Algorithms of discrete logarithm in finite fields · viii INTRODUCTION Motivation Cryptography is concerned with secure communications between two entities, for example two computers

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Algorithms of discrete logarithm in finite fieldsRazvan Barbulescu

To cite this version:Razvan Barbulescu. Algorithms of discrete logarithm in finite fields. Cryptography and Security[cs.CR]. Université de Lorraine, 2013. English. �tel-00925228�

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➱❝♦❧❡ ❞♦❝t♦r❛❧❡ ■❆❊▼ ▲♦rr❛✐♥❡

❆❧❣♦r✐t❤♠❡s ❞❡ ❧♦❣❛r✐t❤♠❡s ❞✐s❝r❡ts❞❛♥s ❧❡s ❝♦r♣s ✜♥✐s

❚❍➮❙❊♣rés❡♥té❡ ❡t s♦✉t❡♥✉❡ ♣✉❜❧✐q✉❡♠❡♥t ❧❡ 5 ❞é❝❡♠❜r❡ 2013

♣♦✉r ❧✬♦❜t❡♥t✐♦♥ ❞✉

❉♦❝t♦r❛t ❞❡ ❧✬❯♥✐✈❡rs✐té ❞❡ ▲♦rr❛✐♥❡✭♠❡♥t✐♦♥ ✐♥❢♦r♠❛t✐q✉❡✮

♣❛r

❘❛③✈❛♥ ❇❛r❜✉❧❡s❝✉

❈♦♠♣♦s✐t✐♦♥ ❞✉ ❥✉r②

❘❛♣♣♦rt❡✉rs ✿ ❏❡❛♥✲▼❛r❝ ❈♦✉✈❡✐❣♥❡s Pr♦❢✳ ❯♥✐✈❡rs✳ ❇♦r❞❡❛✉①❆❧❢r❡❞ ▼❡♥❡③❡s Pr♦❢✳ ❯♥✐✈❡rs✳ ❲❛t❡r❧♦♦✱ ❈❛♥❛❞❛

❊①❛♠✐♥❛t❡✉rs ✿ ◆✐❝♦❧❛s ❇r✐s❡❜❛rr❡ ❈❘ ❈◆❘❙❊♠♠❛♥✉❡❧ ❏❡❛♥❞❡❧ Pr♦❢✳ ❯♥✐✈❡rs✳ ▲♦rr❛✐♥❡❆♥t♦✐♥❡ ❏♦✉① Pr♦❢✳ ❯♥✐✈❡rs✳ P❛r✐s ✻❋r❛♥ç♦✐s ▼♦r❛✐♥ Pr♦❢✳ ➱❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡❋r❡❞❡r✐❦ ❱❡r❝❛✉t❡r❡♥ ❑❯ ▲❡✉✈❡♥✱ ❇❡❧❣✐ë

❉✐r❡❝t❡✉r ✿ P✐❡rr✐❝❦ ●❛✉❞r② ❉❘ ❈◆❘❙

▲❛❜♦r❛t♦✐r❡ ▲♦rr❛✐♥ ❞❡ ❘❡❝❤❡r❝❤❡ ❡♥ ■♥❢♦r♠❛t✐q✉❡ ❡t s❡s ❆♣♣❧✐❝❛t✐♦♥s

❈♦♥t❡♥ts

■♥tr♦❞✉❝t✐♦♥ ✐✐✐

■ ❙♠♦♦t❤♥❡ss ❛♥❞ ❊❈▼ ✶

✶ ❙♠♦♦t❤♥❡ss Pr♦❜❛❜✐❧✐t✐❡s ✸✶✳✶ ❙♠♦♦t❤ ♥✉♠❜❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶✳✷ ❚❤❡ L ♥♦t❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶✳✸ ❙♠♦♦t❤ ♣♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✷ ❚❤❡ ❊❧❧✐♣t✐❝ ❈✉r✈❡ ▼❡t❤♦❞ ♦❢ ❢❛❝t♦r✐③❛t✐♦♥ ✾✷✳✶ ❊❧❧✐♣t✐❝ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷✳✷ ❚❤❡ ❊❈▼ ❛❧❣♦r✐t❤♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷✳✷✳✶ ❈♦♠♣❧❡①✐t② ❛♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✷✳✸ ❈❧❛ss✐❝❛❧ ✐♠♣r♦✈❡♠❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✷✳✸✳✶ ❆r✐t❤♠❡t✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✷✳✸✳✷ ❚♦rs✐♦♥ ♣♦✐♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✸ ❋✐♥❞✐♥❣ ❊❈▼✲❢r✐❡♥❞❧② ❝✉r✈❡s ✷✺✸✳✶ ●❛❧♦✐s ♣r♦♣❡rt✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✸✳✶✳✶ ❚♦rs✐♦♥ ♣r♦♣❡rt✐❡s ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺✸✳✶✳✷ ❊✛❡❝t✐✈❡ ❝♦♠✲

♣✉t❛t✐♦♥s ♦❢ Q(E[m]) ❛♥❞ ρm(Gal(Q(E[m])/Q)) ❢♦r ♣r✐♠❡♣♦✇❡rs✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✸✳✶✳✸ ❉✐✈✐s✐❜✐❧✐t② ❜② ❛ ♣r✐♠❡ ♣♦✇❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶✸✳✷ ❆♣♣❧✐❝❛t✐♦♥s t♦ s♦♠❡ ❢❛♠✐❧✐❡s ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✸✳✷✳✶ ●❡♥❡r✐❝ ●❛❧♦✐s ❣r♦✉♣ ♦❢ ❛ ❢❛♠✐❧② ♦❢ ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺✸✳✷✳✷ ❇❡tt❡r t✇✐st❡❞ ❊❞✇❛r❞s ❝✉r✈❡s ✇✐t❤ t♦rs✐♦♥ Z/2Z× Z/4Z

✉s✐♥❣ ❞✐✈✐s✐♦♥ ♣♦❧②♥♦♠✐❛❧s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻✸✳✷✳✸ ❇❡tt❡r ❙✉②❛♠❛ ❝✉r✈❡s ❜② ❛ ❞✐r❡❝t ❝❤❛♥❣❡ ♦❢ t❤❡ ●❛❧♦✐s ❣r♦✉♣ ✸✾✸✳✷✳✹ ❈♦♠♣❛r✐s♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✸✳✸ ❙♦♠❡ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✹ ■♠♣r♦✈❡♠❡♥ts t♦ t❤❡ s♠♦♦t❤✐♥❣ ♣r♦❜❧❡♠ ✹✸✹✳✶ ❊①♣♦s✐t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✹✳✶✳✶ ❚❤❡ ❞✐r❡❝t ❛♣♣r♦❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹✹✳✶✳✷ Pr❛❝t✐❝❛❧ ✐♠♣r♦✈❡♠❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✐

✐✐ ❈❖◆❚❊◆❚❙

✹✳✷ ❙tr♦♥❣❡r s♠♦♦t❤♥❡ss r❡s✉❧ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻✹✳✸ ❙❡❧❡❝t✐♦♥ ✇✐t❤ ♦♥❡ ❛❞♠✐ss✐❜✐❧✐t② t❡st ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼✹✳✹ ❚❤❡ ❛❞♠✐ss✐❜✐❧✐t② str❛t❡❣② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾✹✳✺ ❇❡st ♣❛r❛♠❡t❡rs ✐♥ t❤❡ ❛❞♠✐ss✐❜✐❧✐t② str❛t❡❣② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

■■ ❉✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s ✐♥ ✜♥✐t❡ ✜❡❧❞s ✺✺

✺ ❇❛s✐❝ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ❛❧❣♦r✐t❤♠s ✺✼

✺✳✶ ●❡♥❡r✐❝ ❛❧❣♦r✐t❤♠s ❢♦r ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼✺✳✷ ■♥❞❡① ❈❛❧❝✉❧✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

✺✳✷✳✶ ❚❤❡ ❛❧❣♦r✐t❤♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾✺✳✷✳✷ ❆♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵✺✳✷✳✸ ■♥❞✐✈✐❞✉❛❧ ❧♦❣❛r✐t❤♠ st❛❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶

✺✳✸ ■♥❞❡① ❈❛❧❝✉❧✉s ✐♥ s♠❛❧❧ ❝❤❛r❛❝t❡r✐st✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷✺✳✹ ❚❤❡ ✐❞❡❛ ♦❢ ❤❛❧❢✲r❡❧❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

✺✳✹✳✶ ❚❤❡ ❛❧❣♦r✐t❤♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸✺✳✹✳✷ ❍❡✉r✐st✐❝ ❝♦♠♣❧❡①✐t② ❛♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹

✻ ❖✈❡r✈✐❡✇ ♦♥ ◆❋❙ ❛♥❞ ❋❋❙ ✻✼

✻✳✶ Pr❡r❡q✉✐s✐t❡s ♦♥ ❢✉♥❝t✐♦♥ ❛♥❞ ♥✉♠❜❡r ✜❡❧❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼✻✳✷ ❚❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵✻✳✸ ❉❡t❛✐❧❡❞ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ◆❋❙ st❛❣❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸

✻✳✸✳✶ P♦❧②♥♦♠✐❛❧ s❡❧❡❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹✻✳✸✳✷ ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥✿ t❤❡ s✐❡✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹✻✳✸✳✸ ❋✐❧t❡r✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼✻✳✸✳✹ ❚❤❡ ❧✐♥❡❛r ❛❧❣❡❜r❛ st❛❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼✻✳✸✳✺ ■♥❞✐✈✐❞✉❛❧ ❧♦❣❛r✐t❤♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵

✻✳✹ ❱✐rt✉❛❧ ❧♦❣❛r✐t❤♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶✻✳✹✳✶ ❉❡✜♥✐♥❣ ✈✐rt✉❛❧ ❧♦❣❛r✐t❤♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶✻✳✹✳✷ ❈♦♠♣✉t✐♥❣ ❙❝❤✐r♦❦❛✉❡r ♠❛♣s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷✻✳✹✳✸ ❯s✐♥❣ ✈✐rt✉❛❧ ❧♦❣❛r✐t❤♠s ✐♥ ◆❋❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸✻✳✹✳✹ ❘❡♠♦✈✐♥❣ t❤❡ ❝❧❛ss ♥✉♠❜❡r ❝♦♥❞✐t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹

✻✳✺ ❈♦♠♣✉t✐♥❣ ✈❛❧✉❛t✐♦♥s ❛t ♣r♦❜❧❡♠❛t✐❝ ♣r✐♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺✻✳✻ ❚❤❡ ❋✉♥❝t✐♦♥ ❋✐❡❧❞ ❙✐❡✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻

✻✳✻✳✶ ❉✐✛❡r❡♥❝❡s ✇✐t❤ ◆❋❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻✻✳✻✳✷ ❚❤❡ ❋❋❙ ❛❧❣♦r✐t❤♠✿ st❛❣❡ ❜② st❛❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵✻✳✻✳✸ ❘❡♣❧❛❝✐♥❣ ❙❝❤✐r♦❦❛✉❡r ♠❛♣s ❜② ✈❛❧✉❛t✐♦♥s ❛t ✐♥✜♥✐t② ✳ ✳ ✳ ✳ ✾✶

✼ ❖❧❞ ❛♥❞ ♥❡✇ ❝♦♠♣❧❡①✐t✐❡s ❢♦r ◆❋❙ ✾✺

✼✳✶ ❈❧❛ss✐❝❛❧ ◆❋❙ ♦✈❡r ♣r✐♠❡ ✜❡❧❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺✼✳✷ ❉✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ❢❛❝t♦r② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✼✼✳✸ ❈♦♠♣❧❡①✐t② ♦❢ ❞❡s❝❡♥t st❡♣s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾✼✳✹ ❩♦♦❧♦❣② ♦❢ ◆❋❙ ✈❛r✐❛♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶✼✳✺ ❚❤❡ ❢✉♥❝t✐♦♥ ✜❡❧❞ s✐❡✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷

❈❖◆❚❊◆❚❙ ✐✐✐

✽ ■♠♣r♦✈❡♠❡♥ts t♦ ◆❋❙ ❛♥❞ ❋❋❙ ✶✵✼✽✳✶ ❚❤❡ ❧❛tt✐❝❡ s✐❡✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼

✽✳✶✳✶ ❚❤❡ ❧❛tt✐❝❡ s✐❡✈❡ t❡❝❤♥✐q✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼✽✳✶✳✷ ❈♦♠♣✉t✐♥❣ s❤♦rt ✈❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✾✽✳✶✳✸ ❊✈❛❧✉❛t✐♥❣ t❤❡ s♣❡❡❞✲✉♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵

✽✳✷ P❛r❛❧❧❡❧✐③❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡❛r ❛❧❣❡❜r❛ st❛❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷✽✳✷✳✶ ●❡♥❡r❛❧ ❝♦♥s✐❞❡r❛t✐♦♥s ♦♥ t❤❡ ♣❛r❛❧❧❡❧✐s♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷✽✳✷✳✷ ❇❧♦❝❦ ❲✐❡❞❡♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷

✽✳✸ P♦❧②♥♦♠✐❛❧ s❡❧❡❝t✐♦♥ ❢♦r ♥♦♥✲♣r✐♠❡ ✜❡❧❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✹✽✳✸✳✶ ❚❤❡ ♠❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✺✽✳✸✳✷ ❆♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✻✽✳✸✳✸ ❈♦♠♣❛r✐s♦♥ ✇✐t❤ ♦t❤❡r ♠❡t❤♦❞s ♦❢ ♣♦❧②♥♦♠✐❛❧ s❡❧❡❝t✐♦♥ ✳ ✳ ✳ ✶✶✻

✽✳✹ ❙♠♦♦t❤✐♥❣ ✇✐t❤ t✇♦ ♥♦♥✲❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✼✽✳✹✳✶ ■♥❡✣❝✐❡♥❝② ♦❢ t❤❡ ♥❛✐✈❡ ❛♣♣r♦❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✽✽✳✹✳✷ ❘❛t✐♦♥❛❧ r❡❝♦♥str✉❝t✐♦♥ ♦✈❡r ♥✉♠❜❡r ✜❡❧❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✽✽✳✹✳✸ ❚❤❡ ❡✛❡❝t ♦❢ r❛t✐♦♥❛❧ r❡❝♦♥str✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾

✾ ❙❡❧❡❝t✐♥❣ ♣♦❧②♥♦♠✐❛❧s ❢♦r ❋❋❙ ✶✷✶✾✳✶ ■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✶✾✳✷ ◗✉❛♥t✐✜❝❛t✐♦♥ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✷

✾✳✷✳✶ ❙✐③❡ ♣r♦♣❡rt② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✷✾✳✷✳✷ ❘♦♦t ♣r♦♣❡rt② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✹✾✳✷✳✸ ❈❛♥❝❡❧❧❛t✐♦♥ ♣r♦♣❡rt②✲▲❛✉r❡♥t r♦♦ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✾

✾✳✸ ❈♦♠❜✐♥✐♥❣ s✐③❡✱ r♦♦t ❛♥❞ ❝❛♥❝❡❧❧❛t✐♦♥ ♣r♦♣❡rt✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✸✾✳✸✳✶ ❆❞❛♣t✐♥❣ ▼✉r♣❤②✬s E t♦ t❤❡ ❋❋❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✸✾✳✸✳✷ ❊①♣❡r✐♠❡♥t❛❧ ✈❛❧✐❞❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✺✾✳✸✳✸ ❈♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ f ❛♥❞ g ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✻✾✳✸✳✹ ❆ s✐❡✈✐♥❣ ♣r♦❝❡❞✉r❡ ❢♦r ❛❧♣❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✼

✾✳✹ ■♥s❡♣❛r❛❜❧❡ ♣♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✽✾✳✹✳✶ P❛rt✐❝✉❧❛r✐t✐❡s ♦❢ t❤❡ ✐♥s❡♣❛r❛❜❧❡ ♣♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✽✾✳✹✳✷ ❙♣❡❡❞✲✉♣ ✐♥ t❤❡ ❋❋❙ ❞✉❡ t♦ t❤❡ ✐♥s❡♣❛r❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✾✾✳✹✳✸ ❘♦♦t ♣r♦♣❡rt② ♦❢ ✐♥s❡♣❛r❛❜❧❡ ♣♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✵

✾✳✺ ❆♣♣❧✐❝❛t✐♦♥s t♦ s♦♠❡ ❡①❛♠♣❧❡s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✷✾✳✺✳✶ ❚❤♦♠é✬s r❡❝♦r❞ ✉s✐♥❣ t❤❡ ❈♦♣♣❡rs♠✐t❤ ❛❧❣♦r✐t❤♠ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✷✾✳✺✳✷ ❏♦✉①✲▲❡r❝✐❡r✬s ❝❧❛ss✐❝❛❧ ❋❋❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✷✾✳✺✳✸ ❏♦✉①✲▲❡r❝✐❡r✬s t✇♦ r❛t✐♦♥❛❧ s✐❞❡ ✈❛r✐❛♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✸✾✳✺✳✹ ❘❡❝♦r❞s ♦♥ ♣❛✐r✐♥❣✲❢r✐❡♥❞❧② ❝✉r✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✸

✾✳✻ ❈♦♥❝❧✉s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✺

✶✵ ❆ q✉❛s✐✲♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ ✶✹✼✶✵✳✶ ❘❡❝❡♥t ❉▲P ♣r♦❣r❡ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✼✶✵✳✷ ❖✉r r❡s✉❧ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✾✶✵✳✸ ❙❡tt✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✵✶✵✳✹ ▲♦❣❛r✐t❤♠s ♦❢ t❤❡ ❢❛❝t♦r ❜❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✵✶✵✳✺ ▼❛✐♥ r❡s✉❧t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✶✶✵✳✻ ❈♦♥s❡q✉❡♥❝❡s ❢♦r ✈❛r✐♦✉s r❛♥❣❡s ♦❢ ♣❛r❛♠❡t❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✷✶✵✳✼ ❆❧❣♦r✐t❤♠ ❢♦r ♦♥❡ ❞❡s❝❡♥t st❡♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✸

✐✈ ❈❖◆❚❊◆❚❙

✶✵✳✽ ❙✉♣♣♦rt✐♥❣ t❤❡ ❤❡✉r✐st✐❝ ❛r❣✉♠❡♥t ✐♥ t❤❡ ♣r♦♦❢ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✹✶✵✳✾ ❆♥ ✐♠♣r♦✈❡♠❡♥t ❜❛s❡❞ ♦♥ ❛❞❞✐t✐♦♥❛❧ ❤❡✉r✐st✐❝s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✻✶✵✳✶✵ ❈♦♥❝❧✉s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✻

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s ✶✺✾

❘és✉♠é ✶✻✷

❇✐❜❧✐♦❣r❛♣❤② ✶✼✶

❈❖◆❚❊◆❚❙ ✈

✈✐ ❈❖◆❚❊◆❚❙

■♥tr♦❞✉❝t✐♦♥

✈✐✐

✈✐✐✐ ■◆❚❘❖❉❯❈❚■❖◆

▼♦t✐✈❛t✐♦♥

❈r②♣t♦❣r❛♣❤② ✐s ❝♦♥❝❡r♥❡❞ ✇✐t❤ s❡❝✉r❡ ❝♦♠♠✉♥✐❝❛t✐♦♥s ❜❡t✇❡❡♥ t✇♦ ❡♥t✐t✐❡s✱ ❢♦r❡①❛♠♣❧❡ t✇♦ ❝♦♠♣✉t❡rs ❝♦♥♥❡❝t❡❞ t♦ t❤❡ ■♥t❡r♥❡t✳ ❚❤❡ ❡❛s✐❡st ♠❡t❤♦❞ r❡q✉✐r❡st❤❛t t❤❡ t✇♦ ❡♥t✐t✐❡s ❤❛✈❡ ♣r❡✈✐♦✉s❧② ❡①❝❤❛♥❣❡❞ ❛ s❡❝r❡t ❦❡② ✇❤✐❝❤ ❛❧❧♦✇s t❤❡♠ t♦❡♥❝r②♣t ❛♥❞ t❤❡♥ t♦ ❞❡❝r②♣t t❤❡ ♠❡ss❛❣❡✳ ❲❤❡♥ t❤✐s ✐s ♥♦t ♣♦ss✐❜❧❡✱ ♦♥❡ ✉s❡s ❛♠❡t❤♦❞ ✇❤✐❝❤ ❛❧❧♦✇s t❤❡ t✇♦ ❡♥t✐t✐❡s t♦ ❛❣r❡❡ ♦♥ ❛ ❝♦♠♠♦♥ s❡❝r❡t ❦❡② ✇❤✐❧❡ ✉s✐♥❣❛♥ ✐♥s❡❝✉r❡ ❝❤❛♥♥❡❧✳ ❚❤✐s ✐❞❡❛✱ ✇❤✐❝❤ ❣♦❡s ❜❛❝❦ t♦ ❉✐✣❡ ❛♥❞ ❍❡❧❧♠❛♥ ✐♥ 1976✐s t❤❡ ❜❛s✐s ♦❢ ♣✉❜❧✐❝ ❦❡② ❝r②♣t♦❣r❛♣❤②✳ ■ts ♠❛✐♥ t♦♦❧✱ t❤❡ ♦♥❡✲✇❛② ❢✉♥❝t✐♦♥s✱ ❛r❡♠❛t❤❡♠❛t✐❝❛❧ ❢✉♥❝t✐♦♥s ✇❤✐❝❤ ❛r❡ ❡❛s② t♦ ❝♦♠♣✉t❡ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥ ❛♥❞ ❤❛r❞ ✐♥ t❤❡♦t❤❡r✳ ❚❤❡ ✜rst ❡①❛♠♣❧❡ ❬❉❍✼✻❪ ♦❢ s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ✇❛s t❤❡ ❡①♣♦♥❡♥t✐❛t✐♦♥ ♦❢ ❛♥✐♥t❡❣❡r ♠♦❞✉❧♦ ❛ ♣r✐♠❡ ❜❡❝❛✉s❡ ✐ts ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥✱ ❝❛❧❧❡❞ t❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠✱s❡❡♠❡❞ t♦ t❛❦❡ ❛♥ ❡①♣♦♥❡♥t✐❛❧ t✐♠❡✳ ■t ✐s t❤❡ ❞✐✣❝✉❧t② ♦❢ t❤✐s ♣r♦❜❧❡♠ t❤❛t ✇❡❛♥❛❧②③❡ ✐♥ t❤✐s t❤❡s✐s✳

❚❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✐s ❝✉rr❡♥t❧② ✐♥ ✉s❡✱ ❜✉t ♦t❤❡r ♦♥❡✲✇❛② ❢✉♥❝t✐♦♥s ❛r❡♠♦r❡ ♣♦♣✉❧❛r✳ ❚❤❡ ♠♦st ♥♦t✐❝❡❛❜❧❡ ❛r❡ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ✐♥t❡❣❡rs ❬❘❙❆✼✽❪✱❤❛✈✐♥❣ ❛s ✐♥✈❡rs❡ t❤❡ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥✱ ❛♥❞ t❤❡ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♥ ❛♥❡❧❧✐♣t✐❝ ❝✉r✈❡ ❬▼✐❧✽✻✱ ❑♦❜✽✼❪✱ ✇❤♦s❡ ✐♥✈❡rs❡ ✐s ❝❛❧❧❡❞ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❞✐s❝r❡t❡❧♦❣❛r✐t❤♠✳ ❆♥ ✉♥❡①♣❧❛✐♥❡❞ ❢❛❝t ✐s t❤❛t ❡✈❡r② t✐♠❡ ❛♥ ❛❧❣♦r✐t❤♠✐❝ ✐♠♣r♦✈❡♠❡♥t ❤❛s❜❡❡♥ ♠❛❞❡ ✐♥ t❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ♣r♦❜❧❡♠ ✭❉▲P✮ ✐t ❤❛s ❜❡❡♥ tr❛♥s❧❛t❡❞ t♦ t❤❡❢❛❝t♦r✐♥❣ ♣r♦❜❧❡♠ ❛♥❞ ✈✐❝❡✲✈❡rs❛✱ ♠❛❦✐♥❣ ✐t ✐♥t❡r❡st✐♥❣ t♦ t❛❝❦❧❡ t❤❡ t✇♦ ♣r♦❜❧❡♠st♦❣❡t❤❡r✳ ❚❤❡ ❝❛s❡ ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❝r②♣t♦❣r❛♣❤② ✐s ❞✐✛❡r❡♥t ❜❡❝❛✉s❡✱ ❡①❝❡♣t ❢♦rs♦♠❡ ✇❡❛❦ ❝❛s❡s✱ t❤❡ ❜❡st ❛❧❣♦r✐t❤♠s ❦♥♦✇♥ ❛r❡ ❡①♣♦♥❡♥t✐❛❧✳ ◆❡✈❡rt❤❡❧❡ss ✐t ✐s❛❧s♦ ❛ ♠♦t✐✈❛t✐♦♥ ❢♦r ✉s ❜❡❝❛✉s❡✱ t❤❡ t✇♦ t②♣❡s ♦❢ ❛tt❛❝❦s ♦♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ❛r❡❡✐t❤❡r ✐♥s♣✐r❡❞ ❢r♦♠ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ❛❧❣♦r✐t❤♠s ❬●❍❙✵✷❪ ♦r ❝♦♥s✐st ✐♥ r❡❞✉❝✐♥❣t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ t♦ t❤❡ ❝❧❛ss✐❝❛❧ ♣r♦❜❧❡♠ ♦❢ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✐♥✜♥✐t❡ ✜❡❧❞s ❬▼❖❱✾✸✱ ❋❘✾✹❪✳ ❋♦r ❡①❛♠♣❧❡ ♦♥❡ ❝❛♥ s♦❧✈❡ t❤❡ ❉▲P ♦♥ s✉♣❡r✲s✐♥❣✉❧❛r❡❧❧✐♣t✐❝ ❝✉r✈❡s ❞❡✜♥❡❞ ♦✈❡r F2n ❛♥❞ F3n ✐❢ ♦♥❡ ❝♦♠♣✉t❡s ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s ✐♥ F24·n❛♥❞✱ r❡s♣❡❝t✐✈❡❧②✱ F36·n ✳

❲❤❡♥ ❝♦♠♣❛r✐♥❣ ✈❛r✐♦✉s ❛❧❣♦r✐t❤♠s ❢♦r t❤❡ ❉▲P ✐t ✐s ❝♦♥✈❡♥✐❡♥t t♦ ❞❡✜♥❡ t❤❡❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥✿

Lx(α, c) = exp(c(log x)α(log log x)1−α

),

✇❤❡r❡ 0 ≤ α ≤ 1 ❛♥❞ c > 0✳ ❋♦r ❡①❛♠♣❧❡ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❛❧❣♦r✐t❤♠s t❛❦❡ ❛t✐♠❡ Lx(1, c) ❢♦r s♦♠❡ ❝♦♥st❛♥t c✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠s ♦❢❝♦♠♣❧❡①✐t② (log x)k ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s Lx(0, k)✳

❈♦♠♣✉t✐♥❣ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s ✐♥ ❛ ♣r✐♠❡ ✜❡❧❞ Fp ❤❛s ❛❧r❡❛❞② ❜❡❡♥ ❛❞❞r❡ss❡❞❜② ❑r❛✐t❝❤✐❦ ✐♥ ❬❑r❛✷✷❪✳ ❚❤❡ ❦❡② ♥♦t✐♦♥ ✇❛s t❤❛t ♦❢ s♠♦♦t❤ ♥✉♠❜❡rs✿ ❛♥ ✐♥t❡❣❡r ✐sB✲s♠♦♦t❤ ✐❢ ❛❧❧ ✐ts ♣r✐♠❡ ❞✐✈✐s♦rs s♠❛❧❧❡r t❤❛♥ B✳ ❚❤✐s ♥♦t✐♦♥ ✇❛s t❤❡♥ ✉s❡❞ ✐♥ t❤❡✇♦r❧❞ ♦❢ ❢❛❝t♦r✐③❛t✐♦♥ ✇❤❡r❡ ▲❡❤♠❡r ❛♥❞ P♦✇❡rs ❬▲P✸✶❪ ♣r♦♣♦s❡❞ ❛ ♠❡t❤♦❞ ✇❤✐❝❤♣r♦✈❡❞ t♦ ❜❡ ✈❡r② ❡✛❡❝t✐✈❡ ✐♥ ❬▼❇✼✺❪✳ ■ts ❝♦♠♣❧❡①✐t② ✇❛s LN(1/2, c)✱ c > 0✱ ❜✉tt❤❡ t♦♦❧s t♦ ♣r♦✈❡ ✐t ✇❡r❡ ♦♥❧② ❞❡s✐❣♥❡❞ ❛ ❝♦✉♣❧❡ ♦❢ ②❡❛rs ❧❛t❡r ❬❈❊P✽✸❪✳ ❚❤❡ ❜❡st❛❧❣♦r✐t❤♠ ❢♦r ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ✐♥ 1976 ✇❛s ❙❤❛♥❦s✬❜❛❜②✲st❡♣✲❣✐❛♥t✲st❡♣ ♦❢ ❝♦♠♣❧❡①✐t②

√p = Lp(1,

12)✱ ❛s ✐t ✐s ♥♦t❡❞ ✐♥ ❬❉❍✼✻❪✳

❚❤❡ ✜rst s✉❜✲❡①♣♦♥❡♥t✐❛❧ ❛❧❣♦r✐t❤♠ ❢♦r ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s✱ ■♥❞❡① ❈❛❧❝✉❧✉s✱✐s ✐♥s♣✐r❡❞ ❢r♦♠ ❑r❛✐t❝❤✐❦✬s ♠❡t❤♦❞ ❛♥❞ ✇❛s ✐♥❞❡♣❡♥❞❡♥t❧② ❞✐s❝♦✈❡r❡❞ ❜② t✇♦

■◆❚❘❖❉❯❈❚■❖◆ ✐①

t❡❛♠s ❬❆❞❧✼✾❪ ❛♥❞ ❬P♦❤✼✼❪✳ ■t ❤❛s ❛ ❝♦♠♣❧❡①✐t② ♦❢ t②♣❡ Lp(1/2, ·)✱ ✇❤✐❝❤ ✐s ❛s❢❛st ❛s t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❦♥♦✇♥ ❛t t❤❛t t✐♠❡✳ ❲❡ st❛rt ❜② ❝❤♦♦s✐♥❣ ❛♥✐♥t❡❣❡r B > 0 ❛♥❞ ❜② ♠❛❦✐♥❣ ❛ ❧✐st ♦❢ ❛❧❧ t❤❡ ♣r✐♠❡s ❧❡ss t❤❛♥ B✱ ✇❤♦s❡ s❡t ✐s❝❛❧❧❡❞ t❤❡ ❢❛❝t♦r ❜❛s❡✳ ❚❤❡ ✜rst st❛❣❡ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ❝♦♥s✐sts ✐♥ ♣✐❝❦✐♥❣ r❛♥❞♦♠♥✉♠❜❡rs ❢r♦♠ ❛ ❧✐st ❛♥❞ t❡st✐♥❣ t❤❡✐r B✲s♠♦♦t❤♥❡ss✱ ✉♥t✐❧ ♦♥❡ ❝♦❧❧❡❝t B ♥✉♠❜❡rs✳❲❡ ✇✐❧❧ s❡❡ ❤♦✇ ❡❛❝❤ s♠♦♦t❤ ♥✉♠❜❡r ♣r♦❞✉❝❡s ❛ ❧✐♥❡❛r ❡q✉❛t✐♦♥ ❛♠♦♥❣ t❤❡ ❞✐s❝r❡t❡❧♦❣❛r✐t❤♠s ♦❢ t❤❡ ♣r✐♠❡s ✐♥ t❤❡ ❢❛❝t♦r ❜❛s❡✳ ❆ s❡❝♦♥❞ st❡♣ ❝♦♥s✐sts ✐♥ s♦❧✈✐♥❣ ❛ ❧❛r❣❡❧✐♥❡❛r s②st❡♠✱ ✇❤✐❝❤ ❣✐✈❡s ✉s t❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s ♦❢ t❤❡ ❢❛❝t♦r ❜❛s❡ ❡❧❡♠❡♥ts✳ ❆t❤✐r❞ st❛❣❡ ❝❛❧❧❡❞ ✐♥❞✐✈✐❞✉❛❧ ❧♦❣❛r✐t❤♠ st❛❣❡ ❡①♣r❡ss❡s t❤❡ ❞❡s✐r❡❞ ❧♦❣❛r✐t❤♠ ✇✐t❤r❡s♣❡❝t t♦ t❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s ♦❢ t❤❡ ❢❛❝t♦r ❜❛s❡ ❡❧❡♠❡♥ts✳

▼❛♥② ❛❧❣♦r✐t❤♠s ✇❤✐❝❤ ❢♦❧❧♦✇❡❞ ❤❛✈❡ t❤❡ s❛♠❡ ♠❛✐♥ st❛❣❡s✱ ✐♥ ♣❛rt✐❝✉❧❛r t❤❡♠♦❞❡r♥ ❛❧❣♦r✐t❤♠s ♦❢ ❝♦♠♣❧❡①✐t② L(1/3, ·)✳ ■♥ t❤❡s❡ ❛❧❣♦r✐t❤♠s ♦♥❡ ❤❛s ❛♥ ❛❞❞✐✲t✐♦♥❛❧ st❛❣❡ ✇❤✐❝❤ s❡❧❡❝ts t✇♦ ❛♣♣r♦♣r✐❛t❡ ♣♦❧②♥♦♠✐❛❧s f ❛♥❞ g ✐♥ Q[x] ♦r Fq[t][x]❛❝❝♦r❞✐♥❣ t♦ t❤❡ t②♣❡ ♦❢ ✜♥✐t❡ ✜❡❧❞s ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥✿ ❧❛r❣❡ ♦r s♠❛❧❧ ❝❤❛r❛❝t❡r✲✐st✐❝✳ ❚❤❡ ❢♦✉r st❛❣❡s t♦ ❦❡❡♣ ✐♥ ♠✐♥❞ ❛r❡ t❤❡♥✿

• P♦❧②♥♦♠✐❛❧ s❡❧❡❝t✐♦♥✳ ◆♦t ♣r❡s❡♥t ✐♥ ❛❧❧ t❤❡ ❛❧❣♦r✐t❤♠s✱ t❤✐s st❛❣❡ ❝♦rr❡✲s♣♦♥❞s t♦ t❤❡ s❡❧❡❝t✐♦♥ ♦❢ t✇♦ ♣♦❧②♥♦♠✐❛❧s f ❛♥❞ g s✉❜❥❡❝t t♦ ❛ s❡t ♦❢ ❝♦♥✲❞✐t✐♦♥s✳

• ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥✳ ❲❡ ❝♦❧❧❡❝t ❛ ❧✐st ♦❢ ♥✉♠❜❡rs✱ r❡s♣❡❝t✐✈❡❧②✱ ♦❢ ♣♦❧②♥♦♠✐❛❧s✇❤✐❝❤ ❛r❡ s♠♦♦t❤✳ ❋♦r ❡①❛♠♣❧❡ ✇❡ ❝♦❧❧❡❝t ♣❛✐rs ♦❢ ✐♥t❡❣❡rs (a, b) s✉❝❤ t❤❛tF (a, b) ❛♥❞ G(a, b) ❛r❡ B✲s♠♦♦t❤ ✇❤❡r❡ F (x, y) = ydeg ff(x/y) ❛♥❞ G(x, y) =ydeg gg(x/y)✳

• ▲✐♥❡❛r ❛❧❣❡❜r❛ st❛❣❡✳ ❲❡ s♦❧✈❡ ❛ ❧❛r❣❡ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✇✐t❤ ❝♦❡❢✲✜❝✐❡♥ts ♠♦❞✉❧♦ t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ❣r♦✉♣✳

• ■♥❞✐✈✐❞✉❛❧ ❧♦❣❛r✐t❤♠✳ ❲❡ ✉s❡ t❤❡ ♣r❡✈✐♦✉s r❡s✉❧t t♦ ❝♦♠♣✉t❡ t❤❡ ❞❡s✐r❡❞❧♦❣❛r✐t❤♠✳ ◆♦t❡ t❤❛t ✇❡ ❞♦ ♥♦t ♥❡❡❞ t♦ r❡♣❡❛t t❤❡ ♣r❡✈✐♦✉s st❛❣❡s ✐❢ ♠♦r❡t❤❛♥ ♦♥❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✐s ♥❡❡❞❡❞✳

■♥ ❛❧❣♦r✐t❤♠s ♦❢ L(1/3) ❝♦♠♣❧❡①✐t② ❬●♦r✾✸✱ ❙❝❤✾✸✱ ❏▲✵✸✱ ❏▲❙❱✵✻❪✱ t❤❡ s♠♦♦t❤✲♥❡ss t❡sts ❝♦♥t✐♥✉❡ t♦ ♣❧❛② ❛♥ ✐♠♣♦rt❛♥t r♦❧❡✳ ❖♥ t❤❡ ♦♥❡ ❤❛♥❞✱ ❛❧t❤♦✉❣❤ t❤❡t❤❡♦r❡t✐❝❛❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ✉s❡s ❛ t❡❝❤♥✐q✉❡ ❝❛❧❧❡❞ s✐❡✈✐♥❣✱ ✐♥ ♣r❛❝t✐❝❡❞✉❡ t♦ t❤❡ ♠❡♠♦r② ❝♦♥str❛✐♥ts ♦♥❡ ♣r♦❝❡❡❞s ✐♥ t✇♦ st❡♣s✿ ♦♥❡ ❝♦❧❧❡❝ts ♣❛✐rs ✇❤✐❝❤❛r❡ ❧✐❦❡❧② t♦ ❜❡ s♠♦♦t❤ ❛♥❞ t❤❡♥ ♦♥❡ t❡sts t❤❡✐r s♠♦♦t❤♥❡ss✳

❚❤❡ ❢❛st❡st s♠♦♦t❤♥❡ss t❡st ❦♥♦✇♥ t♦❞❛② ✐s ▲❡♥str❛✬s ❬▲❡♥✽✼❪ ❡❧❧✐♣t✐❝ ❝✉r✈❡♠❡t❤♦❞ ♦❢ ❢❛❝t♦r✐③❛t✐♦♥ ✭❊❈▼✮✳ ❚❤✐s ❤❡✉r✐st✐❝ ❛❧❣♦r✐t❤♠ ❤❛s ❛ ♣r♦✈❡♥ ❝♦✉♥t❡r♣❛rt✉s✐♥❣ ❤②♣❡r❡❧❧✐♣t✐❝ ❝✉r✈❡s ✭❍❊❈▼✮✱ ❜✉t ❊❈▼ s❡❡♠s t♦ ❤❛✈❡ ❜❡tt❡r ♣❡r❢♦r♠❛♥❝✐❡s✐♥ ♣r❛❝t✐❝❡ ❬❄❪✳ ❊❈▼ ❤❛s ❜❡❡♥ t❤❡ ♦❜❥❡❝t ♦❢ ♠❛♥② ✐♠♣r♦✈❡♠❡♥ts✱ t❤❡ ♠♦st ♥♦✲t✐❝❡❛❜❧❡ ❜❡✐♥❣ t❤❡ ❙t❛❣❡ 2 ❝♦♥t✐♥✉❛t✐♦♥✱ t❤❡ ❝✉r✈❡ ❛r✐t❤♠❡t✐❝ ❛❝❝❡❧❡r❛t✐♦♥ ❛♥❞ t❤❡s❡❧❡❝t✐♦♥ ♦❢ ❝✉r✈❡s ✇✐t❤ ❧❛r❣❡r t♦rs✐♦♥ ♦✈❡r Q✳ ❍❡♥❝❡ ✐t ✇❛s ♣r♦♣♦s❡❞ t♦ ♣✉t t❤❡❡❧❧✐♣t✐❝ ❝✉r✈❡s ✐♥ ♥❡✇ ❢♦r♠s s♦ t❤❛t t❤❡ ❝✉r✈❡ ❛r✐t❤♠❡t✐❝ ♥❡❡❞s ❢❡✇❡r ✜❡❧❞ ♦♣❡r❛✲t✐♦♥s ❬▼♦♥✾✷❪✱❬❇▲✵✼✱ ❇❇▲P✶✸❪✳ ❆ ❞✐✛❡r❡♥t ❞✐r❡❝t✐♦♥ ♦❢ ✐♠♣r♦✈❡♠❡♥t ✇❛s t♦ s❡❧❡❝t❝✉r✈❡s ✇✐t❤ ❜❡tt❡r s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t✐❡s ❬❙✉②✽✺❪✱ ❬❆▼✾✸❪✱ ❬❇❇▲✶✵❪✳ ❚❤✐s ♠♦t✐✈❛t❡❞✉s t♦ ✜♥❞ ❛ ✉♥✐✜❡❞ t❡❝❤♥✐q✉❡ ✇❤✐❝❤ s❡❧❡❝ts ❣♦♦❞ ❝✉r✈❡s ♦♥ ❛♥② ❢❛♠✐❧② ♦❢ ❡❧❧✐♣t✐❝❝✉r✈❡s ❛♥❞ t❤❡♥ t♦ ❞❡♠♦♥str❛t❡ ✐ts ❡✣❝✐❡♥❝② ✇✐t❤ ❝♦♥❝r❡t❡ ❡①❛♠♣❧❡s✳

① ■◆❚❘❖❉❯❈❚■❖◆

▲❡t ✉s r❡t✉r♥ t♦ t❤❡ ♠❛✐♥ t♦♣✐❝ ♦❢ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s✳ ❖♥❡ t❛❝❦❧❡s t❤❡ ✜❡❧❞s Fp✇✐t❤ p ♣r✐♠❡ ✉s✐♥❣ t❤❡ ♥✉♠❜❡r ✜❡❧❞ s✐❡✈❡ ✭◆❋❙✮ ❬▲▲✾✸✱ ●♦r✾✸✱ ❙❝❤✾✸✱ ❏▲✵✸✱ ❙❝❤✵✺✱❈❙✵✻❪✳ ❱❡r② s✐♠✐❧❛r ✐s t❤❡♥ t❤❡ ❝❛s❡ ♦❢ ✜❡❧❞s ♦❢ s♠❛❧❧ ❝❤❛r❛❝t❡r✐st✐❝ ✇❤❡r❡ ♦♥❡ ✉s❡st❤❡ ❢✉♥❝t✐♦♥ ✜❡❧❞ s✐❡✈❡ ✭❋❋❙✮ ❬❆❞❧✾✹✱ ❆❍✾✾✱ ❏▲✵✷❪✳ ❚❤❡ ✐♥t❡r♠❡❞✐❛t❡ ❝❛s❡✱ ❝❛❧❧❡❞t❤❡ ♠✐❞❞❧❡ ♣r✐♠❡ ❝❛s❡✱ r❡♠❛✐♥❡❞ ❧❡ss ✉♥❞❡rst♦♦❞ ❢♦r ♦✈❡r ❛ ❞❡❝❛❞❡✳ ❋✐rst ✐t ❤❛s❜❡❡♥ s❤♦✇♥ t❤❛t ❋❋❙ ❡①t❡♥❞s t♦ ❛ s❤❛r♣ ❞♦♠❛✐♥ ♦❢ t❤❡ ♠✐❞❞❧❡ ♣r✐♠❡ ❝❛s❡ ❬❏▲✵✻❪✳❚❤❡♥ ◆❋❙ ✇❛s ❡①t❡♥❞❡❞ t♦ ❛ ❧❛r❣❡ ❞♦♠❛✐♥ ♦❢ ✜❡❧❞s ♦❢ ❧❛r❣❡ ❝❤❛r❛❝t❡r✐st✐❝ ❬❏▲❙❱✵✻❪❛♥❞✱ ✐♥ t❤❡ s❛♠❡ ❛rt✐❝❧❡✱ t❤❡ r❡♠❛✐♥✐♥❣ ✜❡❧❞s ✇❡r❡ ❛tt❛❝❦❡❞ ❜② ❛ ♥❡✇ ✈❛r✐❛♥t ♦❢◆❋❙✳ ❍❡♥❝❡ ✇❡ ❝❛♥ ♥♦✇ ❝♦♠♣✉t❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s ✐♥ ❛♥② ✜♥✐t❡ ✜❡❧❞ FQ ✐♥t✐♠❡ LQ(1/3, c)✱ c > 0✳ ■♥ t❤✐s t❤❡s✐s ✇❡ ❤❛✈❡ s❡❛r❝❤❡❞ ♦♥ t❤❡ ♦♥❡ ❤❛♥❞ ❢♦r✐♠♣r♦✈❡♠❡♥ts ✐♥ ❛❧❧ t❤❡s❡ ❛❧❣♦r✐t❤♠s ❛♥❞ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ t♦ ✉♥❞❡rst❛♥❞ t❤❡✐♥t❡r❛❝t✐♦♥s ❜❡t✇❡❡♥ ◆❋❙ ❛♥❞ ❋❋❙✳

❚❤❡ ❋❋❙ ❛❧❣♦r✐t❤♠ ♠✐❣❤t s❡❡♠ ❧❡ss ✐♥t❡r❡st✐♥❣ t❤❛♥ ◆❋❙ ❜❡❝❛✉s❡ ✐♥ ❝r②♣t♦❣✲r❛♣❤② ♦♥❡ ❤❛s ❛✈♦✐❞❡❞ t❤❡ ✜♥✐t❡ ✜❡❧❞s ♦❢ s♠❛❧❧ ❝❤❛r❛❝t❡r✐st✐❝✳ ■♥❞❡❡❞✱ ✐♥ 1984✇❤❡♥ ■♥❞❡① ❈❛❧❝✉❧✉s ✇❛s t❤❡ ❜❡st ❛❧❣♦r✐t❤♠ ❢♦r ✜❡❧❞s Fp ✇✐t❤ ❛ ❝♦♠♣❧❡①✐t② ♦❢Lp(1/2, c)✱ c > 0✱ ❈♦♣♣❡rs♠✐t❤ ♣✉❜❧✐s❤❡❞ ❛♥ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ s♦❧✈❡s t❤❡ ❉▲P ✐♥❜✐♥❛r② ✜❡❧❞s FQ ✐♥ t✐♠❡ LQ(1/3, c′)✱ c′ > 0❀ t❤❡② ✇❡r❡ ❤❡♥❝❡ ♠✉❝❤ ✇❡❛❦❡r t❤❛♥ t❤❡♣r✐♠❡ ✜❡❧❞s ❬❈♦♣✽✹❪✳ ◆♦✇❛❞❛②s ❛❧❧ t❤❡ ✜♥✐t❡ ✜❡❧❞s ❛s ✇❡❧❧ ❛s t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ❝❛♥❜❡ t❛❝❦❧❡❞ ✇✐t❤ ❛❧❣♦r✐t❤♠s ♦❢ t②♣❡ L(1/3, ·)✳ ❋✉rt❤❡r♠♦r❡✱ ❋❋❙ r❡❣❛✐♥❡❞ ✐♠♣♦r✲t❛♥❝❡ ✐♥ 2000 ✇✐t❤ t❤❡ ✐♥✈❡♥t✐♦♥ ♦❢ ♣❛✐r✐♥❣ ❜❛s❡❞ ❝r②♣t♦❣r❛♣❤②✱ tr❛♥s❢♦r♠✐♥❣ t❤❡♣❛✐r✐♥❣s ❢r♦♠ ❛♥ ❛tt❛❝❦ ✐♥t♦ ❛ ❝r②♣t♦❣r❛♣❤✐❝ t♦♦❧ ❬❏♦✉✵✵❪✳ ■♥ t❤❡ ❧❛st ❢❡✇ ②❡❛rs✱ ✐t❤❛s ❜❡❡♥ ❛♥ ❛❝t✐✈❡ ❛r❡❛ ♦❢ r❡s❡❛r❝❤ t♦ ❛❝❝❡❧❡r❛t❡ t❤❡ ❋❋❙ ❛❧❣♦r✐t❤♠✱ ✇✐t❤ ❛ s♣❡✲❝✐❛❧ ❝♦♥❝❡r♥ ♦♥ ♣r❛❝t✐❝❛❧ ❡✣❝✐❡♥❝② ❬❍❙❲+✶✵✱ ❍❙❙❚✶✷❪✳ ❲❡ ❤❛✈❡ ❢♦❝✉s❡❞ ♦♥ t❤❡♣♦❧②♥♦♠✐❛❧ s❡❧❡❝t✐♦♥ ❛♥❞ s❤♦✇❡❞ t❤❛t ❛ ❣♦♦❞ ❝❤♦✐❝❡ ❝❛♥ ❞✐✈✐❞❡ t❤❡ ♦✈❡r❛❧❧ t✐♠❡ ❜②❛ ❢❛❝t♦r ♦❢ 2✳

❘❡❝❡♥t❧②✱ ❛ s✉r♣r✐s✐♥❣ ❜r❡❛❦t❤r♦✉❣❤ ✇❛s ♠❛❞❡ ❜② ❏♦✉① ✐♥ ❬❏♦✉✶✸❛❪✳ ❲❤✐❧❡❦❡❡♣✐♥❣ t❤❡ s❡tt✐♥❣ ♦❢ ❋❋❙✱ ❤❡ s❤♦✇❡❞ t❤❛t ✐♥ t❤❡ ♠✐❞❞❧❡ ♣r✐♠❡ ❝❛s❡ ♦♥❡ ❝❛♥ r❡❞✉❝❡t❤❡ ❝♦st ♦❢ t❤❡ r❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ st❛❣❡ ✉s✐♥❣ ❛ t❡❝❤♥✐q✉❡ ❝❛❧❧❡❞ ♣✐♥♣♦✐♥t✐♥❣✳ ❍✐s✐❞❡❛ ✇❛s t❤❡♥ ❛♣♣❧✐❡❞ t♦ ✜♥✐t❡ ✜❡❧❞s ♦❢ ✈❡r② s♠❛❧❧ ❝❤❛r❛❝t❡r✐st✐❝ ✐♥ t✇♦ ✐♥❞❡♣❡♥❞❡♥t✇♦r❦s ❬❏♦✉✶✸❜❪ ❛♥❞ ❬●●▼❩✶✸❪✳ ■t ❜r♦❦❡ t❤❡ ❜❛rr✐❡r ♦❢ t❤❡ L(1/3, ·) ❝♦♠♣❧❡①✐t②s✐♥❝❡ ❏♦✉① ♦❜t❛✐♥❡❞ ❛ ❝♦♠♣❧❡①✐t② ♦❢ t②♣❡ L(1/4+ o(1))✳ ❖♥❡ ♠✉st ♥♦t❡ t❤❛t ❜♦t❤❛❧❣♦r✐t❤♠s ❝♦✉❧❞ ❝♦♠♣✉t❡ t❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s ♦❢ t❤❡✐r r❡s♣❡❝t✐✈❡ ❢❛❝t♦r ❜❛s❡s✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ❍❡♥❝❡ ❏♦✉① ❛s❦❡❞ ✐❢ t❤❡ r❡♠❛✐♥✐♥❣ ♣❛rt ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✱ t❤❡✐♥❞✐✈✐❞✉❛❧ ❧♦❣❛r✐t❤♠ st❛❣❡✱ ❝❛♥ ❜❡ ❛❝❝❡❧❡r❛t❡❞✳ ❆❞❞✐t✐♦♥❛❧❧②✱ t❤❡ ♥❡✇ ✐❞❡❛ ❝♦✉❧❞♥♦t ❜❡ ✉s❡❞ ❢♦r ❛❧❧ t❤❡ ✜❡❧❞s ✐♥ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ❞♦♠❛✐♥ ♦❢ ❋❋❙✳ ❲❡ ✇✐❧❧ ❣✐✈❡ ❛♥s✇❡rst♦ t❤❡s❡ t✇♦ ✐ss✉❡s ❜② s❤♦✇✐♥❣ t❤❛t✱ ❢♦r ❛❧❧ ✜♥✐t❡ ✜❡❧❞s ✇❤❡r❡ ❋❋❙ ✇♦r❦s✱ ❡①❝❡♣t❢♦r t❤❡ ♠✐❞❞❧❡ ♣r✐♠❡ ❝❛s❡✱ t❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t✐♠❡ L(α, ·)✇✐t❤ α < 1/3✳ ■♥ ♣❛rt✐❝✉❧❛r ✐♥ t❤❡ ❝❛s❡ ♦❢ ✜❡❧❞s FQ ✇✐t❤ Q = qk s✉❝❤ t❤❛t q < k+2❛♥❞ q ≈ k ♦♥❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✐♥ t✐♠❡ (logQ)O(log logQ) ✇❤✐❝❤✐♥ ❝♦♠♣❧❡①✐t② t❤❡♦r② ✐s ❝❛❧❧❡❞ q✉❛s✐✲♣♦❧②♥♦♠✐❛❧✱ ❛♥❞ ✐s s♠❛❧❧❡r t❤❛♥ L(ǫ, ·) ❢♦r ❛♥②ǫ > 0✳

❙✉♠♠❛r② ♦❢ ❝♦♥tr✐❜✉t✐♦♥s

P❛r❛♠❡tr✐③❛t✐♦♥s ❚❤❡ s❡❛r❝❤ ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ✇❤✐❝❤ ❛r❡ ❜❡st s✉✐t❡❞ ❢♦r ❊❈▼✐s ❛♥ ❛❝t✐✈❡ t♦♣✐❝ ✐♥ ❛❧❣♦r✐t❤♠✐❝ ♥✉♠❜❡r t❤❡♦r② ❬❙✉②✽✺✱ ▼♦♥✾✷✱ ❆▼✾✸✱ ❇❇▲✶✵✱

■◆❚❘❖❉❯❈❚■❖◆ ①✐

❇❈✶✵✱ ❘❛❜✶✵❪✱ ❜✉t t❤❡ ♠❡t❤♦❞s s❡❡♠ t♦ ❜❡ ❛❞✲❤♦❝✳ ❲❡ ❣✐✈❡ ❛ ✈✐❡✇♣♦✐♥t ✇❤✐❝❤❡♥❝♦♠♣❛ss ❛❧❧ t❤❡ ♣r❡✈✐♦✉s ♠❡t❤♦❞s ❛♥❞ ✇❤✐❝❤ ❛❧❧♦✇❡❞ ✉s t♦ ✜♥❞ ♥❡✇ ❢❛♠✐❧✐❡s ♦❢❝✉r✈❡s ❬❄❪✳ ■♥ ♣❛rt✐❝✉❧❛r ❣✐✈❡♥ ❛ ❝✉r✈❡ ✇❡ ❝❛♥ ♠❡❛s✉r❡ ✐ts ❡✣❝✐❡♥❝② ❜② ❝♦♠♣✉t✐♥❣t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ❋r♦❜❡♥✐✉s ❛♣♣❧✐❝❛t✐♦♥✱ ❢♦r ✇❤✐❝❤ ❢❛st ❛❧❣♦r✐t❤♠s ❡①✐st ❬❙✉t✶✷❪✳❚❤❡ ♥❡✇ ❢❛♠✐❧✐❡s t❤❛t ✇❡ ❞✐s❝♦✈❡r❡❞ ✇❡r❡ ✉s❡❞ ❜② ❇♦✉✈✐❡r ✐♥ t❤❡ ●P❯ ❝♦❞❡ ♦❢●▼P✲❊❈▼✱ ❛ ✈❡r② ❝♦♠♣❡t✐t✐✈❡ s♦❢t✇❛r❡ ♦❢ ❢❛❝t♦r✐③❛t✐♦♥ ✇✐t❤ ❊❈▼✳

❙♠♦♦t❤✐♥❣ P❛rt ♦❢ t❤❡ ❝r②♣t♦❧♦❣② ❝♦♠♠✉♥✐t② ❝♦♥s✐❞❡rs t❤❛t ❛♥ ❛tt❛❝❦❡r ❝❛♥♣❡r❢♦r♠ ❛ s❧✐❣❤t❧② ❧♦♥❣❡r ❝♦♠♣✉t❛t✐♦♥ ❜❡❢♦r❡ t❤❡ ♦♣❡♥✐♥❣ ♦❢ t❤❡ ❝❤❛❧❧❡♥❣❡✳ ■♥ t❤✐s❝❛s❡ t❤❡ s❡❝✉r✐t② ♦❢ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ❝r②♣t♦s②st❡♠s ✇♦✉❧❞ ❞r♦♣ t♦ t❤❡ ❝♦♠♣❧❡①✐t②♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ❧♦❣❛r✐t❤♠ st❛❣❡✳ ■t ✐s ❦♥♦✇♥ t❤❛t t❤✐s st❛❣❡ ✐s ❞♦♠✐♥❛t❡❞ ❜② ✐ts✜rst st❡♣✱ ❝❛❧❧❡❞ s♠♦♦t❤✐♥❣✳ ■♥ ♦✉r ✇♦r❦✱ ✇❡ ✐♠♣r♦✈❡❞ ✐t ✉s✐♥❣ ❛ str❛t❡❣② ✐♥t✇♦✱ ❛♥❞ t❤❡♥ ♠♦r❡✱ st❡♣s✳ ❚❤❡ ♣r❡✈✐♦✉s❧② ❦♥♦✇♥ ❝♦♠♣❧❡①✐t② ♦❢ Lp(1/3, 1.44) ✇❛sr❡❞✉❝❡❞ t♦ Lp(1/3, 1.232)✳

❉▲ ❢❛❝t♦r② ❚❤❡ ③♦♦❧♦❣② ♦❢ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ❛❧❣♦r✐t❤♠s ❢♦r ♣r✐♠❡ ✜❡❧❞s ♣❛r✲❛❧❧❡❧s t❤❛t ♦❢ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠s✱ ✇✐t❤ ♦♥❡ ❡①❝❡♣t✐♦♥✳ ❚❤✐s ✐s ✇❤② ✇❡♣r♦♣♦s❡❞ t❤❡ ❉▲ ❢❛❝t♦r② ❜② tr❛♥s❧❛t✐♥❣ t❤❡ ✐❞❡❛ ♦❢ ❈♦♣♣❡rs♠✐t❤✬s ❢❛❝t♦r✐③❛t✐♦♥❢❛❝t♦r② t♦ ♦✉r ♣r♦❜❧❡♠✳ ❚❤❡ ♠❛✐♥ ❞✐✣❝✉❧t② ✇❛s t♦ s❤♦✇ t❤❛t t❤❡ ✐♥❞✐✈✐❞✉❛❧ ❧♦❣❛✲r✐t❤♠ st❛❣❡ r❡♠❛✐♥s ♥❡❣❧✐❣✐❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦t❤❡r st❛❣❡s✱ ❞❡s♣✐t❡ ✐ts ✐♥❝r❡❛s❡✐♥ ❝♦♠♣❧❡①✐t②✳ ❚❤❡ s♣❡❝✐✜❝✐t② ♦❢ t❤❡ ❉▲ ❢❛❝t♦r② ✐s t❤❛t ♦♥❡ ❝❛♥ s❤❛r❡ t❤❡ ♣r❡✲❝♦♠♣✉t❡❞ ✐♥❢♦r♠❛t✐♦♥ ❢♦r ❛❧❧ t❤❡ ♣r✐♠❡s ♦❢ ❛ ❣✐✈❡♥ ❜✐t✲s✐③❡✳ ❍❡♥❝❡✱ ❛❢t❡r s♦♠❡♣r❡✲❝♦♠♣✉t❛t✐♦♥s ♦❢ ❝♦♠♣❧❡①✐t② Lp(1/3, 2.007)✱ t❤❡ ♠❛✐♥ ♣❤❛s❡ ♦❢ ◆❋❙ ❢♦r ❡❛❝❤♣r✐♠❡ t❛❦❡s t✐♠❡ Lp(1/3, 1.639)✱ ❛♥❞ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ❧♦❣❛r✐t❤♠ st❛❣❡ t❛❦❡s t✐♠❡Lp(1/3, 1.232)✳ ❚❤❡ ❞r❛✇❜❛❝❦ ✐s t❤❡ ✉s❡ ♦❢ ❛ ❞✐s❦✲s♣❛❝❡ ♦❢ Lp(1/3, 1.639)✳

P♦❧②♥♦♠✐❛❧ s❡❧❡❝t✐♦♥ ❢♦r ❋❋❙ ❚❤❡ ♣♦❧②♥♦♠✐❛❧ s❡❧❡❝t✐♦♥ st❛❣❡ ♦❢ ♠♦❞❡r♥ ❛❧✲❣♦r✐t❤♠s ❛s ◆❋❙ ♦r ❋❋❙ ✇❡r❡ ❞❡✈❡❧♦♣❡❞ ✐♥ ❛♥ ♥♦♥✲✉♥✐✜❡❞ ♠❛♥♥❡r✳ ■♥❞❡❡❞✱ ❢♦r❋❋❙ ✐t ✇❛s ♣r♦♣♦s❡❞ t♦ ✉s❡ ♣✉r❡❧② ✐♥s❡♣❛r❛❜❧❡ ♣♦❧②♥♦♠✐❛❧s ❬❈♦♣✽✹✱ ❚❤♦✵✸❪✱ ❝❧❛s✲s✐❝❛❧ ❋❋❙ ♣♦❧②♥♦♠✐❛❧s ❬❏▲✵✷✱ ❏▲✵✼❪✱ t✇♦ r❛t✐♦♥❛❧ s✐❞❡ ❋❋❙ ❬❏▲✵✻❪ ❛♥❞ ✐♥s❡♣❛r❛❜❧❡♣♦❧②♥♦♠✐❛❧s ❬❍❙❲+✶✵✱ ❍❙❙❚✶✷❪✳ ❖♥❡ ❝❛♥ ❡①t❡♥❞ t❤❡ ❧✐st ✇✐t❤ ♠♦r❡ ❛♥❞ ♠♦r❡✐❞❡❛s ❛♥❞✱ ❛s ✐♥ t❤❡ ◆❋❙ ❝❛s❡✱ ♦♥❡ ❝❛♥ ✐♥tr♦❞✉❝❡ ✈❛r✐♦✉s ❢✉♥❝t✐♦♥s ❧✐❦❡ ▼✉r♣❤②✬sα ❛♥❞ E ❬▼✉r✾✾✱ ❇❛✐✶✶❪✳ ■♥ ❬❇❛r✶✸❪✱ ✐♥st❡❛❞ ♦❢ ❛❞❞✐♥❣ ♥❡✇ ♣r♦♣❡rt✐❡s✱ ✇❡ ❝♦♠✲♣❛r❡❞ t❤❡ ✈❛r✐♦✉s ♠❡t❤♦❞s t♦ ❡❛❝❤ ♦t❤❡r✳ ❲❡ s❤♦✇❡❞ t❤❛t t❤❡ t✇♦ r❛t✐♦♥❛❧ s✐❞❡❋❋❙ ♦✛❡rs ❛ s♠❛❧❧ s❡t ♦❢ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ ✇❡ ❣❛✈❡ t❤❡ ❡①❛❝t ❛❞✈❛♥t❛❣❡ ♦❢ ✐♥s❡♣✲❛r❛❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳ ❋✐♥❛❧❧②✱ ✇❡ ❞❡✜♥❡❞ t❤❡ ǫ ❢✉♥❝t✐♦♥✱ ♠✐①✐♥❣ α ❛♥❞ E✱ ✇❤✐❝❤❝♦♠♣❛r❡s ❛r❜✐tr❛r② s❡♣❛r❛❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳ ❲❡ ❣❛✈❡ ❡①♣❡r✐♠❡♥t❛❧ ❡✈✐❞❡♥❝❡ t❤❛t ǫ♣r❡❞✐❝ts t❤❡ s✐❡✈❡ ❡✣❝✐❡♥❝② ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ ✉♣ t♦ ❛ 5% ❡rr♦r✳ ■t ✇❛s r❛♣✐❞ ❡♥♦✉❣❤t♦ ❛❧❧♦✇ t❤❡ s❡❧❡❝t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧s ❢♦r t✇♦ r❡❝♦r❞ ❝♦♠♣✉t❛t✐♦♥s ✇✐t❤ ❋❋❙ ✐♥❝❤❛r❛❝t❡r✐st✐❝ 2 ❬❇❇❉+✶✷✱ ❇❇❉+✶✸❪✳

◗✉❛s✐✲♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ ❚❤❡ r❡❝❡♥t ❛❧❣♦r✐t❤♠s ♦❢ ❏♦✉① ❬❏♦✉✶✸❜❪ ❛♥❞●r❛♥❣❡r ❛♥❞ ♦t❤❡rs ❬●●▼❩✶✸❪ ❤❛❞ t❤❡ ♣❛rt✐❝✉❧❛r✐t② t❤❛t t❤❡ ♠❛✐♥ ♣❤❛s❡ ❝♦♠✲♣✉t❛t✐♦♥ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ❚❤❡ ❢❛st❡r ♠❡t❤♦❞ ❢♦r t❤❡ ✐♥❞✐✈✐❞✉❛❧❧♦❣❛r✐t❤♠ st❛❣❡✱ ♣r❡s❡♥t❡❞ ✐♥ ❬❏♦✉✶✸❜❪✱ ♣r♦❝❡❡❞s ✐♥ t❤r❡❡ st❡♣s✿ ❛ st❡♣ ❝♦rr❡s♣♦♥❞✲✐♥❣ t♦ t❤❡ s♠♦♦t❤✐♥❣ ✐♥ ◆❋❙✱ ❛ ❝❧❛ss✐❝❛❧ ❞❡s❝❡♥t ✇❤✐❝❤ ✇❛s ❛❧r❡❛❞② ✉s❡❞ ✐♥ ❋❋❙

①✐✐ ■◆❚❘❖❉❯❈❚■❖◆

❛♥❞ ❛ ♥❡✇ t❡❝❤♥✐q✉❡ ✉s✐♥❣ ●rö❜♥❡r ❜❛s✐s✳ ■♥ ❛ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ●❛✉❞r②✱ ❏♦✉① ❛♥❞❚❤♦♠é ❬❇●❏❚✶✸❪ ✇❡ ♣r♦♣♦s❡❞ ❛♥ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ❛❧❧♦✇s t♦ ❝♦♠♣✉t❡ logP ❢♦r❛♥② ♣♦❧②♥♦♠✐❛❧ P ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ logQi ❢♦r ❛ s❡t ♦❢ ♣♦❧②♥♦♠✐❛❧s Qi♦❢ ❞❡❣r❡❡ ❧❡ss t❤❛♥ degP/2✳ ❍❡♥❝❡✱ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ❧♦❣❛r✐t❤♠ st❛❣❡ ❝♦♥s✐sts ♦♥❧②✐♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ ❞❡s❝❡♥t tr❡❡✳ ❲❡ s❤♦✇❡❞ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ♥♦❞❡s ✐♥t❤❡ tr❡❡ ✐s q✉❛s✐✲♣♦❧②♥♦♠✐❛❧ ❛♥❞ t❤❛t t❤❡ ❝♦♠♣✉t❛t✐♦♥s ❞♦♥❡ ❛t ❡✈❡r② st❡♣ t❛❦❡ ❛♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

❑❛r❛ts✉❜❛✲❧✐❦❡ ❢♦r♠✉❧❛❡ ❆♥ ❛❞❞✐t✐♦♥❛❧ ❝♦♥tr✐❜✉t✐♦♥ ✇❤✐❝❤ ❣♦❡s ❜❡②♦♥❞ t❤❡s❝♦♣❡ ♦❢ t❤✐s ❞♦❝✉♠❡♥t ✐s t❤❡ r❡s❡❛r❝❤ ♦❢ ❑❛r❛ts✉❜❛✲❧✐❦❡ ❢♦r♠✉❧❛❡✳ ❆❧❧ t❤❡ ✐♥t❡❣❡r❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❤❛✈❡ ❡①♣❧✐❝✐t st❛❣❡s ♦❢ ❡✈❛❧✉❛t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥❛♥❞ ✐♥t❡r♣♦❧❛t✐♦♥✳ ❆ s❡♠✐♥❛❧ ♣❛♣❡r ♦❢ ▼♦♥t❣♦♠❡r② ❬▼♦♥✵✺❪ s❤♦✇❡❞ t❤❛t ♦t❤❡r✉♥❡①♣❧❛✐♥❡❞ ❢♦r♠✉❧❛❡ ❡①✐sts✱ ❛♥❞ t❤❡② ❛r❡ ✈❡r② ❡✛❡❝t✐✈❡ ✐♥ t❤❡ ❛r✐t❤♠❡t✐❝ ♦❢ ✜♥✐t❡✜❡❧❞s ❬❆❧❜✶✶✱ ❆❧❜✶✷❪✳ ❆ s❡r✐❡s ♦❢ ♥❡✇ ❢♦r♠✉❧❛❡ ❢♦❧❧♦✇❡❞ ✐♥ t❤❡ ♥❡①t ②❡❛rs ❬❈❍✵✼✱❋❍✵✼✱ ❖s❡✵✽✱ ❈Ö✵✽✱ ❈❖✵✾✱ ❈❑❖✵✾✱ ❈Ö✶✵✱ ❈Ö✶✶✱ ❈❇❍✶✶❪✱ ❜✉t ✈✐rt✉❛❧❧② ♥♦t❤✐♥❣✇❛s ❞✐s❝♦✈❡r❡❞ ❛❢t❡r ✷✵✶✶✱ s♦ t❤❛t ♦♥❡ ❝♦✉❧❞ ❛s❦ ✐❢ t❤❡s❡ ❢♦r♠✉❧❛❡ ❛r❡ ♦♣t✐♠❛❧✳❲❡ st❛rt❡❞ ❛ ♣r♦❥❡❝t ✇✐t❤ ❉❡tr❡②✱ ❊st✐❜❛❧s ❛♥❞ ❩✐♠♠❡r♠❛♥♥ ✇❤✐❝❤ ❛✐♠❡❞ t♦r❡♣r♦❞✉❝❡ ❛♥❞ ❡①t❡♥❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ♠❛❞❡ ❜② ▼♦♥t❣♦♠❡r②✳ ❚❤❡ ♦✉t❝♦♠❡ ✇❛s❞✐✛❡r❡♥t s✐♥❝❡ ✇❡ ❞✐s❝♦✈❡r❡❞ ❛ ❢❛st❡r ❛❧❣♦r✐t❤♠✳ ■t ❛❧❧♦✇❡❞ ✉s ✐♥ ❬❇❉❊❩✶✷❪ t♦❝♦♠♣❧❡t❡ t❤❡ ❡①❤❛✉st✐✈❡ s❡❛r❝❤ ♦❢ ▼♦♥t❣♦♠❡r② ❛♥❞ t♦ ♣r♦✈❡ t❤❡ ♦♣t✐♠❛❧✐t② ♦❢ ❤✐s❢♦r♠✉❧❛❡✱ ❛s ✇❡❧❧ ❛s ❢♦r ♠♦st ♦❢ t❤❡ ♠❛❣✐❝❛❧ ❢♦r♠✉❧❛❡ ❦♥♦✇♥ t♦❞❛②✳ ❆s ❛ ❜♦♥✉s✱✇❡ ❢♦✉♥❞ ♥❡✇ ❢♦r♠✉❧❛❡ ✐♥ ❝❤❛r❛❝t❡r✐st✐❝ 3 ✇❤✐❝❤ ❝❛♥ ❜❡ ✉s❡❞ ✐♥ ❝r②♣t♦❣r❛♣❤②❀ ❢♦r❡①❛♠♣❧❡ ❆❧❣♦r✐t❤♠ 1 ✐♥ ❬❊st✶✵❪ ✉s❡s 12 ♣r♦❞✉❝ts ✇❤❡r❡❛s ✇❡ ❞✐s❝♦✈❡r❡❞ ❛ ❢♦r♠✉❧❛✇✐t❤ 11 ♣r♦❞✉❝ts✳ ◆♦t❡ ❤♦✇❡✈❡r t❤❛t ❜❡❝❛✉s❡ ♦❢ t❤❡ q✉❛s✐✲♣♦❧②♥♦♠✐❛❧ t❤❡ ♠❛✐♥❛♣♣❧✐❝❛t✐♦♥ ♦❢ ♦✉r ❢♦r♠✉❧❛✱ t❤❡ ♣❛✐r✐♥❣✲❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ✐♥ ❝❛r❛❝t❡r✐st✐❝ 3✱ ✐s ♥♦❧♦♥❣❡r ✐♥t❡r❡st✐♥❣✳

❈❤❛♣t❡r ♦r❣❛♥✐③❛t✐♦♥

❚❤❡ t❤❡s✐s ❤❛s t✇♦ ♣❛rts✿ ♦♥❡ ✐♥ ✇❤✐❝❤ ✇❡ st✉❞② ❊❈▼ ❛s t❤❡ ❜❡st s♠♦♦t❤♥❡sst❡st t♦❞❛② ❛♥❞ ♦♥❡ ✐♥ ✇❤✐❝❤ ✇❡ ❛❞❞r❡ss t❤❡ ❣❡♥❡r❛❧ ♣r♦❜❧❡♠ ♦❢ ❝♦♠♣✉t✐♥❣ ❞✐s❝r❡t❡❧♦❣❛r✐t❤♠s ✐♥ ❛♥② ✜♥✐t❡ ✜❡❧❞✳

■♥ ❈❤❛♣t❡r ✶ ✇❡ ♠❛❦❡ ❛ ❧✐st ♦❢ t❤❡ s♠♦♦t❤♥❡ss r❡s✉❧ts ✇❤✐❝❤ ✇✐❧❧ ❜❡ ♥❡❡❞❡❞t❤r♦✉❣♦✉t t❤❡ ❞♦❝✉♠❡♥t✳ ■♥ ❈❤❛♣t❡r ✷ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❊❈▼ ❛❧❣♦r✐t❤♠✱ ✇❤✐❝❤s❡r✈❡s ❛s ❜❛s✐s ❢♦r t❤❡ ♥❡①t t✇♦ ❝❤❛♣t❡rs✳ ■♥ ❈❤❛♣t❡r ✸ ✇❡ ✐♠♣r♦✈❡ t❤❡ ❛❧❣♦r✐t❤♠✐ts❡❧❢ ❛♥❞ ✐♥ ❈❤❛♣t❡r ✹ ✇❡ ♣✉t ❊❈▼ ❛t ✇♦r❦ ✐♥ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ❧♦❣❛r✐t❤♠ st❛❣❡ ♦❢◆❋❙✳

❲❡ st❛rt t❤❡ s❡❝♦♥❞ ♣❛rt ✇✐t❤ ❈❤❛♣t❡r ✺ ✇❤✐❝❤ ♣r❡s❡♥ts ❜❛s✐❝ ❛❧❣♦r✐t❤♠s✱st✐❧❧ ✉s❡❞ t♦❞❛② t♦ s✐♠♣❧✐❢② t❤❡ ♠♦❞❡r♥ ❛❧❣♦r✐t❤♠s✳ ■♥ ❈❤❛♣t❡r ✻ ✇❡ ♣r❡s❡♥t ❛♠✐❞❞❧❡❜r♦✇ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ◆❋❙ ❛♥❞ ❋❋❙ ❛❧❣♦r✐t❤♠s✳ ■♥ t❤❡ ♥❡①t ❝❤❛♣t❡r ✇❡❝♦♠♣✉t❡ t❤❡ ❝♦♠♣❧❡①✐t✐❡s ♦❢ ♣r❡✈✐♦✉s❧② ✐♥tr♦❞✉❝❡❞ ❛❧❣♦r✐t❤♠s✱ ✇❤✐❝❤ ❣✐✈❡s ✉s t❤❡♦❝❝❛s✐♦♥ t♦ ✐♥tr♦❞✉❝❡ t❤❡ ❉▲ ❢❛❝t♦r②✳ ❙♦♠❡ ♦❢ t❤❡ ❞❡t❛✐❧s t❤❛t ✇❡ s❦✐♣♣❡❞ ✐♥ t❤❡✜rst ❞❡s❝r✐♣t✐♦♥ ♦❢ ◆❋❙ ❛♥❞ ❋❋❙ ✇❡r❡ ✐♥s❡rt❡❞ ✐♥ ❈❤❛♣t❡r ✽ ✇❤❡r❡ ✇❡ ❛❧s♦ ♣r❡s❡♥tt✇♦ ♥❡✇ ✐♠♣r♦✈❡♠❡♥ts✳ ■♥ ❈❤❛♣t❡r ✾ ✇❡ ❢♦❝✉s ♦♥ t❤❡ ♣♦❧②♥♦♠✐❛❧ s❡❧❡❝t✐♦♥ st❛❣❡♦❢ ❋❋❙✱ ♦❜t❛✐♥✐♥❣ ❛ ♥❡✇ ❢✉♥❝t✐♦♥ t♦ r❛♥❦ ♣♦❧②♥♦♠✐❛❧s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡✐r s✐❡✈❡❡✣❝✐❡♥❝②✳ ❚❤❡ ❧❛st ❝❤❛♣t❡r ♦❢ t❤❡ s❡❝♦♥❞ ♣❛rt ✐s ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ t❤❡ r❡st✳ ■t

■◆❚❘❖❉❯❈❚■❖◆ ①✐✐✐

♣r❡s❡♥ts ❛ q✉❛s✐✲♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ ✐♥ s♠❛❧❧ ❝❤❛r❛❝t❡r✐st✐❝ ❛s ❛ r❡s✉❧t ♦❢ t❤❡r❡❝❡♥t ❞❡✈❡❧♦♣♠❡♥t ♦♥ t❤❡ ♣r♦❜❧❡♠✳

P❡rs♦♥❛❧ ✇♦r❦s

❬❇❛r✶✸❪ ❘✳ ❇❛r❜✉❧❡s❝✉✳ ❙❡❧❡❝t✐♥❣ ♣♦❧②♥♦♠✐❛❧s ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ✜❡❧❞ s✐❡✈❡✱ ✷✵✶✸✳❆✈❛✐❧❛❜❧❡ ❛t ❈r②♣t♦❧♦❣② ❡Pr✐♥t ❆r❝❤✐✈❡ ❘❡♣♦rt ✷✵✶✸✴✷✵✵✱ ❆❝❝❡♣t❡❞ ❢♦r ♣✉❜❧✐❝❛✲t✐♦♥ ✐♥ ▼❛t❤✳ ❈♦♠♣✳

❬❄❪ ❘✳ ❇❛r❜✉❧❡s❝✉✱ ❏✳ ❲✳ ❇♦s✱ ❈✳ ❇♦✉✈✐❡r✱ ❚✳ ❑❧❡✐♥❥✉♥❣✱ ❛♥❞ P✳ ▲✳ ▼♦♥t❣♦♠❡r②✳❋✐♥❞✐♥❣ ❊❈▼✲❢r✐❡♥❞❧② ❝✉r✈❡s t❤r♦✉❣❤ ❛ st✉❞② ♦❢ ●❛❧♦✐s ♣r♦♣❡rt✐❡s✳ ■♥❆❧❣♦r✐t❤♠✐❝◆✉♠❜❡r ❚❤❡♦r②✕❆◆❚❙ ❳✱ ♣❛❣❡s ✻✸✕✽✻✱ ✷✵✶✸✳

❬❇❇❉+✶✸❪ ❘✳ ❇❛r❜✉❧❡s❝✉✱ ❈✳ ❇♦✉✈✐❡r✱ ❏✳ ❉❡tr❡②✱ P✳ ●❛✉❞r②✱ ❍✳ ❏❡❧❥❡❧✐✱ ❊✳ ❚❤♦♠é✱▼✳ ❱✐❞❡❛✉✱ ❛♥❞ P✳ ❩✐♠♠❡r♠❛♥♥✳ ❉✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✐♥ ●❋✭2809✮ ✇✐t❤ ❋❋❙✱✷✵✶✸✳ ❆✈❛✐❧❛❜❧❡ ❛t ❈r②♣t♦❧♦❣② ❡Pr✐♥t ❆r❝❤✐✈❡ ❘❡♣♦rt ✷✵✶✸✴✶✾✼✱ ❆❝❝❡♣t❡❞ ❢♦r♣r❡s❡♥t❛t✐♦♥ ❛t t❤❡ P✉❜❧✐❝ ❑❡② ❈r②♣t♦❣r❛♣❤② ✷✵✶✹ ❝♦♥❢❡r❡♥❝❡✳

❬❇❉❊❩✶✷❪ ❘✳ ❇❛r❜✉❧❡s❝✉✱ ❏✳ ❉❡tr❡②✱ ◆✳ ❊st✐❜❛❧s✱ ❛♥❞ P✳ ❩✐♠♠❡r♠❛♥♥✳ ❋✐♥❞✐♥❣♦♣t✐♠❛❧ ❢♦r♠✉❧❛❡ ❢♦r ❜✐❧✐♥❡❛r ♠❛♣s✳ ■♥ ❆r✐t❤♠❡t✐❝ ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s✕❲❆■❋■ ✷✵✶✷✱✈♦❧✉♠❡ ✼✸✻✾ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t✳ ❙❝✐✳✱ ♣❛❣❡s ✶✻✽✕✶✽✻✳ ❙♣r✐♥❣❡r✱ ✷✵✶✷✳

❬❇●❏❚✶✸❪ ❘✳ ❇❛r❜✉❧❡s❝✉✱ P✳ ●❛✉❞r②✱ ❆✳ ❏♦✉①✱ ❛♥❞ ❊✳ ❚❤♦♠é✳ ❆ q✉❛s✐✲♣♦❧②♥♦♠✐❛❧❛❧❣♦r✐t❤♠ ❢♦r ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✐♥ ✜♥✐t❡ ✜❡❧❞s ♦❢ s♠❛❧❧ ❝❤❛r❛❝t❡r✐st✐❝✱ ✷✵✶✸✳ ❈r②♣✲t♦❧♦❣② ❡Pr✐♥t ❆r❝❤✐✈❡ ❘❡♣♦rt ✷✵✶✸✴✹✵✵✱ ❙✉❜♠✐tt❡❞ t♦ ❊✉r♦❝r②♣t ✷✵✶✹✳

①✐✈ ■◆❚❘❖❉❯❈❚■❖◆

P❛rt ■

❙♠♦♦t❤♥❡ss ❛♥❞ ❊❈▼

✶

❈❤❛♣t❡r ✶

❙♠♦♦t❤♥❡ss Pr♦❜❛❜✐❧✐t✐❡s

▼♦st ♦❢ t❤❡ s✉❜✲❡①♣♦♥❡♥t✐❛❧ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ❛❧❣♦r✐t❤♠s r❡❧② ♦♥ t❤❡♥♦t✐♦♥ ♦❢ s♠♦♦t❤♥❡ss✱ ✇❤✐❝❤ ✐s ❛ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦ t❤❛t ✇❡ ✇✐❧❧ ♠❡❡t ❡✈✲❡r②✇❤❡r❡ ✐♥ t❤✐s ❞♦❝✉♠❡♥t✳ ❋✐rst ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ✐♥t❡❣❡rs✱s♠♦♦t❤♥❡ss ✐s ❛❧s♦ ✐♠♣♦rt❛♥t ✐♥ t❤❡ ❝❛s❡ ♦❢ ♣♦❧②♥♦♠✐❛❧s✳ ❚❤✐s s❤♦rt❝❤❛♣t❡r s✉♠♠❛r✐③❡s ❛❧❧ t❤❡ ❜❛s✐❝ t❤❡♦r❡t✐❝❛❧ r❡s✉❧ts ❛❜♦✉t t❤❡ s♠♦♦t❤✲♥❡ss ♣r♦❜❛❜✐❧✐t② ✇❤✐❝❤ ❝♦♠❡ ❢r♦♠ ❛♥❛❧②t✐❝ ♥✉♠❜❡r t❤❡♦r②✳❲❡ ♣r♦❝❡❡❞ ❛s ❢♦❧❧♦✇s✳ ❆❢t❡r ❣✐✈✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ s♠♦♦t❤ ♥✉♠❜❡rs✇❡ r❡❝❛❧❧ t❤❡ ♠❛✐♥ r❡s✉❧t ♦❢ ❬❈❊P✽✸❪✳ ❲❡ t❤❡♥ r❡✐♥t❡r♣r❡t ✐t ✉s✐♥❣ t❤❡ L✲♥♦t❛t✐♦♥✱ ✇❤✐❝❤ ✐s ❝♦♥✈❡♥✐❡♥t ❢♦r t❤❡ ❛♥❛❧②s✐s ♦❢ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ❛♥❞❢❛❝t♦r✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠s✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❡ ❝❤❛♣t❡r ✇✐t❤ t❤❡ ❛♥❛❧♦❣♦✉sr❡s✉❧ts ❢♦r ♣♦❧②♥♦♠✐❛❧s✳

✶✳✶ ❙♠♦♦t❤ ♥✉♠❜❡rs

▼❛♥② ❝r②♣t♦❣r❛♣❤✐❝ ❛❧❣♦r✐t❤♠s ❣❡♥❡r❛t❡ r❛♥❞♦♠ ♥✉♠❜❡rs ❜❡❧♦✇ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛♥❞ t❡st ✐❢ ❛❧❧ t❤❡✐r ♣r✐♠❡ ❢❛❝t♦rs ❛r❡ s♠❛❧❧✳ ▼♦r❡ ❢♦r♠❛❧❧②✱ ✐❢ B ✐s ❛♥ ✐♥t❡❣❡r✱ ✇❡s❛② t❤❛t ❛♥ ✐♥t❡❣❡r ✐s B✲s♠♦♦t❤ ✐❢ ✐ts ♣r✐♠❡ ❢❛❝t♦rs ❛r❡ ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ B✳❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ❡✈❛❧✉❛t✐♥❣ t❤❡ ♥✉♠❜❡r ψ(x, y) ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❧❡ss t❤❛♥♦r ❡q✉❛❧ t♦ x ✇❤✐❝❤ ❛r❡ y✲s♠♦♦t❤✱ ✐✳❡✳✱ t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t

Ψ(x, y) ={n ∈ [1, x] | n ✐s y✲s♠♦♦t❤

}.

❲❡ ♥❡①t ❞❡✜♥❡ ❢♦r♠❛❧❧② t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ ♥✉♠❜❡r ✐♥ [1, x] t♦ ❜❡ s♠♦♦t❤✿

Ps♠♦♦t❤(x, y) = ψ(x, y)/x.

♥♦t❛t✐♦♥ ♠❡❛♥✐♥❣f = O(g) ∃c > 0, x0 > 0 x ≥ x0 ⇒ |f | ≤ c|g|f = o(g) ∀ǫ > 0 ∃xǫ x ≥ xǫ ⇒ |f | ≤ ǫ|g|f = Θ(g) f = O(g) ❛♥❞ g = O(f)f = Õ(g) ∃k ∈ N f = O

((log g)kg

)✳

❚❛❜❧❡ ✶✳✶✿ ❆ ❧✐st ♦❢ ♥♦t❛t✐♦♥s✳

✸

✹ ❈❍❆P❚❊❘ ✶✳ ❙▼❖❖❚❍◆❊❙❙ P❘❖❇❆❇■▲■❚■❊❙

❚❤❡ ✜rst ❛s②♠♣t♦t✐❝ ❢♦r♠✉❧❛ ❢♦r ψ(x, y) ✇❛s t❤❛t ♦❢ ❉✐❝❦♠❛♥ ✐♥ 1930 ✭s❡❡❬❍❚✾✸❪✮ ✇❤♦ ♣r♦✈❡❞ t❤❛t ❢♦r ❛♥② ✜①❡❞ u > 0

limx→∞

ψ(x, x1/u)/x = ρ(u),

✇❤❡r❡ ρ(u) ✐s ❞❡✜♥❡❞ ❛s t❤❡ ✭✉♥✐q✉❡✮ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ s✉❝❤ t❤❛t ρ(u) = 1 ❢♦ru ∈ [0, 1] ✇❤✐❝❤ s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥

uρ′(u) = −ρ(u− 1) (u > 1).

◆♦t❡ t❤❛t t❤✐s ❡q✉❛t✐♦♥ ❛❧❧♦✇s t♦ ❡st✐♠❛t❡ ❉✐❝❦♠❛♥✬s r❤♦ t♦ ❛♥② ♣r❡❝✐s✐♦♥ ♦♥ ❛♥②✜①❡❞ ✐♥t❡r✈❛❧ [0, c]✳ ▼♦r❡♦✈❡r✱ ✇❤❡♥ u ✐s ❧❛r❣❡ ❡♥♦✉❣❤✱ ρ(u) ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞❜② u−u ✭❈♦r♦❧❧❛r② ✶✳✸✱❬❍❚✾✸❪✮✿

limu→∞

ρ(u) = u−u(1+o(1)).

■♥ ♣r❛❝t✐❝❡✱ t❤❡ r❡❧❡✈❛♥t ✐♥t❡r✈❛❧ ❢♦r u ✐s [1, 10] ♦♥ ✇❤✐❝❤ u−u ❤❛s t❤❡ ❣♦♦❞ ♦r❞❡r♦❢ ♠❛❣♥✐t✉❞❡✳

❚❤❡ ❞r❛✇❜❛❝❦ ♦❢ ❉✐❝❦♠❛♥✬s r❡s✉❧t ✐s t❤❛t ✐t ❞♦❡s ♥♦t ❝♦✈❡r t❤❡ ❝❛s❡ ψ(x, x1/u)✇❤❡♥ u ❞❡♣❡♥❞s ♦♥ x✱ ❡✳❣✳ u =

√log x✳ ❈❛♥✜❡❧❞✱ ❊r❞ös ❛♥❞ P♦♠❡r❛♥❝❡ ♣r♦✈❡❞ ❛

str♦♥❣❡r r❡s✉❧t✿

❚❤❡♦r❡♠ ✶✳✶✳✶ ✭❬❈❊P✽✸❪✮✳ ■❢ ǫ > 0 ✐s ✜①❡❞ ❛♥❞ 3 ≤ u ≤ (1 − ǫ) log x/ log log x✱t❤❡♥

ψ(x, x1/u) = x exp{−u(log u+ log log u− 1 + o(1)

)}.

■♥ s❤♦rt Psmooth = u−u(1+o(1))✱ ✇❤❡r❡ o(1) ✐s ❛ ❢✉♥❝t✐♦♥ ❞❡♣❡♥❞✐♥❣ ♦♥ x ❛♥❞ u✇❤✐❝❤ t❡♥❞s t♦ 0 ✉♥✐❢♦r♠❧② ✇❤❡♥ x t❡♥❞s t♦ ✐♥✜♥✐t②✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❡ ♠❛❦❡ t❤❡❧✐st ♦❢ t❤❡ ✇❡❧❧ ❦♥♦✇♥ ♥♦t❛t✐♦♥s ✐♥ ❚❛❜❧❡ ✶✳✶✳

■♥ t❤❡ ❝♦♠♣❧❡①✐t② ❛♥❛❧②s✐s ♦❢ s❡✈❡r❛❧ ❛❧❣♦r✐t❤♠s ✇❡ ✇✐❧❧ ❡st✐♠❛t❡ t❤❡ ♣r♦❜❛❜✐❧✐t②t❤❛t t✇♦ ♥✉♠❜❡rs ❛r❡ s✐♠✉❧t❛♥❡♦✉s❧② s♠♦♦t❤✳ ■t t✉r♥s ♦✉t t❤❛t ✇❡ ✇✐❧❧ ♦❜t❛✐♥ ❣♦♦❞❢♦r♠✉❧❛❡ ✉s✐♥❣ t❤❛t✱ ❢♦r x1, x2, y > 0✱ ♦♥❡ ❤❛s t❤❡ ✐♥❡q✉❛❧✐t②

Psmooth(x1, y)Psmooth(x2, y) ≥ Psmooth(x1x2, y)1+o(1).

◆❡✈❡rt❤❡❧❡ss✱ ❛ s❧✐❣❤t❧② ❧♦♥❣❡r ❛r❣✉♠❡♥t s❤♦✇s t❤❛t t❤❡ ❡q✉❛❧✐t② ❤♦❧❞s✳

❈♦r♦❧❧❛r② ✶✳✶✳✷✳ ▲❡t x ❛♥❞ y ❜❡ s✉❝❤ t❤❛t u = log xlog y

s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥s ✐♥❚❤❡♦r❡♠ ✶✳✶✳✶✳ ❚❤❡♥ ❢♦r x1 ❛♥❞ x2 s✉❝❤ t❤❛t y ≤ x1, x2 ≤ x ✇❡ ❤❛✈❡

Psmooth(x1, y)Psmooth(x2, y) = Psmooth(x1x2, y)1+o(1).

Pr♦♦❢✳ P✉t u1 = (log x1)/(log y) ❛♥❞ u2 = (log x2)/(log y) ❛♥❞✱ ✇✐t❤♦✉t ❧♦ss ♦❢❣❡♥❡r❛❧✐t②✱ ❛ss✉♠❡ u2 ≥ u1✳ ❙✐♥❝❡ u1, u2 ≥ 1✱ t❤❡ ❧♦❣❛r✐t❤♠s ♦❢ u1✱ u2 ❛♥❞ u1 +u2 ❛r❡ ♣♦s✐t✐✈❡✳ P✉t L(u1, u2) = u2(log u2) + u1(log u1) ❛♥❞ R(u1, u2) = (u1 +u2) log(u1 + u2)✳ ❲❡ ❤❛✈❡ t♦ s❤♦✇ t❤❛t L = R(1 + o(1))✳ ❖♥ t❤❡ ♦♥❡ ❤❛♥❞ ♦♥❡❤❛s L ≤ R✳ ❋♦r t❤❡ s❡❝♦♥❞ ✐♥❡q✉❛❧✐t② ✇❡ t❛❦❡ ǫ > 0 ❛♥❞ ✇❡ ❞✐st✐♥❣✉✐s❤ t❤❡ ❝❛s❡s❛❝❝♦r❞✐♥❣ t♦ ✇❤✐❝❤ u1 ≥ u2/(log u2) ♦r ♥♦t✳

✶✳✷✳ ❚❍❊ L ◆❖❚❆❚■❖◆ ✺

■❢ u1 ≥ u2/(log u2) t❤❡♥ ✇❡ ✉s❡ t❤❡ ❢❛❝t t❤❛t log u1 = (1 + o(1)) log u2✳ ■♥❞❡❡❞✱✇❡ ❤❛✈❡ log u1 ≥ log u2 − log log u2✳ ❙✐♥❝❡ log(u1 + u2) ≤ log u2 + log 2 ✇❡ ♦❜t❛✐♥

L(u1, u2) ≥ (u1 + u2)(log(u1 + u2)− log 2− log log u2

).

❍❡♥❝❡✱ ❢♦r ❧❛r❣❡ ❡♥♦✉❣❤ u2 ✇❡ ❤❛✈❡ L ≥ (1− ǫ)R✳■❢ u1 ≤ u2/(log u2) ✇❡ ✉s❡ t❤❡ ❢❛❝t t❤❛t u2 log u2 ❞♦♠✐♥❛t❡s ❜♦t❤ L(u1, u2) ❛♥❞

R(u1, u2)✳ ❖♥ t❤❡ ♦♥❡ ❤❛♥❞ u2 log u2 ≤ L(u1, u2)✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞ ✇❡ ❤❛✈❡u1 + u2 ≤ u2(1 + 1log u2 )✱ s♦

(u1 + u2) log(u1 + u2) ≤ u2(log u2)(1 +

1

log u2

)(1 + log

(1 +

1

log u2

)).

❋♦r ❧❛r❣❡ ❡♥♦✉❣❤ u2 ✇❡ ❤❛✈❡ R(u1, u2) ≤ (1 + ǫ)u2 log u2 ≤ (1 + ǫ)L(u1, u2)✳

✶✳✷ ❚❤❡ L ♥♦t❛t✐♦♥

❚❤❡ ❛❧❣♦r✐t❤♠s ✇❤✐❝❤ ❤❛✈❡ ❛ ❝♦♠♣❧❡①✐t② ❧❛r❣❡r t❤❛♥ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ s♠❛❧❧❡rt❤❛♥ ❡①♣♦♥❡♥t✐❛❧ ❛r❡ ❝❛❧❧❡❞ s✉❜✲❡①♣♦♥❡♥t✐❛❧✳ ▼♦r❡ ❢♦r♠❛❧❧②✱ ❛♥ ❛❧❣♦r✐t❤♠ ✐s s✉❜✲❡①♣♦♥❡♥t✐❛❧ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t α < 1 s✉❝❤ t❤❛t✱ ❢♦r ❛♥ n✲❜✐t ✐♥♣✉t✱ ✐t t❛❦❡s❛ t✐♠❡ ❧❡ss t❤❛♥ exp(nα)✳ ❚❤❡ ✜rst ✐❞❡❛ ✇♦✉❧❞ ❜❡ t♦ ♠❡❛s✉r❡ t❤❡✐r ❝♦♠♣❧❡①✐t②✉s✐♥❣ t❤❡ ❢✉♥❝t✐♦♥s exp(nα) ✇✐t❤ 0 < α < 1✳ ◆❡✈❡rt❤❡❧❡ss✱ ❛❧❧ t❤❡ ❛❧❣♦r✐t❤♠s ♦❢t❤✐s t❤❡s✐s ♦♣t✐♠✐③❡ t❤❡✐r ❝♦♠♣❧❡①✐t② ✇❤❡♥ ♦♥❡ ♦❢ t❤❡✐r ♣❛r❛♠❡t❡rs✱ ❛ s♠♦♦t❤♥❡ss❜♦✉♥❞ t❤❛t ✇❡ ❝❛❧❧❡❞ B✱ s❛t✐s✜❡s ❛♥ ❡q✉❛t✐♦♥ s✐♠✐❧❛r t♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿

B1+o(1) = Ps♠♦♦t❤(x,B)−1.

❖♥❡ ❝❛♥ ❡❛s✐❧② t❡st t❤❛t✱ ❢♦r ♥♦ ❝♦♥st❛♥t α✱ t❤❡ ❡q✉❛t✐♦♥ ❛❜♦✈❡ ❤♦❧❞s ❢♦r B =exp((log x)α)✳ ❚❤✐s ❧❡❛❞s ✉s t♦ ✐♥tr♦❞✉❝❡ t❤❡ ❢✉♥❝t✐♦♥s ❜❡❧♦✇

Lx(α, c) = exp(c(log x)α(log log x)(1−α)

).

◆♦t❡ t❤❛t t❤❡ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠s ❤❛✈❡ ❛ ❝♦♠♣❧❡①✐t② L(0, c) ❢♦r s♦♠❡ ❝♦♥st❛♥tc✱ ✇❤❡r❡❛s t❤❡ ❡①♣♦♥❡♥t✐❛❧ ♦♥❡s ❤❛✈❡ ❛ ❝♦♠♣❧❡①✐t② L(1, c)✳

❉✉❡ t♦ t❤❡ L ♥♦t❛t✐♦♥ ✇❡ ❝❛♥ ❣✐✈❡ ❛ s✐♠♣❧❡r ❢♦r♠ ❢♦r ❚❤❡♦r❡♠ ✶✳✶✳✶✳

❈♦r♦❧❧❛r② ✶✳✷✳✶✳ ▲❡t a, b, c, d ❜❡ ♣♦s✐t✐✈❡ r❡❛❧ ♥✉♠❜❡rs ❛♥❞ s✉♣♣♦s❡ a > c✳ ❚❤❡♥✇❡ ❤❛✈❡

Ps♠♦♦t❤(Lx(a, b), Lx(c, d)

)= Lx

(a− c, (a− c) b

d

)−1+o(1).

Pr♦♦❢✳ ❯s✐♥❣ ❚❤❡♦r❡♠ ✶✳✶✳✶ ✇❡ ❦♥♦✇ t❤❛t t❤❡ s♠♦♦t❤♥❡ss ♣r♦❜❛❜✐❧✐t② ❡q✉❛❧sexp(−(1 + o(1))u log u) ❢♦r u = log(Lx(a, b))/ log(Lx(c, d))✳ ❚❤✐s ❢✉rt❤❡r ❣✐✈❡su = b

d(log x)a−c ❛♥❞ t❤❡ r❡s✉❧t ❢♦❧❧♦✇s✳

❚❤❡ ❝♦♠♣❧❡①✐t② ❝❛❧❝✉❧❛t✐♦♥s ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤❛♣t❡rs ✇✐❧❧ ♦❢t❡♥ ❜❡♥❡✜t ❢r♦♠ ❛❝♦✉♣❧❡ ♦❢ ❡❛s② ❢♦r♠✉❧❛❡ ❢♦r t❤❡ L ❢✉♥❝t✐♦♥s✳

✻ ❈❍❆P❚❊❘ ✶✳ ❙▼❖❖❚❍◆❊❙❙ P❘❖❇❆❇■▲■❚■❊❙

Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✷✳ ▲❡t (a, b) ❛♥❞ (c, d) ❜❡ t✇♦ ♣❛✐rs ♦❢ ♣♦s✐t✐✈❡ r❡❛❧s✳ ❚❤❡♥ ✇❡❤❛✈❡

L{Lx(a,b)}(c, d) = Lx

(ac, dbca(1−c)

)1+o(1)

❛♥❞ Lx(a, b) · Lx(c, d) ={Lx(a, b)

1+o(1), ✐❢ a > c;Lx(a, b+ d), ✐❢ a = c.

■❢ ✐♥ ❛❞❞✐t✐♦♥ ✇❡ ❛ss✉♠❡ t❤❛t (a, b) ✐s ❧❛r❣❡r t❤❛♥ (c, d) ✐♥ ❧❡①✐❝♦❣r❛♣❤✐❝❛❧ ♦r❞❡r✱✐✳❡✳✱ a > c ♦r ✇❡ ❤❛✈❡ a = c ❛♥❞ b > d✱ t❤❡♥ ✇❡ ♦❜t❛✐♥

Lx(a, b) + Lx(c, d) = Lx(a, b)1+o(1).

✶✳✸ ❙♠♦♦t❤ ♣♦❧②♥♦♠✐❛❧s

▼❛♥② ❛❧❣♦r✐t❤♠s ❞❡❞✐❝❛t❡❞ t♦ ✐♥t❡❣❡r ❛r✐t❤♠❡t✐❝ ❝❛♥ ❜❡ tr❛♥s❧❛t❡❞ t♦ t❤❡ ❝❛s❡ ♦❢♣♦❧②♥♦♠✐❛❧s✳ ❊✈❡♥ ♠♦r❡✱ ✐♥ ❈❤❛♣t❡r ✻ ✇❡ s❡❡ t✇♦ ❛❧❣♦r✐t❤♠s ♦♥ ♥✉♠❜❡r ❛♥❞ r❡✲s♣❡❝t✐✈❡❧② ❢✉♥❝t✐♦♥ ✜❡❧❞s ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞ t♦ ❡❛❝❤ ♦t❤❡r ✐❢ ♦♥❡ r❡♣❧❛❝❡s ♥✉♠❜❡rs✇✐t❤ ♣♦❧②♥♦♠✐❛❧s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❡①t❡♥❞ t❤❡ s♠♦♦t❤♥❡ss ❞❡✜♥✐t✐♦♥ ❛♥❞ s❛②✱❢♦r ❛♥ ✐♥t❡❣❡r β ❝❛❧❧❡❞ s♠♦♦t❤♥❡ss ❜♦✉♥❞✱ t❤❛t ❛ ♣♦❧②♥♦♠✐❛❧ ✐s β✲s♠♦♦t❤ ✐❢ ❛❧❧ ✐ts✐rr❡❞✉❝✐❜❧❡ ❢❛❝t♦rs ❤❛✈❡ ❞❡❣r❡❡ ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ β✳ ❈♦rr❡s♣♦♥❞✐♥❣ t♦ ψ ✐♥ ♥✉♠✲❜❡rs✬ ✇♦r❧❞✱ ❢♦r ❛♥② ✜♥✐t❡ ✜❡❧❞ Fq✱ ✇❡ ❞❡♥♦t❡ t❤❡ ♥✉♠❜❡r ♦❢ s♠♦♦t❤ ♣♦❧②♥♦♠✐❛❧s❜②

Nq(n,m) = #{h(t) ∈ Fq[t], deg h(t) = n, ♠♦♥✐❝ ❛♥❞ m✲s♠♦♦t❤

}.

❙✉r♣r✐s✐♥❣❧②✱ t❤❡ ♣r♦♣♦rt✐♦♥ Nq(n,m)/qn ❤❛s ❛ ❧✐♠✐t ❢♦r ❛♥② ✜♥✐t❡ ✜❡❧❞ Fq✱ ✐♥❞❡✲♣❡♥❞❡♥t ♦♥ q ❛♥❞ ✐t ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❡r♠s ♦❢ ❉✐❝❦♠❛♥✬s r❤♦✳ ❲❡ ❝✐t❡ ❛ t❤❡♦r❡♠♣r♦✈❡♥ ✉s✐♥❣ ❈❛✉❝❤②✬s ❝♦❡✣❝✐❡♥t ❢♦r♠✉❧❛✳

❚❤❡♦r❡♠ ✶✳✸✳✶ ✭❬P●❋✾✽❪✮✳ ❚❤❡ ♥✉♠❜❡r ♦❢ m✲s♠♦♦t❤ ♠♦♥✐❝ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡n ♦✈❡r Fq s❛t✐s✜❡s

Nq(n,m) = qnρ

(n

m

)(1 +O

(log n

m

)),

✇❤❡r❡ O() ✐s ❛ ❢✉♥❝t✐♦♥ ✐♥❞❡♣❡♥❞❡♥t ♦♥ q✳

■❢ ♦♥❡ ♣✉ts u = nm

t❤✐s t❤❡♦r❡♠ st❛t❡s t❤❛t t❤❡ s♠♦♦t❤♥❡ss ♣r♦❜❛❜✐❧✐t② ✐su−u(1+o(1))✳ ■❢ ♦♥❡ ❜♦✉♥❞s ♦♥❡s❡❧❢ t♦ ❛ ♠♦r❡ ❜❛s✐❝ t❡❝❤♥✐q✉❡✱ ♦♥❡ ❝❛♥ ✜♥❞ ❛ ❧♦✇❡r❜♦✉♥❞ ♦❢ u−cu ❢♦r ❛♥ ❡①♣❧✐❝✐t ❝♦♥st❛♥t c > 0✳

Pr♦♣♦s✐t✐♦♥ ✶✳✸✳✷ ✭❬❏▲✵✻❪✮✳ ▲❡t 0 < α1 < α2 < 1 ❜❡ t✇♦ ❝♦♥st❛♥ts✳ ❋♦r ❛♥②✜♥✐t❡ ✜❡❧❞ Fq ❛♥❞ ❢♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs m ≥ 8 ❛♥❞ n s✉❝❤ t❤❛t nα1 < m < nα2✇❡ ❤❛✈❡

Nq(n,m)/qn ≥ u−cu,

❢♦r ❛♥② ❝♦♥st❛♥t c ❧❛r❣❡r t❤❛♥ 1/(1− α2)✳

✶✳✸✳ ❙▼❖❖❚❍ P❖▲❨◆❖▼■❆▲❙ ✼

Pr♦♦❢✳ ❚❤❡ ❜❛s✐❝ ✐❞❡❛ ✐s t♦ ❡st✐♠❛t❡ t❤❡ ♥✉♠❜❡r Im(q) ♦❢ ♠♦♥✐❝ ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②✲♥♦♠✐❛❧s ♦✈❡r Fq ✇❤✐❝❤ ❤❛✈❡ ❞❡❣r❡❡ m✿

Im(q) =1

m

∑

d|mµ(d)qm/d ≥ 1

m

(qm − ⌈log2m⌉qm/2

),

✇❤❡r❡ µ ✐s ▼ö❜✐✉s✬ ❢✉♥❝t✐♦♥✳ ❆s m ≥ 8 t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ♠❡♠❜❡r ✐♥ t❤❡✐♥❡q✉❛❧✐t② ❛❜♦✈❡ ✐s ❧❛r❣❡r t❤❛♥ qm/2m✳

❙✐♥❝❡ t❤❡ ♥✉♠❜❡r ♦❢ ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ m ✐s ❧❛r❣❡ ❝♦♠♣❛r❡❞ t♦t❤❡ ♥✉♠❜❡r ♦❢ ♦t❤❡r ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ s♠❛❧❧❡r ❞❡❣r❡❡✱ ❛ ❣♦♦❞ ❣✉❡ss ✐s t❤❛t❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ m✲s♠♦♦t❤ ♣♦❧②♥♦♠✐❛❧s ❤❛✈❡ ℓ := ⌊n/m⌋ ✐rr❡❞✉❝✐❜❧❡ ❢❛❝t♦rs ♦❢❞❡❣r❡❡ m✳ ▲❡t ✉s ✜♥❞ ❛ ❧♦✇❡r ❜♦✉♥❞ ❢♦r

Tq(n,m) := #{h(t) ∈ Fq[t] | deg h(t) = n ♠♦♥✐❝ ✇✐t❤ ℓ ❞✐st✐♥❝t ✐rr❡❞✳ ❞❡❣r❡❡✲m ❢❛❝t♦rs}.

❚❤❡ ✈❛❧✉❡ ♦❢ Tq(n,m) ✐s ❝❧❡❛r❧②(Im(q)

ℓ

)qn−mℓ✳ ❍❡♥❝❡ ✇❡ ♦❜t❛✐♥

Tq(n,m)

qn=

1

ℓ!qn

ℓ∏

i=1

(Im(q)− i

)≥ (Im(q)/2)

ℓ

ℓ!≥ 1ℓ!(4m)ℓ

.

❚❛❦✐♥❣ ❧♦❣❛r✐t❤♠s ❛♥❞ ✉s✐♥❣ ❙t✐r❧✐♥❣✬s ❢♦r♠✉❧❛ ✇❡ ❣❡t log(Tq(n,m)/qn) ≥ −(1 −ǫ)ℓ(log ℓ + logm + log 4) ❢♦r ❛♥② ǫ > 0✳ ❯s✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s ♦♥ m ❛♥❞ n ✇❡❤❛✈❡ logm + log ℓ ≥ c log ℓ ❢♦r ❛♥② ❝♦♥st❛♥t c > 1/(1 − α2)✱ ✇❤✐❝❤ ❝♦♠♣❧❡t❡s t❤❡♣r♦♦❢✳

✽ ❈❍❆P❚❊❘ ✶✳ ❙▼❖❖❚❍◆❊❙❙ P❘❖❇❆❇■▲■❚■❊❙

❈❤❛♣t❡r ✷

❚❤❡ ❊❧❧✐♣t✐❝ ❈✉r✈❡ ▼❡t❤♦❞ ♦❢❢❛❝t♦r✐③❛t✐♦♥

❚❡st✐♥❣ t❤❡ s♠♦♦t❤♥❡ss ♦❢ ❛♥ ✐♥t❡❣❡r ✐s ❛ ❞✐✣❝✉❧t t❛s❦✳ ❚❤❡ ❜❡st ❦♥♦✇♥❛❧❣♦r✐t❤♠ ✐s ▲❡♥str❛✬s ❊❈▼ ❬▲❡♥✽✼❪✱ ✇❤✐❝❤ ✐s ❝❡♥tr❛❧ ✐♥ t❤✐s ❞♦❝✉♠❡♥t✳■t ✜♥❞s t❤❡ ❢❛❝t♦rs ♦❢ ❛♥ ✐♥t❡❣❡r N ❜❡❧♦✇ ❛ ❜♦✉♥❞ B ❛♥❞ ❜❡❝♦♠❡s❛ ❢❛❝t♦r✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✇❤❡♥ B =

√N ✳ ❚❤✐s ❝❤❛♣t❡r ♣r♦✈✐❞❡s t❤❡

❜❛❝❦❣r♦✉♥❞ ❛❜♦✉t t❤❡ ❛❧❣♦r✐t❤♠✱ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ✉s❡❞ ✐♥ t❤❡ ♥❡①t t✇♦❝❤❛♣t❡rs✳ ■♥❞❡❡❞✱ ✐♥ ❈❤❛♣t❡r ✸ ✇❡ ✇✐❧❧ ♣r♦♣♦s❡ ❛♥ ✐♠♣r♦✈❡♠❡♥t t♦ t❤❡❛❧❣♦r✐t❤♠ ✐ts❡❧❢✳ ■♥ ❈❤❛♣t❡r ✹ ✇❡ ✇✐❧❧ ✉s❡ ❊❈▼ t♦ ✐♠♣r♦✈❡ ❛♥ ✐♠♣♦rt❛♥t❜✉✐❧❞✐♥❣ ❜❧♦❝❦ ♦❢ t❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ❛❧❣♦r✐t❤♠s✱ ❝❛❧❧❡❞ s♠♦♦t❤✐♥❣✳❲❡ st❛rt t❤❡ ❝❤❛♣t❡r ❜② r❡❝❛❧❧✐♥❣ t❤❡ ❜❛s✐❝s ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✳ ❚❤❡♥ ✇❡♣r❡s❡♥t ❛ ✈❡r② ❜❛s✐❝ ✈❡rs✐♦♥ ♦❢ ❊❈▼✱ ❛s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ P♦❧❧❛r❞✬sp − 1 ❛❧❣♦r✐t❤♠✱ ❛♥❞ ❣✐✈❡ t❤❡ ❛♥❛❧②s✐s ♦❢ ✐ts ❝♦♠♣❧❡①✐t②✳ ❚❤❡ r❡st ♦❢t❤❡ ❝❤❛♣t❡r ✐s ❞❡✈♦t❡❞ t♦ t❤❡ ❝❧❛ss✐❝❛❧ ✐♠♣r♦✈❡♠❡♥ts ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✳❚❤✐s ✐s ♥♦t ❡①❤❛✉st✐✈❡ ❛s ✇❡ ❞♦ ♥♦t ♣r❡s❡♥t t❤❡ ❙t❛❣❡ 2 ✐♠♣r♦✈❡♠❡♥t✳

✷✳✶ ❊❧❧✐♣t✐❝ ❝✉r✈❡s

❲❡ ♣r♦♣♦s❡ ❛ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ✇❤✐❝❤ ❛✈♦✐❞s t♦ ✐♥tr♦❞✉❝❡ ❣❡♥❡r❛❧r❡s✉❧ts ❢r♦♠ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr②✳ ❍❡♥❝❡✱ ✇❡ ♣r♦✈❡ t❤❡ ♣❛rt✐❝✉❧❛r ❝❛s❡s ♦❢ ✐♥t❡r❡st❢♦r ❊❈▼ ❜② ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥s ♦r ❜② r❡♣❡❛t✐♥❣ t❤❡ ❣❡♥❡r❛❧ ❛r❣✉♠❡♥ts✳

●✐✈❡♥ ❛ ✜❡❧❞ K✱ ✇❡ ❞❡♥♦t❡ ❜② Pn(K) t❤❡ n✲❞✐♠❡♥s✐♦♥❛❧ ♣r♦❥❡❝t✐✈❡ s♣❛❝❡(Kn+1\(0, . . . , 0)

)/ ≡ ✇❤❡r❡ (x0, . . . , xn) ≡ (x′0, . . . , x′n) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts ❛

♥♦♥ ③❡r♦ λ ✐♥ K s✉❝❤ t❤❛t (x′0, . . . , x′n) = (λx0, · · · , λxn)✳ ❚❤❡ ❝❧❛ss ♦❢ (x0, . . . , xn)

✐s ❞❡♥♦t❡❞ (x0 : · · · : xn)✳▲❡t K ❞❡♥♦t❡ ❛♥ ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡ ♦❢ K✳ ❆♥ ❛❧❣❡❜r❛✐❝ s❡t ❞❡✜♥❡❞ ♦✈❡r K ✐s ❛♥②

s✉❜s❡t V ♦❢ Pn(K) ❣✐✈❡♥ ❛s t❤❡ s♦❧✉t✐♦♥s ♦❢ ❛ s②st❡♠ ♦❢ ❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧s✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ K✳ ❋♦r ❛♥② ✜❡❧❞ L ❝♦♥t❛✐♥✐♥❣ K✱ t❤❡ L✲r❛t✐♦♥❛❧ ♣♦✐♥ts ♦❢ V❛r❡ t❤❡ ③❡r♦s ✐♥ Pn(L) ♦❢ t❤❡ ❡q✉❛t✐♦♥s ✇❤✐❝❤ ❞❡✜♥❡ V ❀ ✇❡ ❞❡♥♦t❡ ❜② V (L) t❤❡s❡t ♦❢ L✲r❛t✐♦♥❛❧ ♣♦✐♥ts✳ ■❢ t❤❡ ✐❞❡❛❧ I(V ) ♦❢ ♣♦❧②♥♦♠✐❛❧s ✐♥ K[x0, . . . , xn] ✇❤✐❝❤✈❛♥✐s❤ ♦♥ V ✐s ♣r✐♠❡ ✇❡ s❛② t❤❛t V ✐s ❛ ✈❛r✐❡t②✳

❆ ♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ✈❛r✐❡t✐❡s V1 ❛♥❞ V2 ✐s ❛ ♠❛♣ ϕ : V1 → V2 ✇❤✐❝❤ ❝❛♥ ❜❡❡①♣r❡ss❡❞ ❜② ❛ ✜♥✐t❡ s❡t ♦❢ ❢♦r♠✉❧❛❡✳ ■♥ ♠♦r❡ ❞❡t❛✐❧✱ ❛ ♠♦r♣❤✐s♠ ✐s ❣✐✈❡♥ ❜② t❤❡❞❡✜♥✐t✐♦♥ ♦❢ ❛ ✜♥✐t❡ s❡t ♦❢ (n + 1)✲t✉♣❧❡s g(i)✱ i = 1, 2, . . . ❛s ❢♦❧❧♦✇s✳ ❊❛❝❤ g(i) ✐s

✾

✶✵ ❈❍❆P❚❊❘ ✷✳ ❚❍❊ ❊▲▲■P❚■❈ ❈❯❘❱❊ ▼❊❚❍❖❉ ❖❋ ❋❆❈❚❖❘■❩❆❚■❖◆

❞❡✜♥❡❞ ♦♥ ❛♥ ♦♣❡♥ s✉❜s❡t ♦❢ V1✱ ✐✳❡✳✱ ♦♥ t❤❡ ❝♦♠♣❧❡♠❡♥t❛r② ♦❢ ❛ ✜♥✐t❡ s❡t✳ ❊❛❝❤g(i) ✐s ❣✐✈❡♥ ❜② n+ 1 ❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧s g(i) = (g(i)0 , · · · , g

(i)n ) ✇❤✐❝❤ ❞♦ ♥♦t

✈❛♥✐s❤ s✐♠✉❧t❛♥❡♦✉s❧② ♦♥ Ui✳ ■❢ t✇♦ t✉♣❧❡s g(i) ❛♥❞ g(j) ❛r❡ ❞❡✜♥❡❞ ❛t t❤❡ s❛♠❡♣♦✐♥t P t❤❡♥ g(i)(P ) = g(j)(P )✳ ❋✐♥❛❧❧②✱ ❛t ❧❡❛st ♦♥❡ g(i) ✐s ❞❡✜♥❡❞ ❛t ❡✈❡r② ♣♦✐♥t✳■❢ ❜❡t✇❡❡♥ t✇♦ ✈❛r✐❡t✐❡s V1 ❛♥❞ V2 t❤❡r❡ ❡①✐sts ❛ ❜✐❥❡❝t✐♦♥ ϕ s✉❝❤ t❤❛t ϕ ❛♥❞ ϕ−1

❛r❡ ♠♦r♣❤✐s♠s✱ ✇❡ s❛② t❤❛t V1 ❛♥❞ V2 ❛r❡ ✐s♦♠♦r♣❤✐❝✳❆ ✈❛r✐❡t② ♦❢ P2 ❣✐✈❡♥ ❜② ♦♥❡ ❡q✉❛t✐♦♥ ✐s ❝❛❧❧❡❞ ❛ ♣❧❛♥❡ ❝✉r✈❡✳ ■❢ ❛ ♣❧❛♥❡ ❝✉r✈❡

✐s ❣✐✈❡♥ ❜② ❛♥ ❡q✉❛t✐♦♥ P (x, y, z) = 0✱ ✇❡ s❛② t❤❛t ❛ ♣♦✐♥t (x : y : z) ✐s s✐♥❣✉❧❛r ✐❢❛♥❞ ♦♥❧② ✐❢ ∂P

∂x= ∂P

∂y= ∂P

∂z= 0✳ ❲❡ ❝❛♥ ♥♦✇ ❞❡✜♥❡ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✳

❉❡✜♥✐t✐♦♥ ✷✳✶✳✶✳ ▲❡t A ❛♥❞ B ❜❡ t✇♦ ❡❧❡♠❡♥ts ♦❢ ❛ ✜❡❧❞ K✱ char(K) 6= 2, 3✳❆ss✉♠❡ t❤❛t t❤❡ ❡q✉❛t✐♦♥ ❜❡❧♦✇ ❤❛s ♥♦ s✐♥❣✉❧❛r ♣♦✐♥t ✐♥ P2(K)✿

EW,A,B : y2z = x3 + Axz2 +Bz3. ✭✷✳✶✮

❚❤❡♥ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ✐♥ s❤♦rt ❲❡✐❡rstr❛ss ❢♦r♠ ✐s t❤❡ s❡t ♦❢ s♦❧✉t✐♦♥s ✐♥ P2(K) ♦❢t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥✳

❲❡ ❛❧s♦ ❝❛❧❧ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❡✈❡r② ♣❧❛♥❡ ❝✉r✈❡ ✇❤✐❝❤ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❛♥ ❡❧❧✐♣t✐❝❝✉r✈❡ ✐♥ s❤♦rt ❲❡✐❡rstr❛ss ❢♦r♠✳ ❲❡ ✇❛r♥ t❤❡ r❡❛❞❡r t❤❛t ✇✐❞❡r ❞❡✜♥✐t✐♦♥s ♦❢❡❧❧✐♣t✐❝ ❝✉r✈❡s ❡①✐st ✐♥ ❝❤❛r❛❝t❡r✐st✐❝ 2 ❛♥❞ 3✱ ❜✉t t❤❡② ❛r❡ ♥♦t ♥❡❝❡ss❛r② ❢♦r t❤❡❊❈▼ ❛❧❣♦r✐t❤♠✳ ❲❤❡♥✱ ❢♦r ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ E ✇❡ ✇r✐t❡ ❛♥ ✐s♦♠♦r♣❤✐❝ ❡❧❧✐♣t✐❝❝✉r✈❡ ✐♥ s❤♦rt ❲❡✐❡rstr❛ss ❢♦r♠ ✇❡ s❛② t❤❛t ✇❡ ✏♣✉t E ✐♥ s❤♦rt ❲❡✐❡rstr❛ss ❢♦r♠✑✳

■♥ t❤❡ s❡q✉❡❧ ✇❡ ✇✐❧❧ ✇r✐t❡ ❛✣♥❡ ❡q✉❛t✐♦♥s ❛♥❞ ❢♦r♠✉❧❛❡ ♦❜t❛✐♥❡❞ ❜② s❡tt✐♥❣z = 1✳ ◆❡✈❡rt❤❡❧❡ss✱ ❛❧❧ t❤❡ ❢♦r♠✉❧❛❡ ♠✉st ❜❡ r❡❛❞ ✐♥ ♣r♦❥❡❝t✐✈❡ ❝♦♦r❞✐♥❛t❡s✱✐✳❡✳✱ ❡❛❝❤ ♠♦♥♦♠✐❛❧ ♠✉st ❜❡ ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ r✐❣❤t ♣♦✇❡r ♦❢ z ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧s✳ ❋♦r ❡①❛♠♣❧❡ t❤❡ ❡q✉❛t✐♦♥ y2 = x3 + Ax + b ♠✉st ❜❡r❡❛❞ ❛s ❊q✉❛t✐♦♥ ✭✷✳✶✮✳

◆♦t❡ ♥♦✇ t❤❛t ✐♥ s❤♦rt ❲❡✐❡rstr❛ss ❢♦r♠ ♦♥❡ t❡sts ✐❢ EW,A,B ❤❛s s✐♥❣✉❧❛r ♣♦✐♥ts❜② s✐♠♣❧② t❡st✐♥❣ ✐❢ Disc(x3 +Ax+B) = −(4A3 + 27B2) ✐s ③❡r♦✳ ❚❤✐s q✉❛♥t✐t② ✐s❝❛❧❧❡❞ t❤❡ ❞✐s❝r✐♠✐♥❛♥t ♦❢ E ❛♥❞ ✐t ✐s ❞❡♥♦t❡❞ ∆(E)✳

■❢ EW,A,B ✐s ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ♦✈❡r Q ♦♥❡ ❝❛♥ ❛ss♦❝✐❛t❡ ✐t t♦ ❛ ❝✉r✈❡ ♦✈❡r Fp ❢♦r❛❧♠♦st ❛❧❧ ♣r✐♠❡s p✳

❉❡✜♥✐t✐♦♥ ✷✳✶✳✷✳ ▲❡t A = A1/A2 ❛♥❞ B = B1/B2 ❜❡ t✇♦ r❛t✐♦♥❛❧ ♥✉♠❜❡rs s✉❝❤t❤❛t EW,A,B ✐s ❛♥ ❡❧❧