t❤❛t

π(x) ✐s ♦❢ ♠❛❣♥✐t✉❞❡ x/ log x✳ ❆♥❞

❘✐❡♠❛♥♥ ❧❛✐❞ ♦✉t ❛ ♣❧❛♥ t♦ ♣r♦✈❡ t❤❡

• ❛✉ss✕▲❡❣❡♥❞r❡ ❝♦♥❥❡❝t✉r❡✳

❋✐❢t② ②❡❛rs ❧❛t❡r✿ ❍❛❞❛♠❛r❞ ❛♥❞ ❞❡ ❧❛ ❱❛❧❧❡❡ P♦✉ss✐♥ ♣r♦✈❡❞ ✐t✳

✼✺

❋✐❢t② ②❡❛rs ❧❛t❡r✿ ❊r❞➤s ❛♥❞ ❙❡❧❜❡r❣ ❣❛✈❡ ❛♥ ❡❧❡♠❡♥t❛r② ♣r♦♦❢✳ ❲❡✬r❡ ❛ ❜✐t ♦✈❡r❞✉❡ ❢♦r t❤❡ ♥❡①t ✐♥st❛❧❧♠❡♥t . . . ✳

✼✻

■ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤✐♥❦ t❤❛t ❜❡②♦♥❞ t❤❡ ✏Pr✐♠❡ ◆✉♠❜❡r ❚❤❡♦r❡♠✑✱ ❊r❞➤s ✇❛s s❡❛r❝❤✐♥❣ t♦♦ ❢♦r t❤❡ ✏❆♠✐❝❛❜❧❡ ◆✉♠❜❡r ❚❤❡♦r❡♠✑✱ t❤❡ ✏P❡r❢❡❝t ◆✉♠❜❡r ❚❤❡♦r❡♠✑✱ ❛♥❞ s♦ ♦♥✳ ■♥ ❛❧❧ ♦❢ t❤❡s❡ ♣r♦❜❧❡♠s ❛♥❞ r❡s✉❧ts ✇❡ ❝❛♥ s❡❡ ❡❝❤♦❡s ♦❢ t❤❡ ♣❛st ❛t t❤❡ ❞❛✇♥ ♦❢ ♥✉♠❜❡r t❤❡♦r② ❛♥❞ ♠❛t❤❡♠❛t✐❝s✳

✼✼

P❡r❤❛♣s t❤❡ ❛♥❝✐❡♥t ♣r♦❜❧❡♠s ✇✐❧❧ ♥❡✈❡r ❜❡ ❝♦♠♣❧❡t❡❧② s♦❧✈❡❞✱ ❜✉t t❤✐♥❦✐♥❣ ❛❜♦✉t t❤❡♠ st❛t✐st✐❝❛❧❧② ❤❛s ♠❛❞❡ ❛❧❧ t❤❡ ❞✐✛❡r❡♥❝❡✳ ❆♥❞ ❧❡❛❞✐♥❣ t❤❡ ✇❛②✱ ✇❛s P❛✉❧ ❊r❞➤s✳

❑ös③ö♥ö♠

✼✽

✼✾