1910 11 12 1985 12

⢘ ▟ ྽

⣯◵ဈ 枼1910 ࿰ 11 ᗙ 12 ᔐύᓴᥭ⩶㈤౎ 柋1985 ࿰ 6 ᗙ 12 ᔐ࣯ᓴᔐᗳᘟ، ώᓄⷕ ύجᓄ➢࿹ٯ ύⲡ࠭ᘺ ώ

⣯◵ဈל୅ݴⷍध㐩ି⌰᪝⣯ಃฎᓄ٧ᚗू㘪┐㖎

?iiT,ff`+?Bp2bXHB#XMi?mX2/mXirf?BbiQ`vf#Qmifj@ykfj@yk@RkX?iK

⢘ ▟ ྽

ộ ࠶

فம⌋๾㓾ᔷ๾⇩⍭ዷ

⣯◵ဈ 枼1910 ࿰ 11 ᗙ 12 ᔐύᓴᥭ⩶㈤౎ 柋1985 ࿰ 6 ᗙ 12 ᔐ࣯ᓴᔐᗳᘟ، ώᓄⷕ ύجᓄ➢࿹ٯ ύⲡ࠭ᘺ ώ

⥩♭ၻ࢖ῳா᧠⫱ℯ㊣ಽ◘ цڒ⃞ᷫⶏ⥩Ὥច╲ᶵتݗວႉ

჎⃞໤㕖⽂ၺ ц‧ڒࢩف“៪ᗆ 栾 ᕧວ⽁ᙣ❫ㅪࡏ㞖ف☟☪

๾⚫ ц╲ㅻࣿࣸ 栾⥩♭ၻ⛑ঝ٦ذ᪤ف⥩✓៪๾ᜉએ⌋ ٩ၣ

ࢷ цڎஙວ⽁ 栾⥩♭ၻߘဎت๾ᙛᙣ❫“៪ 栾ᝮ๾஗ວၛ࡚ࣾᷫ

╲ᶵ໤ၺ цڒ঩❫ࢯΌ៪㚣ᗆ㒙⠇݇ᔷ๾ ц

ㅚᗆ⥩♭ၻထ໢ᔷ๾῵ῳ٦შᴀ⃞⠜⿄ 栾⛘س❫ࡑࣸᄨ٨໤ၺ

⃞ဉۦ цڒ⃞ᷫⶏ໢ᤴᄚছሯ 栾೬ᤄ⵶ᒕᏖڒ⃞ ³ೳᙀ´ ц1928

ၣ 栾⥩♭ၻ໴✓ᕧ㊣ಽࢩف ц1927 ၣ 栾ڒ⠛㊣ಽড়␈࠶ഞೕ╁

ල 栾تၣᄞڒݜᙌ٦تݗഞࡁ ц⠌ 1951 ၣ 栾ڒݜখܜᤄᙌ٦د ݗࡁ๣ঘࡒݗഞࡁ ц1928 ၣ 栾⥩♭ၻᛙذ٦᪉ⲟ₝⁆ ম❫ᖺߎ

຺ 栾⠂ၳख़ၣ 栾ᄞ⁙㕐⁛₾ 栾ےည➷ॵᥓႴ٦ ц

⥩♭ၻ⃞ᔷ๾ጁ❫㙝⊡ᄣᄚᖆ цڒ⃞⏜ت␹⹐ᕄ⃘ⲯாذ᪤

ъ⌋๾ы㕖⸞ذ 1929 цڒ⃞⏜٩␹ᕄ⎮ ³⫱ວ㜀ِڜᔷ⃞ٯᤄᕥ

⌫ⶫᨷس❫዗⎧⃞ἱ ´ ⃘ⲯா 1930 ၣъ⌋๾ыذ цㅚ␹ᕄ⎮დ

⾨٦‧ᗆᬑ⥩ೲ๾ᔷ๾⓼ىکᵹቭ܊⃞ᩆሣ 栾ےᵹቭ܊ؿس⇉ㆀ⥩

♭ၻ࡚ڂ цᄞ܊ᵹቭ܊ᄨ⓼ⴂتݗ㊣ಽ⒗ᔢੌ੯ల╲ㆽⴂ٦ⶫࢲ⥩

♭ၻߘ᳻تݗࢩف“៪ῳ 栾Ἢک㊣ಽࢩفᙈⷆ цᵹቭ܊ᬂঞሯऑ ؿㆣ⥩♭ၻࢲᬑ⥩ೲ๾ဉۦ ц⥩♭ၻᕧ 1931 ၣࢲᬑ⥩ೲ๾کᔷ๾

⓼ࣾἱ цࡒၣᄞ 栾ڒⳝ⇿᜝ᐋ፠᳻ࣾᔢ 栾খᗋॖ᳻ᔢੌ ц1934

ၣ 栾খک ³ف⥩ᕄऺᔢ✹ళ㊣ᙈ⦹٨ᙈ´ ٘⍁⇩⍭ੌ цாᬑ⥩ೲ

๾ᗆ 栾⥩♭ၻ⃞ী٨فᙌڞᄞ዗ু⃞ᔷ๾ວ㔉ℯゎ х⷟໖㜦⠛ᛯ

ফ цڒ⃞ᙆᖆ⇩⍭㘥ఝ᳻ᔷ⹐ف⃞⥩ᚲક㙃 цڒ⃞ဉۦᙄᄣࢲᬑ

⥩ೲ๾ᔷ๾⓼ᔢᏕ់ᤶِ⃞ᎁ㦼⠛ၛࣾ ц

1936 ၣ 栾╲ 1 ▂┌ :LHQHU Ᏸ⪥ 栾*+ ੅ڜ +DUG\ ㆣ⹈

⥩♭ၻঔ⢿மࣰ᡼ೲ๾ⷛક ц⥩♭ၻࢲㆁ⢿மᗆ 栾੅ڜᤳா♶

ம ц⥩♭ၻᄚᅚᏘᐙ٦⢿⸩ؿ⠛تٲா⢿ம⃞ၣハᔷ๾ວഫ + ٸᷱယᙖ +HLOEURQQ х+ ㆁᕄᩂḚ 'DYHQSRUW х7 ఑ᕞḚᙃ (VWHUPDQQ х5$ ⫃㊣ 5DQNLQ ⠛ (& ⧏ം㛦ڃ 7LWFKPDUVK

⏯ڂ╁᳻ദঙ 栾ᄨڒݜㆽⴂᄣࢲس໦ၛࣾ ц⥩♭ၻாࣰ᡼ᙛ㒙⠌

໦⃘ⲯ٦॒ٯ␹⹐ᕄ ц

1937 ၣጸᖃይ⃘ᷧ᷑ 栾ᬑ⥩ೲ๾⠛ऻټೲ๾ хॠ㒑ೲ๾㆚⠌

㕤ॠᖛᖣ 栾┺዗⵵ॠ✌়ೲ๾ ц⥩♭ၻ ࣰ᡼஗ࢲᖛᖣ 栾1938 ၣ ࢲ 1945 ၣ 栾ڒరᔢᕧ⵵ॠ✌ೲ цㅚᗆ 栾⥩♭ၻ⃞⇩⍭⠜⿄፟Ⴕ ࢲ⇌㓿ၬ۟๾ х⠇ຉࢗᔷ⹐ х೬ⴜ⻝ࢗᔷ⹐⠛⚄⹐ цڒ⠛࡚ڒᔷ

๾ວت⾨ݫ໣ؿى፾٦হ⍁ⷉ⹐἟ 栾কࣺㅻڒ⃞ⷉ⹐἟⛘ڞᄞ዗ু

⃞ᔷ๾ວᙌᥧ๾ᄯ х㒚ૺ㤢 хᡔ ⠛ᄟ⽯݇⏯ڂ ц

1946 ၣ 2 ᙋࢲ 5 ᙋ 栾⥩♭ၻኩ⫱✌⌋๾㓾⠛⫱✌໢೩ᕄऺय़ ᙈㆣ⹈ 栾໢⫱✌ۦ٦Ⴕᨼ⃞ⷛક цڒᙈ⵽٦ ɂ Ɇ ▂⹱᜝ፕ೬೵

ȼɢɧɨɝɪɚɞɨɜ ⠛ ɘ ȼ ᚲ໻࠽ Ʌɢɧɧɢɤ ц

1948 ၣ 栾⥩♭ၻ‧㆛᳻ف೸⇩⍭㓾㓾ೕ ц1947 ၣ⠌ 1948

ၣ 栾⥩♭ၻکᗣᚲᕞ㘢㞖┙⇩⍭㓾ⷛક⇩⍭ੌ 栾খாᗣᚲᕞ㘢 ೲ๾ᔢᏕᔷ⹐⸵ ц1948 ၣ⠌ 1950 ၣ 栾⥩♭ၻኩܜࢯ⹱ڳೲ๾ِ

✄ 栾کᤳᔢᏕ цாܜࢯ⹱ڳᙛ㒙 栾ڒᎁ໣⃞ၬݗ๾ῳڞᄞ௎዗᳻

✓៪ᔷ๾ວ 5 ఑໮ۄ $\RXE 栾- ັဴᷱ 0LWFKHOO ⠛ / ⠡ᆱ

⽒ᷱᄹ 6FKRHQIHOG цㅚᙛ㒙 栾㔀ᔷ⹐೩ 栾⥩♭ၻㆧ᪵⿏ᙌ㓳ఝ ذ⃞ᕥ⌫⹐ х࡜௳⚄⠛ఝ⹐⏯㘥ఝ ц

1950 ၣ 栾⥩♭ၻ⠛ڒ⃞൐๣ࡁഞت⾨஗ம 栾কࣺჃ⎧فம⌋

๾㓾ᔷ๾⇩⍭ዷ⃞⒖޻ဉۦ 栾1952 ၣⳝک਍᳻ዷ㒆 栾ᄨᤴڒ९ ࡑࣸጶゎᕧჃⷝ⇩⍭ዷ⃞ဉۦ цᎃᵥ⥩♭ၻ⃞ሣ⵽ 栾⇩⍭ዷभ

፩┓ⓔᔷ๾ хኩΌᔷ๾⠛ⷆ␡ᢍጦⲣ⃞تٲ࢛⌋ цڒㆧ໢ల㚙㖤

ၣᔷ๾ວဉۦ═٧Ḛࢭ㊟ⶁ цா㖤ၣᔷ๾ວف 栾ᔷ⹐ᕥ㖭ᙌ㔉ᗤ ᰄ х᯺ጥᩪ⠛ộ࠶ 栾ڜᔷᕥ㖭ᙌ⦍ਖ਼࠻ 栾ⴜ࢛ᚬᕥ㖭ᙌ㨻ᖜ⠛㔎 ટ㐐 ц᳻٦ڒݜঘفமၣハᔷ๾ວᗣㅺঞ℉ 栾⥩♭ၻ໎٦ت⓼ࢧ ᙀ 桌 ъఽೈ┟ᔷ⹐ыفᕄ᷹ 栾1957 х ъᔷ⹐໣დы1957 х ъ೬ⴜ⻝ᔷࢗᔷ⹐ف⃞࡜௳ఝ⃞⸾ਘ࢛ᚬы1958 х ъᎁᔷਘ⃞

ۅⷆঘ࡚ாᔷ⹐ف⃞ኩΌыفᕄ᷹ 栾1963 х ъ㞖⏯ᔷ๾დ⹐ы 1963 ⠛ъ࡜௳⚄ы1963 栾⠛⦍ਖ਼࠻়⦰ ц1955 ၣ 栾⥩♭ၻ‧

㆛᳻فம⌋๾㓾๾㇩൦ੌ ц

1966 ၣ 栾⃘ῳ٦ ³ᕄऺೲ㖰਍´ ц⥩♭ၻ⃞ວⳝ ³┄ⲩ࡙´ ጩㅻ ദၬᤄ 栾ጀ⍋ᔧ೺᥆№ 栾⠌ڌᨈᙌر⦝ 栾ڒٜ㆑ࢲጟࢬ㟉ᷧ 栾

໢ڒ܊⸲ 栾ㅚ⍁ᇭᨦ⠌ 1967 ၣጁᙌᖣ㙝⃞ദヷ цㅚᖺ ᕧڒঞࢲ ᦄᰲᚑىဴ⠛ৼᆱ܊◮ἱ⃞Ḛࢭ݁⻏ 栾মڞຊ㖩௅৫ாວⴂ 栾`

⠌মڞࢲဉ៪㇩㒋ঔᗣঘᔷ๾ᕥᨷ ц

1958 ၣ 栾⥩♭ၻⳝک਍᳻فம⌋๾ጦⲣೲ๾ࣚᜉ㒆 цڒ㒑൝ ᄨ٨ኩΌᔷ๾⃞⇩⍭ဉۦ 栾Ḛࢭໜᔷ⹐Όᕧ㞖▂ᔷݽ⍕࢛ᨷ 栾ڒ ㆧࢲဉႲਘဉ៪㇩㒋ᗣঘ ³ࠣ㆛ᨷ´ ᕆᩂㆽഉ )LERQDFFL ᕥᨷ ⠛

³╚⒖ᨷ´&30 ⠛ 3(57 цㄩ٩॒ၣف 栾ڒ⠛ࣾጀ㔉ᄹᨭ хⷆ㕧

⾣ㅺ٦٩॒೬ݗℯမ⠇ᨠ॑ 栾৅ဉڂບ⺋ؿᔢᙈڒݜഫ۟ໜㅚࡒ⍁

ᕥᨷΌᕧڒݜ⠇တ⃞ဉۦفঔ цᄨ⛘ 栾ဉႲ⃞῵㊡ತࣺ٦ 栾῵ਾ

⃞⽶㊡ٜᐋ㞖٦ ц

ᄨ 1979 ၣ㒑൝ 栾فமరⲟ㒑ᔏᔐ⏺ ц⥩♭ၻாڒ⃞๾ῳ⃞य़

ࣾر 栾ຌ዗٦ໝ⦰ъᄨ૕ۗல⹅⾨ы1977 ঘъᔷ⹐ாㄩی࢛ᚬف

⃞ኩΌы1978 栾⠛ộ࠶়⦰ ц  + ੅⼹ᕞ௜ +DOEHUVWDP ى▮

⃞ъ⥩♭ၻ⹐ᕄ㆛㕃ыٜா 1983 ၣ ᕨᗣᚲ᜝࢖᷹⊣࢖᷹ ц⥩♭

ၻா 1978 ၣⳝک਍᳻فம⌋๾㓾ࣚ㓾㒆 ц1980 ၣⳝک਍᳻ኩΌᔷ

๾⇩⍭ዷዷ㒆 цᤴ೩ 栾ڒᄨ 1950 ၣ⠌ 1983 ၣ௎ⳝ㆛⠝᳻فமᔷ

๾ᙈἱ٨㒆 ц

⥩♭ၻۦ᳻ⷛક๾⛓ 栾೬ᤄⷛકᤦ᩺ х♶ம⠛ᖃᙥ ц࠘␩

ၣㆤ㞓ფ 栾ڒڎ఼፾ᔷ๾⇩⍭ঘ࡚ኩΌဉۦ ц1985 ၣ 6 ᙋ 12 ᖃ 栾ڒாᖃᙥᚑټೲ๾ۦ๾ⲣ౛৮ 栾‧⺋ຌᙆᄞتথ⸊ᗆ 栾 ᕧ ᅂ⟽⁙⍶ᵃ⃘ۦ⛘ঔغ ц

⠇ᄨفமరⲟ㒑ᔏᔐ⏺ڞ܊ 栾⥩♭ၻتࡘᄣࢲᨷமॠ㎜ೲ๾

1980 х㛛ᬦفᕄೲ๾ 1983 ⠛♶மܜࢯ⹱ڳೲ๾ 1984 ⃞᠕

⻔ॡೕ๾ۗ цڒㆧⳝ㆛᳻♶ம⌋๾㓾 1982 х⏜دغ‐⌋๾㓾 1983 ⠛ᄹமဓڸࢯٳ⌋๾㓾 1985 㓾ೕ ц

⥩♭ၻ⍕៰কࣺ⊣ᙈᔐᨠ᪃ऑ 栾ᶒሬ⊸ம ц1949 ၣڞࣆ 栾ڒ

ীᇭؿጶゎᕧᷧঝᦩى⠛⠇ ⃞ㅹऑ цჃமᄞ 栾ᓁک⏜ت⠌⏜ࡖ

༂ࡑமڂೲ္൦ᙈ൦ੌ 栾⏜ࡖ༂ࡑமᔐय़ࣚىဴ 栾ڒㆧᖺفமᦩ ىীℕࣚىဴ ц

⥩♭ၻ⃞ى⵶ᔷ๾ဉۦഫرዷㄼ ц

1 ᑸ⵼

1 ֻⴢনٕⵋ ਍ q ᳻تݗᤳᔳᔷঘ f(x) ᳻تݗᔳܫᔷ೬ 㘖჎

f(x) = a_kx^k + · · · + a₁x 栾

ᤴ⬧ (a_k, · · · , a₁, q) = 1 ц⛑ቦຌᔳد⶝ਘ S(q, f(x)) =

q

X

x=1

e(f(x)

q ) 栾e(x) = e^2πix ц

⢵ f(x) = x² 栾ࣁ S(q, x²) ໴ᖺᶅ⇉⃞㞖ᕞਘ ц&) 㞖ᕞ *DXVV

⺯ᖣ٦

|S(q, x²)| = q¹² ц

㓊ᕧ S(q, f(x)) ⃞ۅⷆᖺتݗᙌᇓٍ᤼঱⃞㕜㙃 цா 1940 ၣ⥩♭

ၻຌࡑⶫ᧳ㅚݗક㙃ِࣆ 栾ߘ໢تٲḚ᥉೬㘖჎ᙌٲ╁ᚶ ц⥩♭

ၻΌ㖪္ࠣ♶⃞ᕥᨷ⺯ᖣ٦

|S(q, f(x))| = O(q^1−k1+ε) 栾 (1)

ᤴ⬧ ε ᳻═ຖ⃞ᤳᔷᙌ⠛ ³O´ ᙌ㓊⃞္ᔷߘܜ⽽ᕧ k ⠛ ε ц1938

ၣ 栾⥩♭ၻᙄᄣࢲ჎ 1 栾ے⠛ ³O´ ᙌ㓊⃞္ᔷܜ⽽ᕧ f(x) ⃞ܫ ᔷ a_i (1 ≤ i ≤ k) цᖨ⵽㔀تݗম❫ᔌㅪ⃞ங๣ q^ε ِ೩ 栾ۅⷆ჎

1 ᖺᙆ۷ম❫⃞ ܎ഫ |S(p^k, x^ε)| = p^k(1−1k) 栾࡚ف p ⲯ⊡⽶ᔷঘ p 6 | k ц

ڞᄞ 栾⥩♭ၻখໜ 1 ჎ᏰႵ⠌کሣᤄᔷ᳻ n ⃞ڜᔷᔷఝ K ц 㓊ᕧسຌᔳد⶝ਘ 栾⥩♭ၻ⺯ᖣ٦

p

X

k=1

e

hx^k q

= p q

q

X

x=1

e

hx^k q

+ O(q^12+ε) ц (2) ㅚݗ╁ᚶ໢⥩ᚲક㙃ᙌ㊟⵶ኩΌ ц

⥩♭ၻᔌㅪؿ⑹ऺ٦▂⹱᜝ፕ೬೵㓊ᕧ೩ᷱਘۅⷆ⃞ᕥᨷ 栾ڒ ᎁ࢖▂⹱᜝ፕ೬೵ᕥᨷ⃞᜙ᅂᖺر㖭⃞ၢ௎ݽຖἱ 桌਍

f(x) = α_kx^k + · · · + α₁x ঘ

C_k = C_k(P) =

a+P

X

x=a+1

e(f(x)) ц

਍

t₁ = t₁(k) ≥ 1

4k(k + 1) + lk 栾

ࣁ Z 1

0 · · · Z 1

0 |C_k|^2t1dα₁ · · ·dα_k ≤ (7t₁)^4t1lP^{2t1−k(k+1)2 +δ}(ORJP)^2l 栾

ᤴ⬧

δ = δ(k) = 1

2k (k + 1)

1 − 1 k

l

ц

 ᤴ⎧९Ᏸ࢖ر㖭⃞ຖἱ 桌 ރຖ k ≥ 12 х2 ≤ r ≤ k ঘ

α_r − h q

≤ 1

q² х(h, q) = 1 х1 ≤ q ≤ P^r 栾

ࣁ໢ᕧ P ≤ q ≤ P^r−1 栾ዘݜᙌ S =

p

X

x=1

e(f(x)) ≪ P^{1−σk1 +ε} 栾

ᤴ⬧ σ_k = 2k²(2 ORJk + ORJ ORJk + 3) ц

ாዷᙌ⦰ু⃞ⶫᚬᔷ⹐ໝ⦰ف 栾ၬْㇺᖺᎃᵥ⥩♭ၻ⃞ㄔᨷ܊

ᔡㄼ▂⹱᜝ፕ೬೵ᕥᨷ⃞ ܎ഫ 5& ᧺ᆱ 9DXJKDQ ⦰ъ੅ڜ ± ᙽ Ḛᷱڵᄹᕥᨷы7KH +DUG\/LWWOHZRRG PHWKRG 栾&DPE 7UDFWV 栾 80 栾1981 ц

㓊ᕧḚᄷਘ 栾⥩♭ၻா 1942 ၣ⺯ᖣ٦ۅⷆ჎

1 A+ 1

A

X

a=1 a

X

n=−a

x(n)

≤ √

p − AH

√p 栾1 ≤ A < p

໢ᕧᡤ p ⃞ዷᙌ㖪ىḚᄷ௎዗⎧ цㅚ܃ᄣڒᔌㅪ٦ᡤ p ⃞ᙆ໤উ᜚

ۅⷆঘ۰ᷱᕥ⌫ᙆ໤ⶫۅⷆ ц

2 ⣯ᙀਥ㖿प߭ᗚ㑈ਥ㖿 1770 ၣ 栾⥩ᚲẆላ໢ᕧ᥻ݗᔳ ᔷ k ≥ 2 栾⃟๪ாتݗߘܜ⽽ᕧ k ⃞ᔳᔷ s = s(k) 栾܃᥻تݗ

ᤳᔳᔷ௎মڞⲯ᳻ s ݗ㖪⼻ᔳᔷ⃞ k ᤄᕥࡱِਘ ц⥩ᚲẆላΩࢲ 1900 ၣጁⳝ ' သᷱۄḚ +LOEHUW ⺯ᖣ цᙥغ⓿٩॒ၣڜ 栾੅ڜ

⠛ -( ᙽḚᷱڵᄹ /LWWOHZRRG ࣝ൝ؿ⃘།٦ఽೈᔷ⹐فتݗ㖪္

შᙌࣸ⃞ᕟⶫᚬᕥᨷ 栾९ㅛ္ዷ⍄⃞ ³லᨷ´ 栾 ㅚتᕥᨷমڞᄣ ࢲ⥩ᚲક㙃ᘽⓙ▽⃞╁ᚶ ц਍ G(k) ⲯ⊡ᙆ໤ᔳᔷ s 栾܃᥻ݗ࠸࢛

ೲ⃞ᔳᔷ N ௎মڞⲯ⊡᳻

N = x^k₁ + · · · + x^k_s 栾 (3)

ᤴ⬧ x_i (1 ≤ i ≤ s) ᳻㖪⼻ᔳᔷ ц੅ڜ⠛ᙽḚᷱڵᄹ⺯ᖣ٦ G(k) ≤ (k − 2)2^k−1 + 5 桍ڒݜ້㔦ذ⺯ᖣ٦ᘽშ⃞╁ᚶ 桌‧

s ≥ (k − 2)2^k−1 + 5 ᗆ 栾3 ჎⃞ⶫᔷ r_s,k(N) ᙌتݗᯧㄩⲯㆁ჎

r_s,k(N) = S(N)Γ^s(1 + _k¹)

Γ(_k^s) N ^sk−1(1 + o(1)) 栾 (4)

ᤴ⬧ S(N) ⍄᳻ം‥┙ᔷ 栾຅ᙌتݗ⠛ N ᴸ㓊⃞ᤳر‐ ц⥩♭ၻ

ா 1938 ၣໜ੅ڜ⠛ᙽḚᷱڵᄹ⃞╁ᚶᔌㅪ᳻

G(k) ≤ 2^k + 1 栾

⛘ظڒ⺯ᖣ٦‧ s ≥ 2^k + 1 ᗆ 栾rs,k(N) ⃞ᯧㄩࡕ჎९዗⎧ ц㓊ᕧ ㅚ㘖ဉۦ 栾ڒΌࢲⳝᄞڂዷ⍄⃞⥩ᦧس⏯჎

Z ₁

0

p

X

x=1

e(αx^k)

2k

dα ≪ P^2k−k+ε ц (5)

ݽᄣᎁ࢖⃞ᖺ໢ᕧ໤⃞ k 栾܃ 4 ჎዗⎧⃞᝗ڧ s ≥ 2^k + 1 Ωࢲ ㄩၣጁ࢛ࢭ ᧺ᆱ⠛ '5 သᕞ ± ယᙖ +HDWK%URZQ ᔌㅪ᳻ s ≥ 2^k (k ≥ 3) ঘ s ≥ 7

8 2^k + 1 (k ≥ 6) цᴸ⹐ഫ۟ 栾‧ k > 10 ᗆ 栾

▂⹱᜝ፕ೬೵໢੅ڜ⠛ᙽḚᷱڵᄹ㓊ᕧ⥩ᚲક㙃⃞╁ᚶ═٧ဌೲ⃞

ᔌㅪ 栾ڒ⺯ᖣ٦ 4 ჎‧ s ≥ [10k²ORJk] (k > 10) ᗆ዗⎧ ц⥩♭

ၻໜㅚݗ╁ᚶᔌㅪ᳻ s ≥ 2k²(ORJk+ORJ ORJk + 2.5) (k > 10) цళ ᕧ Ʌ Ƚ ᕨ໻㕧ᷱᙃ ɒɧɢɪɟɥɶɦɚɧ 㓊ᕧᤳᔳᔷࢧັႃ⃞ᕥᨷ 栾ᚲ

໻࠽ᕧ 1943 ၣ═٧သᷱۄḚຖἱتݗᕟ⃞ ³ࢩ⏯´ ⺯ᖣ ц⸪ഫㆁᕄ ᩂḚᎁ࢖ 桌³ㅚت⺯ᖣ⃞ላᨷᴸ‹܊⠇੅ڜ ± ᙽḚᷱڵᄹᕥᨷ⃞ᛖ ٲḚᆄ 栾Ḛࢭᖺ⥩ᦧس⏯჎ ц´

ᙥغ⓿د॒ၣڜ 栾ᄚ೬ᔷ๾ວ⇩⍭ഫ۟ໜ⥩ᚲક㙃ف⃞ x^k Ᏸ Ⴕ᳻ k ᤄ೬㘖჎ ц⥩♭ၻΌ࡚╁ᚶ 1 ࠽ᙏ٦ㅚݗક㙃⃞ى⵶

஡㕜 цသᷱۄḚຖἱ⃞تݗ㖪္Ⴕ⃞ჽ჎ᖺ༝ᕧ⥩♭ၻ⃞ ц਍

i(x) (1 ≤ i ≤ s) ᖺ s ݗ㛘㘖ܫᔷ᳻ᤳ⃞ k ᤄᔳݽ೬㘖჎ 栾⥩♭

ၻா 1937 ၣ⠌ 1940 ၣ㒙⺯ᖣ٦‧ s ≥ 2^k + 1 (1 ≤ k ≤ 10) ঘ‧

s ≥ 2k²(ORJk + ORJ ORJk + 2.5) ᗆ 栾ᕥ⌫

N = f₁(x₁) + · · · + f_s(x_s)

⃞ⶫᔷᙌتݗᯧㄩࡕ჎ цᩆሣாᤴ㓊ᕧം‥┙ᔷ⃞ᤳر‐ક㙃ؿᙣ

ࣺڞ⛑ቦ ц਍ f(x) ᳻تݗᔳݽ೬㘖჎ঘ G(f) ⲯ⊡ᙆ໤⃞ᔳᔷ s

܃໢ዷᙌ࠸࢛ೲ⃞ᔳᔷ N 栾ᕥ⌫

N = f(x₁) + · · · + f(xs)

⃟মⶫ ц਍ ∂⁰f ⲯ⊡ f ⃞ᤄᔷ ц⥩♭ၻா 1940 ၣ⺯ᖣ٦ G(f|∂⁰f = k) ≤ (k − 1)2^k+1 栾

G(f|∂⁰f = 3) ≤ 8

⠛

PD[G(f) ≥ 2^k − 1 栾

ᤴ⬧ f ㅻዷᙌ k ᤄᔳݽ೬㘖჎ 栾࡚ف k ≥ 5 ц

லᨷ⃞⵶㦼মڞᔡㄼᕧر 桌3 ⃞ⶫᔷমڞⲯ⊡዗⍕࢛

r_s,k(N) = R1

0 T(α)^se(−N α)dα 栾 T(α) = Pp

x=1e(αx^k) 栾

ᤴ⬧ P = [N ^1k] цໜ૕ۗ॑㒙 @ ࢛዗⹭ࠣპ⠛⹭ࣻპ 栾຅ݜ⃞

✌㕃࢛ࢭⷒ᳻ M ⠛ m цⓃ‟ⷁِ 栾M भ৖⹭࢛᥺ダ໤⃞࢛ᔷ k q

᳻فᅂ⃞٭س℩ٷ⃞໤॑㒙 栾࡚㚣ࣁ᳻ m ц੅ڜ⠛ᙽḚᷱڵᄹ⺯

ᖣ٦ 栾‧ s ≥ 2k + 1 ᗆ 栾ᯧㄩࡕ჎

Z

m

T(α)^se(−N α)dα = S(N)Γ^s(1 + _k¹)

Γ(_k^s) N ^ks−1(1 + o(1))

዗⎧ ц⥩♭ၻໜㅚݗ╁ᚶᔌㅪ᳻ s ≥ k + 1 цㅚᖺᙆ۷ম❫╁

ᚶ 栾࡚⺯ᖣళᕧڒ⃞ۅⷆ჎ 2 ц

ᙥغ⓿د॒⠌க॒ၣڜ 栾⥩♭ၻ⓼╚௅⇩⍭٦ዷ⹵⥩ᚲ ± ੏ ᄹဓ⾞ક㙃 栾९⇩⍭㓳ࢷ⹭ x_i (1 ≤ i ≤ s) ᳻⽶ᔷᗆ 栾3 ঘ

࡚ᏰႵ⃞মⶫᆄક㙃 ц܎ഫڒ⺯ᖣ٦‧ s ≥ 2^k + 1 (k ≤ 10) ঘ s ≥ 2k²(ORJk + ORJ ORJk + 2.5) (k > 10) ᗆ 栾ᕥ⌫

N = P₁^k + · · · + P_s^k

⃞ⶫᔷᙌتݗᯧㄩࡕ჎ 栾ᤴ⬧ P₁ ⲯ⊡⽶ᔷ цڒٹᄣࢲذㄼ⹭╁ᚶ ڞ⽶ᔷ᳻⻝ᔷ⃞㙒ی╁ᚶ цڒ⃞╁ᚶ◮╁ாڒ⃞ু⦰ъఽೈ┟ᔷ

⹐ыِف ц

౳ࢯક㙃 ц਍ N(k) ⲯ⊡ᙆ໤⃞ᔳᔷ t 栾܃ᕥ⌫┺

x^k₁ + · · · + x^k_t = y₁^h + · · · + y_t^h 栾1 ≤ h ≤ k (6) ᙌتݗ㖪໠္ⶫ 栾९⹭ x_i хy_i ᳻ᤳᔳᔷ 栾ے x₁ х· · · хx_t سᖺ y₁ х· · · хy_t ⃞♞ᐔ ц਍ M(k) ᳻ᙆ໤⃞ᔳᔷ t ܃ 6 মⶫঘ

x^k+1₁ + · · · + x^k+1_t 6= y₁^k+1 + · · · + y_t^k+1 ц 㙝ᵃ 栾ዘݜᙌ

k + 1 ≤ N(k) ≤ M(k) ц

⥩♭ၻா 1938 ၣ⺯ᖣ٦ M(k) ≤ (k + 1)

"

ORJ[^(k+2)₂ ] ORJ(1 + _k¹)

# + 1

!

∼ k² ORJk 栾

ㅚᖺ໢ (0 ⽽Ḛ :ULJKW ダᖆ╁ᚶ M(k) < 7k²(k − 11) · (k +

3)/216 ⃞ᔌㅪ ц⥩♭ၻዷΌ⃞ᕥᨷᖺࢩ⏯⠛ΩᏮ⃞ ц

⥩♭ၻㆧா 1952 ၣᎁ࢖মڞΌ▂⹱᜝ፕ೬೵ᕥᨷ܊⬧ἱ౳ࢯક 㙃 цڒ⺯ᖣ٦ر㖭⃞╁ᚶ 桌਍ t₀  رⲯ═࢖

k 2 3 4 5 6 7 8 9 10

t₀ 3 8 23 55 120 207 336 540 885

k k ≥ 11

t₀ [k²(3ORJk + ORJ ORJk + 9)]

਍ R_k,t(P) ⲯ⊡჎ 6 ㆏়᝗ڧ

1 ≤ x_i хy_i ≤ P х1 ≤ i ≤ t

⃞ⶫᔷ 栾ࣁ‧ t > t₀ ᗆ 栾

pOLP→∞ P ^{2k(k+1)1 −2t}R_k,t(P) = c(k, t) 栾

ᤴ⬧ c(k, t) ᳻تݗߘܜ⽽ᕧ k хt ⃞ᤳ္ᔷ ц

3 ๲ᓄⷕ⁥߭ต⻂Ṓ ਍ q(n) ᳻ໜᤳᔳᔷ n ࢛ፒ዗سীᔳᔷ ዜംᔳᔷ ِਘ⃞࢛ፒݗᔷ ц⥩♭ၻா 1942 ၣࢯΌ੅ڜ⠛ፕ㛦ࣿ

㊣ᕥᨷঘፕᄹ㛦⾞ᷱ⃞ᴸ⎔┙ᨷⴂ࢛ࣛᨷ⺯ᖣ٦ 桌 q(n) = 1

√2

∞

X

k=1

26 |k

X

(h,k)=1

0<h≤k

ω_h,ke

−hn k

· d dnJ₀

iπ

k ((n + 1 24)2

3)¹²

栾

ᤴ⬧ J₀(x) ᳻ 0 ± ┙⼹౻⛱ࢗᔷ ц

਍ A(x) ⲯ⊡ல u² +v² ≤ x ࡐ⃞᜝๣㦼 (u, v) ⃞ݗᔷ ц㞖ᕞல ક㙃᳻໠ጤᙆ໤⃞ θ ܃㓊ܫ჎

A(x) = πx + O(x^θ+ε)

໢ᕧکሣ ε > 0 ⃟዗⎧ 栾ᤴ⬧⠛ ³O´ ᙌ㓊⃞္ᔷܜ⽽ᕧ ε ц㞖ᕞ ᙥڂ⺯ᖣ٦ θ = 1

2 ц1942 ၣ⥩♭ၻ܃⧏ം㛦ڃᕥᨷᘽⓙັ 栾ؿໜ ڒ⃞╁ᚶ  θ = 15

46 ᔌㅪ᳻ θ = 13

40 ц⛘ظாㅚ␹ᕄ⎮ف 栾⥩♭

ၻㆧᎁ࢖▂⹱᜝ፕ೬೵㓊ᕧㅚݗક㙃⃞╁ᚶ⃞⺯ᖣᖺᙌ㎟⃞ ц

⥩♭ၻㆧ⇩⍭٦ᤦᦧ㔀ᨷ ($ ક㙃 ц⛑ቦ້٩ᤄఝ Q(√

d) 栾

ᤴ⬧ d ᳻تݗᴸၢᕥங๣ᔷ ц⥩♭ၻா 1944 ၣ⺯ᖣ٦ 栾‧ d >

e²⁵⁰ ᗆ 栾ࣁس๪ா ($ 栾ڒؿᎁ࢖ 250 মڞ㓰ۘ᳻ 160 ц

1959 ၣ 栾⥩♭ၻ⠛ộ࠶໎٦ت␹ᔷݽ⍕࢛⃞⇎ᕄ 栾ڒݜ⺯ᖣ ٦⢵ ∂²f(x, y)

∂x∂y ா 0 ≤ x хy ≤ 1 ذㅤ☪ 栾ࣁ

Z ₁

0

Z ₁

0

f(x, y)dxdy − 1 F_n

Fn

X

k=1

f

k F_n,

F_n−1k F_n

≤ cORJ3F_n F_n

栾 (7)

ᤴ⬧

F_n = 1

√5

1 + √ 5 2

!n

− 1 − √ 5 2

!n!

(n = 1 х2 х· · ·)

ⲯ⊡ᕆᩂㆽഉၷࢧ ц㔀္ᔷ c ㆧম❫ᔌㅪ೩ 栾7 ჎ᖺᙆ۷ম❫

⃞ 栾ாڞᄞ⃞ت⓼ࢧ⹐ᕄف 栾ڒݜໜ⠇တ⃞ᕥᨷᏰႵࢲ s ± ▂⃞ᇭ ᨦ 栾࡚ف s > 2 цᖨ⇉ F_n−1

F_n (n = 1 х2 х· · ·) ᳻ (

√5 − 1)

2 =

2FRV(2π

5 ) ⃞ᙆ۷ᙌἱㅯㄩ цڒݜዷ܃Ό⃞ᕥᨷᖺாᕧࢯΌ࢛லఝ Q(2 FRV( 2π

m )) (m ≥ 5) ⃞ت┺Ẻ⎧૕ۗ܊ᠫㅡఝ⃞ت┺ళၸ⃞✌

⎧ᙌἱㅯㄩ цڒݜঘ࡚ڒڂாㅚݗ㘥ఝف⃞⼽ỉ 栾भ፩ 2 ▂⠌ 18

▂⃞ᔷݽ݅ᆷ 栾௎भ৖ாڒݜ⃞ໝ⦰ъᔷ⹐ாㄩی࢛ᚬف⃞ኩΌы 1978 ِف ц

1953 ၣ⠌ 1957 ၣ 栾⥩♭ၻாفம⌋๾㓾ᔷ๾⇩⍭ዷ┺☀㘥

໣٦ ³੏ᄹဓ⾞ક㙃´ ⷉ⹐἟ ц੏ᄹဓ⾞ક㙃ᖺ੏ᄹဓ⾞⠍ / ᤦ ፕ (XOHU ⃞݅فᐋ࢖܊⃞ 栾९ 桌$ ᥻تݗ ≥ 6 ⃞ިᔷㇺᖺࡒ ݗം⽶ᔷِਘ 桍% ᥻تݗ ≥ 9 ⃞ംᔷㇺᖺدݗം⽶ᔷِਘ ц㙝 ᵃ % ᖺ $ ⃞Ᏸ⹐ ц1937 ၣ 栾໢ᕧ࠸࢛ೲ⃞ംᔷ 栾▂⹱᜝

ፕ೬೵ⶫ᧳٦ક㙃 % ц1956 ၣ 栾ộ࠶㛘࠻⺯ᖣ᥻ݗೲިᔷㇺ ᖺتݗس⾰ㅻ 3 ݗ⽶ᔷ⃞ٗ⍕ঘتݗس⾰ㅻ 4 ݗ⽶ᔷ⃞ٗ⍕ِ

ਘ 栾⑹ⷒ᳻ (3, 4) 栾ㅚᔌㅪ٦ $$ ယ⾞೨౳ယ Ȼɭɯɲɬɚɛ ⃞╁ᚶ (4, 4) ц1957 ၣộ࠶খໜ࡚╁ᚶᔌㅪ᳻ (2, 3) ц1963 ၣ᯺ጥᩪ⺯

ᖣ٦ (1, 4) цᙆᄞ㔉ᗤᰄ⺯ᖣ٦ (1, 2) 栾ㅚᖺ੏ᄹဓ⾞ક㙃 $ ⃞ ᙆ۷ⷒ㍻ ц

2 רᑸ❊ྮث

4 㜐ⷕ ⠇ᄨ :5 ੅ັ㘢 +DPLOWRQ ⃘Ἢ⏜تݗ㖪ٷᐔ⃞

ম㔀ڜᔷ ± க࠶ᔷڜᔷڞ܊ 栾ম㔀ڜᔷ`ঞ㊟ⶁ ц℩᥿ِر 栾 ᴸ㓳▂ম㔀ڜᔷ 栾९㞓 栾ॵⳝᅦ‟⃞ цΩࢲ 1950 ၣညল 栾⥩♭

ၻڞ៰࡚⑹૕⠛ΩᏮ⃞ᕥᨷ 栾Ꮾㅤ⺯ᖣ٦ㅚݗ㘥ఝ⃞ၬ᝗ຖἱ ц

⢵਍ K ᖺتݗ㞓 栾ظ σ ᖺ K ࢲ຅⠇ゎ⃞تݗتتᖯ໛ 桌a → a^σ цഫᚶ σ ᮹⿏

(a + b)^σ = a^σ + b^σ х(aba)^σ = a^σb^σa^σ х1^σ = 1

ࣁ⍄σ ᳻ख़⠇ীᠫ цᶅ⇉⃞ख़⠇ীᠫ⃞܎๣᳻ীᠫ 桌(ab)^σ = a^σb^σ

⠛ছ⠇ীᠫ 桌(ab)^σ = b^σa^σ ц⦰ু⃞ક㙃᳻㔀⠇ীᠫ⠛ছ⠇ীᠫ ೩ 栾ᖺ৑ㆧ๪ா࡚຅ख़⠇ীᠫ" ⥩♭ၻா 1949 ၣⶫ᧳٦ㅚݗક 㙃 栾ڒ⺯ᖣ 桌

᥻تݗख़⠇ীᠫዜ᳻⠇ীᠫዜ᳻ছ⠇ীᠫ ц

 ᤴমڞᏰ࢖Ḛᄷ 6= 2 ⃞㞓ذ⃞ت▂໛ᄉၬ۟⃞ళᙥຖἱ 桌 ک۟ໜḚᄷ 6= 2 ⃞㞓ذ⃞໛ᄉΩ▤ᖯ໛ࢲ⠇ゎ⃞تت⻝ᐔ 栾ഫ ᚶ݁፾⸾ਘ㓊ܫس⻝ 栾ࣁᅃ᳻تݗ⠇ীᠫዜख़⠇ীᠫ⸦໣⃞ख़▤ᆄ

⻝ᐔ ц

ㅻঔ * ຊᛯ᭾㕂 $QFRFKHD ⠛ , ॥ᗣ⬋ᕞళ .DSODQVN\ ㇺ ߘ❫ாᛖٲ㓳ࢷِر⇩⍭ख़⠇ীᠫક㙃 ц ᕧڒݜዷΌ⃞ᕥᨷళᕧ

▤ᆄڜᔷ⃞ᠫㅡἱ⹐ 栾ዷڞڒݜㇺᙣ❫ຌࡑⶫ᧳ક㙃 ц 1949 ၣ 栾⥩♭ၻ═࢖ر㖭╁ᚶ⃞تݗΩᏮ⺯ᖣ 桌 㞓⃞᥻تݗ⅂ᤳ⵾๣㞓௎भ৖ா຅⃞فᅂِف ц

ㅚݗຖἱாᄞ܊⃞ᕄỉفⳝ⍄᳻ ³଑‧ + &DUWDQ ± ယघᷱ 5

%UDXHU ± ⥩ᦧຖἱ´ цா⥩♭ၻ⠛ယघᷱ⃞⺯ᖣ࢖Ἢِࣆ 栾଑‧Ό ٦㞓⃞ۍ♭῎ἱ⹐⃞ⴜ㕖ጦⲣߘ❫໢ম㔀ڜᔷ⺯ᖣㅚت╁ᚶ ц⥩♭

ၻ⃞⺯ᖣ঩㘙⵶تݗࢩ⏯ᆙ⏯჎ 桌⢵ ab 6= ba 栾ࣁ

a = b⁻¹ − (a − 1)⁻¹b⁻¹(a − 1)(a⁻¹b⁻¹a − (a − 1)⁻¹b⁻¹(a − 1)⁻¹) 1950 ၣ 栾⥩♭ၻㆧ⺯ᖣ٦ ³⢵تݗ㞓سᖺఝ 栾ࣁ຅⃞ٗᨷ⚄س ᖺٳ㓨⼹ᷱ⚄´ ц

5 ☌ⷕ➢⅔㑽࿹ٯฎ ᖆா 1946 ၣ 栾⥩♭ၻ໴⃘ⲯ٦⏜ت

␹㓊ᕧ࡜௳⚄⠇ীᠫ⃞⹐ᕄ цாㅚ␹⹐ᕄⴂ 栾ڒ≌ຖ٦້ㄐ⚄

⃞⠇ীᠫ ц1948 ၣ 栾ڒখ≌ຖ٦Ḛᄷ 6= 2 ⃞کሣఝذㄐ⚄⃞⠇

ীᠫ цڒΌ܊≌ຖㄐ⚄⠇ীᠫ⃞ᕥᨷٜমڞΌ܊≌ຖ࡚຅㙒௳࡜

௳⚄⃞⠇ীᠫ ц㑤ᕧ - ㄸॺ೬ࡐ 'LHXGRQQp ᕧ 1951 ၣ⃘ⲯ٦ 㓊ᕧ࡜௳⚄⠇ীᠫ⃞ໝ⦰ 栾⥩♭ၻ঩ாㄸॺ೬ࡐ⃞ᙀᄞ໎ت␹㓬

㍻ 栾⃘ⲯ٦Όڒ⠇တ⃞ᕥᨷⶫ᧳⃞ㄸॺ೬ࡐ㆝‖ر܊ᨈᙌⶫ᧳⃞

⢵ၡક㙃 ц⥩♭ၻ≌ຖ٦ GL₂(K) хSL₄(K) ⠛ P SL₄(K) ⃞⠇

ীᠫ 栾ᤴ⬧ K ᖺḚᄷ 6= 2 ⃞㞓ঘ O₄⁺(K, f) ⃞⠇ীᠫ 栾ᤴ⬧ K

᳻Ḛᄷ 6= 2 ⃞ఝ 栾⛘ f ᖺᎁᔷ᳻ 2 ⃞٩ᤄ௳ цڞᄞ⥩♭ၻখਘ⦍

ਖ਼࠻≌ຖ٦ SL₂(K) ਘ P SL₂(K) K ᳻Ḛᄷ 6= 2 ⃞㞓 цSL₄(K)

⠛ P SL₄(K) K ᳻Ḛᄷ 6= 2 ⃞㞓 ⃞⠇ীᠫ цীᗆ 栾ڒݜㆧ⺯ᖣ ٦ᛖٲ▤ᆄ⚄ᖺسীᠫ⃞ ц⥩♭ၻாڒዷ໎⃞ㄸॺ೬ࡐໝ⦰⃞㓬

㍻ف 栾⹐ㄼ٦ڒ⃞ᕥᨷ⠛ㄸॺ೬ࡐᕥᨷ⃞᥿ダ 栾ڒ໎ㆀ 桌³ㄸॺ ೬ࡐᕥᨷ໢ n ೲᗆ㘤ᙌᔕ 栾ؿম໢ݗࢭ໤⃞ n ࣺڞኩΌ цᆸഫ ڒڞࣆᙄᎁ࢖ 栾‧ n ᬕ໤ᗆ 栾஡㕜໴ࣺೲ٦ цㄸॺ೬ࡐᕥᨷ໢

ᕧダ໤⃞ n 栾⻝ᄣᄚ⏙፤ 栾ᙌᗆس❫ⶫ᧳ᙆ໤ n ᗆ⃞ᇭᨦ цদ تᕥ㖭 栾⏬⛓⃞ᕥᨷᄨ࠘ম❫໤⃞ n 㒑൝ 栾ㅚ္္ᖺᙆ஡㕜⃞ᇭ ᨦ ц⛘⻗⛓س㕜Ό⏬⛓ᙄΌㅻ⃞ᕥᨷᄨᙥᕄ⃞Ḛ᥉╁ᚶ࢖⃘ 栾Ό

᤽┌ᨷᄣ࢖ت⠴⃞╁ᚶ цㅪ⛘ⷁِ 栾℩᥿ᕧㄸॺ೬ࡐᕥᨷ 栾⏬⛓

঩Ό٦⇌㓿ⷆ␡ ц´

⥩♭ၻ⠛ , ⽽┌ 5HLQHU ≌ຖ٦ GL_n(Z) ਘ P GL_n(Z) ⃞⠇

ীᠫ 栾ㅚᖺᾦذ࡜௳⚄⠇ীᠫဉۦ⃞㒑⎷ цڒݜㆧ⺯ᖣ٦ 栾ഫ ᚶ n ≥ 2 栾ࣁ GL_n(Z)  دݗ࠶┟ῳ዗ 栾SL_n(Z)  ࡒݗ࠶┟ῳ

዗ 栾⛘ SP_2n(Z)  கݗ࠶┟ῳ዗ цாڒݜِࣆ 栾55 ယፕ੅ፕ

%UDKDUD ߘ❫⺯ᖣ SP_2n(Z) ⃞᥻ݗ࠶┟௎মⲯ᳻ᛖᙌ㓳⇌㓿㕃়

ف⃞࠶┟ِ⍕ ц

1940 ၣ 栾⥩♭ၻ⠛ᥧ๾ᄯდㅪ p ± ⚄⌙⃞៷ᅡ цރຖ p ± ⚄ G

⃞㔝ᖺ pⁿ 栾⢵຅⃞࠶┟⃞ᙆೲ㔝᳻ p^n−α 栾ࣁ⍄ G ⃞⌙᳻ α ц܎

ഫڒ⺯ᖣ٦ 栾⢵ G ⃞⌙᳻ α 栾࡚ف p ≥ 3 хn ≥ 2α + 1 栾ࣁ

L G ৖تݗظߘ৖تݗ p^m 㔝⛘⌙᳻ α (2α + 1 ≤ m ≤ n) ⃞

⚄ 栾LL G ৖ p^α ݗ p^m 㔝ᄰᾦ⚄ (α < m < n−α −1) 栾LLL G ف 㔝 ≤ p^m (α ≤ m ≤ n−α) ⃞࠶┟ݗᔷ⏯ᕧ p^m+α 栾࡚ف LL хLLL

࢛ࢭᔌㅪ٦ +$ ⒮ए 0LOOHU ⠛ Ⱥ Ⱥ ႇፕ⌋೵ Ʉɭɥɚɤɨɜ ⃞╁

ᚶ ц

⇌㓿ၬ۟๾ᖺ⥩♭ၻᕧ 1945 ၣ㛘࠻ࣝ൝⃞⇩⍭㘥ఝ ц຅⠛

&/ ⵵᜝ᷱ 6LHJHO 㓊ᕧ࢛჎▤ᆄ⻝ᐔ⃞ဉۦ℩㓊 цா⇌㓿ၬ۟

ف 栾⍰㒙ف⃞㦼ᖺᛖ㙒⇌㓿 栾܎ഫীᡥೲ໤⃞㒆ᕥ⇌㓿 х໢⍄⇌

㓿ዜᕏ໢⍄⇌㓿 цாㅚݗ⍰㒙ف 栾ᙌتݗㅹऑ⚄ цى⵶ક㙃ِت ᖺഫ۟Ό࠘ম❫໦⃞ၬ۟س⻝㊡܊ࢼࣨㅹऑ⚄ ц⥩♭ၻ⃘Ἢߘس⻝

㊡ ³Ⓞ࢜´ ९⿏ڞࢼࣨ⍰㒙⃞ㅹऑ⚄ цڒா 1951 ၣ⺯ᖣ٦㒆ᕥჽ⇌

㓿ګ໛ၬ۟⃞ళᙥຖἱ 桌

ރຖ 1 < n ≤ m 栾ࣁᄨ㞓 K ذ n × m ⇌㓿㕃়ࢲ⠇ゎ⃞ت تᖯ໛݁፾Ⓞ࢜⛓ ⢵ M − N ⃞⌙᳻ 栾ࣁ⍄⇌㓿 M ⠛ N Ⓞ

࢜ 栾ᅃ᳻ڞرჽṊ 桌

Z₁ = P Z^σQ + R 栾 (8)

࡚ف P = P⁽ⁿ⁾ хQ = Q^(m) ᳻মㅎᕥ㓿 栾R ᳻ n×m ⇌㓿 栾⛘ σ

᳻ K ⃞تݗ⠇ীᠫ ц‧ m = n ᗆ 栾ࣁ㔀჎ 8 ِ೩ 栾ㆧ⵶ᬌࣺ

Z₁ = P Z^′τQ + R 栾

࡚ف τ ᖺ K ⃞ছ⠇ীᠫ ц

 ㅚݗຖἱ 栾⥩♭ၻᏰ࢖٦㒆ᕥ⇌㓿໛ᄉၬ۟⃞ళᙥຖἱ 栾Ḛ ᄷ 6= 2 ⃞㞓ذࡑ㓿ᾦ⃞૓وীᠫ 栾ঘḚᄷ 6= 2 х3 ⃞㞓ذࡑ㓿ᾦ⃞

ᙽীᠫ ц

⥩♭ၻ໢⇌㓿ၬ۟⃞⇩⍭⿰ڒ໢೬ⴜ⻝ࢗᔷ⹐⃞⇩⍭ັ࢜℩

㓊 цㅚܬ܃ڒ⇩⍭⇌㓿⃞࢛㙒ક㙃 ц⥩♭ၻா 1944 ၣ⠌ 1946 ၣ

≌ຖ٦ா㈾⚄رⴜ໢⍄⇌㓿ਘᕏ໢⍄⇌㓿⃞࢛㙒 栾ت໢఑ᷱ⒮Ḛ⇌

㓿ா়ীر⃞࢛㙒 栾ঘாᤳٷ⚄ر 栾఑ᷱ⒮Ḛ⇌㓿⃞࢛㙒 ц

1955 ၣ 栾⥩♭ၻாᔷ๾⇩⍭ዷ㘥໣٦تݗڜᔷⷉ⹐἟ 栾ؿ⠛

⦍ਖ਼࠻⃘ⲯ٦ڒݜ়⦰⃞ᙀъ࡜௳⚄ы 栾◮╁٦ڒݜ㓊ᕧ࡜௳⚄ঘ

࡚ᙌ㓊ક㙃⃞዗ᚶ цⷉ⹐἟⃞዗ੌ㔀☟☪⃘། ⥩♭ၻ㒑㓏⃞㘥ఝ ೩ 栾ㆧா࡚຅ᕥ㖭 栾Ḛࢭᖺڜᔷ▮≎ᕥ㖭ސ࢖ㅻ⼽ỉ ц

3 ⱉߪᗬ

6 ߯஄ய 1935 ၣ 栾( ଑‧ &DUWDQ ⺯ᖣ٦ 栾ாⶫᚬᖯ

໛ِر 栾঩ᙌࡖ㙒سম┃ х㨊ᆄ хᙌ‐໢⍄ఝ 栾࡚فࡒ㙒ᖺ܎

೩ఝ 栾࢛ࢭ᳻ 16 ▂⠛ 27 ▂ 栾࡚㚣க㙒⍄᳻ ³࡜௳ఝ 栾মڞΌ⇌

㓿ໜ຅ݜⲯ⊡ഫر 桌

R, = {m × n⇌㓿Z 栾᮹⿏I^(m) − ZZ^∗ > 0} 栾 R,, = {n 㔝໢⍄ᕥ㓿Z 栾᮹⿏I⁽ⁿ⁾ − ZZ^∗ > 0} 栾 R,,, = {n 㔝ᕏ໢⍄ᕥ㓿Z 栾᮹⿏I⁽ⁿ⁾ − ZZ^∗ > 0} 栾 R,9 = {Z = (z₁, · · · , z_n) ∈ &ⁿ ᮹⿏

|Z Z^′|² + 1 − 2Z Z^′ > 0 栾|Z Z^′| < 1} 栾

࡚ف Z ⃞࠶┟᳻ⴜ⻝ᔷ 栾I^(m) ⲯ⊡ m 㔝૕ۗᕥ㓿 栾ঘ Z^∗ ⲯ⊡ Z

⃞ヷ♞ Z^′ ⃞ⴜࡘァ⇌㓿 ц

࡜௳ఝমڞℷۦᗣㅛⴜၢ㖭ذ૕ۗல⠛࡚຅॑ఝா㞖▂⍰㒙⃞㙒 ی цᤴ೩ 栾࡜௳ఝἱ⹐ாᄴ࢛ᕥ⌫ঘⴜၬٜ۟ᙌኩΌ ц

1943 ၣ 栾⵵᜝⃘ᷱⲯ٦ڒ㓊ᕧㄐၬ۟⃞㊟⵶ᕄ⎮ 栾⸋ᕄΌ⇌

㓿ᕥᨷ⇩⍭٦ R,, ц1944 ၣ 栾⥩♭ၻᎁ࢖ 栾࡜௳ఝ⃞⇩⍭মڞ᤽

╁᳻⇌㓿ၬ۟⃞⇩⍭ цẺ⎧ᕧ଑‧ঘ⵵᜝ᷱ 栾⥩♭ၻ═࢖க㙒࡜

௳ఝঘ࡚ㅹऑ⚄⃞⇌㓿ⲯ⊡ ц⥩♭ၻ⃞ᕄ⎮ف㓊ᕧ⠛⵵᜝ᷱဉۦ ᙌ㊟ⴜ⃞㇩࢛ 栾঩ᖺ⑹૕៷ㄼ٦تر цா⸋▮⛓ၷف໎ㆀ 桌³  ᕧ♶ம⠛فமِ㒙ㇵڧ㓥ᮬ 栾ா▮⛓ীሣِر 栾 ⥩♭ၻ⃞⃞ᙎ ঙᥧ๾ᄯ⠛⵵᜝ᷱᔢᏕ໢⸋ᕄۦ٦ت⓼ࢧᄴ໤݇ᔌ ц´ ⥩♭ၻㆧா

ڒ⃞ᕄ⎮فሯ⺍ + ೩ᷱ :H\O х੯ల╲⠛㔉ℯゎ࢛ࢭㅉ═ڒ⵵᜝

ᷱ х* ঽ♭ *LUDXG ⠛଑‧⃞ᕄ⎮ ц

1953 ၣ 栾⥩♭ၻΌ⚄ⲯ⊡⹐ᕥᨷᄣ࢖க㙒࡜௳ఝ⃞ຌᔳᤳٷ

⓼ цⓃ‟௅⸲ 栾ㅚ㙒یᕧாᗣㅛⴜၢ㖭ذ 栾ጤࢲ٦ຌᔳᤳٷ⓼ e(nθ) (n = 0 х ± 1 х· · ·) 栾 ㅚ┺ᤳٷ⓼ 栾ڂݜᖨᄣ࢖૕ۗல

⃞ᛯ⵵᜙ х⵵᜝ᷱ хۄ᜝ᙃ᜙ঘᨮ᚞᜙⏯ ц⥩♭ၻ⃞ᕥᨷḚ㦼᳻

࡛㞓⠛ΩᏮ 栾ؿ໢ⴜ㕖ⷆ␡⃞Ꮨᐙ цࢯΌ࡜௳ఝ⃞ᛯ⵵᜙মڞ⺯

ᖣ঩⵶ⶫᚬࢗᔷݽாᛖݗۘᄴ᪉ჽ Ḛᄷ᪉ჽ ذ═࢖ 栾ࢗᔷா࡜

௳ఝف⃞ݽ໴≌ຖ٦ ц⥩♭ၻໜذㄼဉۦ⠛࡚຅╁ᚶ◮╁ாڒ⃞ໝ

⦰ъ೬ⴜ⻝ࢗᔷ⹐ف⃞࡜௳ఝ⃞⸾ਘ࢛ᚬыِف ц⸋ᙀ⢿ᕄ᷹▮⛓

შ⸾ㅚᙥᙀ໢ᙽ⚄ⲯ⊡⹐ х㨊ᆄ⍰㒙ἱ⹐⠛೬ݗⴜ⻝ᔷ⠇ຉࢗᔷἱ

⹐⃞㊟⵶ᆄ ц⸋ᙀ⃞দت㊟⵶Ḛ㦼᳻⥩♭ၻ⃘།⃞ᔷ๾ጦဋ 栾܎

ഫت㙒ڜᔷᆙ⏯჎⠛⇌㓿⻝࠶ࢗᔷ⃞⍕࢛⃞ⷆ␡ 栾௎࡛ᙌẺ⎧⠜

⿄ ц

ࢯΌ࡜௳ఝ⃞ᨮ᚞᜙ 栾⥩♭ၻ⠛㔎ટ㐐Ⴣ⎧٦࡜௳ఝ⃞⸾ਘ

ࢗᔷ⹐ 栾ؿⶫ᧳٦໢ኩ⃞ፕᗣፕᕞ ± ⼹ᷱḚፕ⒮ᕥ⌫⃞Ṏࢯ࠽㕧 ક㙃 цڒݜ⃘Ἢتٲം‥Ἢ⼋ 桌L ⢵تݗࢗᔷ㆏়تݗᄴ࢛ᕥ

⌫ 栾ࣁᅃ㆏়تݗᄴ࢛ᕥ⌫┺ цLL ঩⵶ࢗᔷݽா࡜௳ఝㆬ‐⃞ت ݗۘᄴ᪉ჽ Ḛᄷ᪉ჽ ذ═࢖ 栾ࣁṎࢯ࠽㕧ક㙃໴ⶫ᧳٦ ц

⥩♭ၻㆧ⃘Ἢت┺࡛ᙌ⠛⸾ਘ␡๣㙒یᆄ⽶⃞ᄴ࢛␡๣ 栾ம㔦 ذ⍄᳻ ³⥩ᦧ␡๣´ ц㔎ટ㐐⇩⍭٦࡜௳ఝ⃞ㆬ‐ᆄ⽶ хၬ۟╁ᠫ

⠛ᙆೲউἱ 栾ؿ⺯ᖣ٦ Cⁿ فᙌ‐॑ఝ⃞ᕨ῎⣪დἱ ц

 ᕧتٲ࡜௳⚄মڞℷۦ࡜௳ఝ⃞Ḛᄷ᪉ჽ 栾⥩♭ၻ⺯ᖣ٦㈾

⚄ذ⃞ޯ㊞⦦┙ᔷমڞ㓨⼹ᷱ᧌ਘ цㅚᖺ࡜௳⚄ذޯ㊞⦦࢛ᚬ⇩⍭

⃞㒑⎷ цㅚ㘖ဉۦ 㨻ᖜ⃞⇩⍭⛘ᄣࢲᄚೲ⼂ີ ц㨻ᖜ⇩⍭٦㈾

⚄ذޯ㊞⦦┙ᔷ⃞㓨⼹ᷱ᧌ਘ х( ⨿ᛦ♭ &HVjUR ᧌ਘ х/ ⽒⛴

ᷱ )HMpU ᧌ਘঘহ⍁᧌ਘ ц㏒ວቭໜ㈾⚄⃞ޯ㊞⦦┙ᔷ⃞ᛖٲ╁

ᚶᏰႵࢲᕰヷ⚄ ц

1954 ၣ 栾⥩♭ၻΌࢩ⏯ᕥᨷ⺯ᖣ٦ 桌ᙌ‐ఝ⃞ 6 ۄ᜝ᙃ

%HUJPDQ ႃ㊡⃞㦯ᙃᘻổ R ᮹⿏ 桌L 2 − R ᳻ၢᕥਘ 桍LL

ாᛖٲ㓳ࢷرᙌ R ≥ −n цㅚᖺ໢ %$ ີ࠽ᕞ )XFKV ຖἱ⃞ᔌ ㅪ цۦ᳻㦯ᙃࡘჽᖯᵥຖἱ⃞ᏰႵ 栾⥩♭ၻ⺯ᖣ٦᥻تݗ္ᘻổ ࡑ┓ఝ௎মڞⶫᚬᖯᵥ᳻૕ۗ⾰Ἦ ц

 ڒ㓊ᕧ೬ⴜ⻝ࢗᔷ⹐⃞⇩⍭ዷდ⾨ 栾⥩♭ၻ⇩⍭٦ދᄴ࢛

ᕥ⌫⹐⃞تٲક㙃 цڒ⃞╁ᚶ◮╁ாڒ⃞⦰ۦъᄨ૕ۗல⹅⾨ы 1977 ঘ⠛ᚲޅ хড়⣪᯼়⦰⃞ᙀъ٩㔝ࡒݗ⠇⻝ᔷࡒݗᙣ⇉ࢗᔷ

⃞္ܫᔷ▤ᆄދᄴ࢛ᕥ⌫┺ы1979 ِف ц

⥩♭ၻᖺᙥغ⓿ᙆີߊംᆄ⃞ᔷ๾ວِت цໜڒ⠛দتۗ⠇๾

዗ጁ⃞६ႃೳጁᔷ๾ວ 6$ ፕ㛦ࣿ㊣ 5DPDQXMDQ ℩᥿ダ 栾ᤳഫ 3 ⼹ᙃ %DWPDQ ዷ⸲ 栾³ࡒڂى⵶ㇺᖺ⠇๾዗ጁ⃞ 栾ㇺᄣ℉ᕧா

੅ڜ㘥໣ِر 栾ா⢿மᄨ٨ㅻتᥧᗆ㒙⃞⇩⍭ဉۦ ц· · · цڒ ݜِ㒙খᙌዩᵃسীِ⬧ ц㛘࠻ 栾ፕ㛦ࣿ㊣ؿᨈᙌࡑ㇩ຌ዗ 

تݗ⠇๾ೳጁࢲتݗ዗ᶅ⃞ хⷎ▸ᙌ┟⃞ᔷ๾ວ⃞ヷ⻝ 栾ڒா

ᛖ⍁⌫ႃذ݁‖٦ᔷ๾⃞উ൝ᆄ 栾`⠌݁‖٦تຖ⌫ႃ⃞Ẇ⹾ᆄ

⽶ цᵃ⛘⥩♭ၻா࡚ᖆᙛᔷ๾ῳ᫇ف 栾໴ထᖺ༁ى᪉௅ۗ⃞ᔷ

๾ວ٦ ц࡚ᤄ 栾ፕ㛦ࣿ㊣⠛੅ڜ⃞ᏮⶻᘽΩᏮ 栾ᘽᙌ᧳ຖᆄሣ

⚉ ц· · · ц㕐ᵃ⥩♭ၻா⢿மဉۦᗆᄣ℉`ೲ 栾ےڒ⠛੅ڜா

ᔷ๾ᕥ㖭⃞Ꮾⶻ㙝ᵃسᖺㅚᡥḚࢭ㕃ف⃞´ ц⥩♭ၻ ³዗ࣹ௅ᄨ⠇๾

ᔷ๾⃞ೳጁ㖤ၣ዗᳻ㅡ⸁㞖ᬂ хᙌ೬ᕥ㖭ࣝㅡ⃞ᔷ๾ೲဳ´ цㅚت

࢜㖫⃞ᖺ⥩♭ၻ㖪ࢎ⃞ࣿࣸ⠛㝪ڂ⃞ᥲࣸ ц

ᒄ Ḃ

ࣕತᒄḂ

[1] ⥩♭ၻ +XD /RR.HQJ 6HOHFWHG SDSHUV 6SULQJHU9HUODJ 1983 [2] ⥩♭ၻ 栾ఽೈ┟ᔷ⹐ 栾⌋๾࢖᷹⊣ 栾1957 ц

[3] ⥩♭ၻ 栾ᔷ⹐໣დ 栾⌋๾࢖᷹⊣ 栾1957 ц

[4] ⥩♭ၻ 栾೬ⴜ⻝ᔷࢗᔷ⹐ف⃞࡜௳ఝ⃞⸾ਘ࢛ᚬ 栾⌋๾࢖᷹

⊣ 栾1958 ц

[5] ⥩♭ၻ 栾ᎁᔷਘ⃞ۅⷆঘ࡚ாᔷ⹐ف⃞ኩΌ 栾⌋๾࢖᷹⊣ 栾 1963 ц

[6] ⥩♭ၻ х⦍ਖ਼࠻ 栾࡜௳⚄ 栾ذ᪤⌋๾ጦⲣ࢖᷹⊣ 栾1963 ц [7] ⥩♭ၻ 栾㞖⏯ᔷ๾დ⹐ 栾⌋๾࢖᷹⊣ 栾⏜تॲ 栾1963 桍

⏜٩ॲ 1981 桍㚣␹ 栾1984 ц

[8] ⥩♭ၻ хộ࠶ 栾ᔷ⹐ாㄩی࢛ᚬف⃞ኩΌ 栾⌋๾࢖᷹⊣ 栾 1978 ц

[9] ⥩♭ၻ 栾ᄨ૕ۗல⹅⾨ 栾⌋๾࢖᷹⊣ 栾1977 ц

[10] ⥩♭ၻ хᚲޅ хড়⣪᯼ 栾٩㔝ࡒݗ⠇⻝ᔷࡒݗᙣ⇉ࢗᔷ⃞္

ܫᔷ▤ᆄދᄴ࢛ᕥ⌫┺ 栾⌋๾࢖᷹⊣ 栾1979 ц [11] ⥩♭ၻ 栾ࠣ㆛๾ 栾⌋๾࢖᷹⊣ 栾1981 ц

[12] ⥩♭ၻ 栾⥩♭ၻ⌋ᗣ⦰ۦ㆛㕃 栾ذ᪤ᔢ✹࢖᷹⊣ 栾1984 ц

℠⊤ᒄḂ

[13] 6 .XQJ DQG .+ /RRN 6 *RQJ DQG .4 /X)XQFWLRQ WKHRU\

⵽ +XD /RR.HQJ 6HOHFWHG SDSHUV 6SULQJHU9HUODJ 1983 634

± 635 ц

[14] + 'DYHQSURW $QDO\WLF PHWKRGV IRU 'LRSKDQWLQH HTXDWLRQV DQG 'LRSKDQWLQH LQHTXDOLWLHV $QQ $UERU 3XEOLVKHUV 1962 8 ц

[15] + +DOEHUVWDP/RR.HQJ +XD 桌2ELWXDU\ $FWD $ULWK L11988 99 ± 117 ц

[16] 6 6DODII$ ELRJUDSK\ RI +XD /RR.HQJ 6FLHQFH DQG WHFKQRORJ\

LQ (DVW $VLD 6LYLQ 1DWKDQ HGLWHG :DWVRQ $FDG 3XEO ,QF 1977 [17] ,0 9LQRJUDGRY 6HOHFWHG ZRUNV 6SULQJHU9HUODJ 1984 ц

[18] =; :DQ $OJHEUD DQG JHRPHWU\ ⵽ +XD /RR.HQJ 6HOHFWHG SDSHUV 6SULQJHU9HUODJ 1983 281 ± 284 ц

[19] 3 %DWPDQ 5HYLHZ RQ VHOHFWHG SDSHUV RI /RR.HQJ +XD 0DWK 0RQWKO\ 1 1986 ц