a 1 = 1 a a a a n n s f() s = Σ log a 1 + a a n log n sup log a n+1 + a n+2 + a n+3 log n sup () s = an /n s s = + t i

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1 0 Dirichlet Serie & Logarithmic Power Serie. Defiitio & Theorem Defiitio.. (Ordiary Dirichlet Serie) Whe,a,,3, are complex umber, we call the followig Ordiary Dirichlet Serie. f() a a a a 3 3 a 4 4 Note Givig log i the geeral Dirichlet erie a e - R, <,,3,, we obtai the ordiary Dirichlet erie. Whe we ay merely Dirichlet erie, it mea ordiary Dirichlet erie i may cae. So, i thi chapter, we follow thi cutom. Theorem.. Whe,t are real umber ad f Oe of the followig hold. a / t i i Dirichlet erie,. f() coverge for arbitrary.. f() diverge for arbitrary. 3. There exit a certai real umber c uch that f() coverge for.t. > c ad f() diverge for.t. < c. Thi c i called the lie of covergece. By covetio, c if f() coverge owhere ad c - if f() coverge everywhere o the complex plae. How to calculate c. Whe ak i diverget, k c lim up. Whe ak i coverget, k c lim up loga a a log loga a a 3 log Example. p-erie ()

2 The, ak k k Sice thi i diverget, c lim up log log Example. Dirichlet Eta erie () The, ak k k Sice thi i diverget, c lim up (-) k- or 0 or 0 lim log log log 0 log Abolute Covergece A Dirichlet erie f() a / i abolutely coverget if the erie a / i coverget. There exit a a 0 a - c uch that f() coverge abolutely for > a ad coverge o-abolutely for < a. Thi a i called the lie of abolute covergece. How to calculate a. Whe k a k i diverget, a lim up. Whe k log a a a log a k i coverget, a lim up loga a a 3 log Example. p-erie () 3 4 The, a k k k Sice thi i diverget, l - -

3 a lim up log log Example. Dirichlet Eta erie () The, a k k k ( -) k- Sice thi i diverget, a lim up log log Uiform Covergece A ee i Theorem.. (. ), geeral Dirichlet erie f() certai domai. Thi i the ame alo about the ordiary Dirichlet erie. a e - coverge uiformly i a There exit a u 0 u - c / uch that f() coverge uiformly for > u ad coverge o-uiformly for < u. Thi u i called the lie of uiform covergece. How to calculate u u lim up log T log where T up t < t < k a k k it up ak cotlogk k ak itlogk k up k kaj a k co j t log j t < (3.) (3.) (3.3) Proof We prove the ecod lie ad the third lie of the provio. Sice k e logk, ak k -it -it logk k ak e k Subtitute e -it logk co( tlogk) - i i( tlogk ) for the right ide, i ak e -it logk k k ak co( tlogk) - i i( tlogk) - 3 -

4 i.e. k a k ak co( tlogk) - k it k i ak i( tlogk) k Traformig the right ide ito a pole form, Where, k a k a co i i k it a ak co( tlogk) k ta - ak co( ) k tlogk, - k ak i( tlogk) k ak i( tlogk) ( ta - x,y i ivere taget fuctio i coideratio of 4 quadrat. ) Takig abolute value, i.e. k k k it a k a co i i a co i i a k it a k ak co( tlogk ) k Expadig the ier of the, j k Here, ice ak i( tlogk) k aj a k co( t log j) co( t log k ) i( t log j) i( t log k) k co( t log j) co( t log k ) i( t log j) i( t log k ) co y log j we obtai (3.3). Example 3. p-erie () 3 4 The, ice a k, T t < up j kco k t log j j k log u limup log Example 3. Dirichlet Eta erie ()

5 Though a k (-) k-, calculatio of T i difficult eve if which formula i ued. So, we put (,t ) log (-) k- k -it k log Calculatiig for 5,800, t 0000, we obtai the follow. Table 5800, t, t, 000, , , , , , , 0.449, , , , , Whe we coider the reult of thee umerical computatio, it eem to be u /. Theorem..3 (Holomorphy) a t i coverge for > c, If Dirichlet erie f f() i holomophic at > c. Ad the derivative of f() i give a follow. f k () (-) k a log k Theorem..4 (Uiquee) Let two Dirichlet erie are a follow. f() a, g() b If both are coverget i a certai domai ad f() g() hold at there, a b for,,3,. Bibliography 数論入門 D.B. ザギヤー著片山考次訳岩波 990 年, etc

6 . Dirichlet Serie & Power Serie Theorem.. ( Melli traform ) Let Dirichlet erie f() ad Power erie F() z are a follow. a f(), F() z a z The, the followig expreio hold i the domai i which f() coverge abolutely a () F(e -t )t - dt (.) 0 Proof Subtitutig z e -w for F() z, a e -w F(e -w ) Itegratig both ide of thi m time with repect to w from to w, w w F(e -w )dw m (-) m From Cauchy formula for repeated itegratio, w w F(e -w )dw m ( ) m So (-) m Givig w 0 to both ide, (-) m a e - w m ( m) a m ( m) a e -w m w F(e -t m- )( w-t) dt w F(e -t m- )( w-t) dt 0F(e -t )(-t ) m- (-) m dt ( m) F(e -t )t m- dt 0 Sice ologer m eed ot be a atural umber, replacig thi with a complex umber, a () F(e -t )t - dt 0 Thi i called Melli traform. (.m) Note I (.m), replacig a atural umber m with a complex umber, ad givig w 0 to both ide, 0 w F(e -w )dw (-) a (-) f() That i, The uper itegral of Fe -w with repect to w from to 0 i equal to the product of (-) ad f(). (.) - 6 -

7 Epecially, whe a,,,3, Left: 0 w F(e -w )dw 0 w Right: (-) Therefore, 0 w (-) () dw (-) () e w - e -w dw 0 w dw e w - Remark A metioed above, the followig were clarified about Melli traform.. i The coefficiet of Dirichlet erie f() ad the coefficiet of Power erie F() z are commo. ii The variable of f() ad the variable t of F(e -t ) are idepedet. becaue, the former i the umber of time of itegratio ad the latter i a variable of the itegrad. Ad ice z e -t, ad z are alo idepedet. Therefore, Dirichlet erie f() ad Power erie F() z are idepedet. Cocluio That i, Power erie F() z i merely medium i the Melli traform. Regrettably, we caot ue Power erie a aalyi tool of Dirichlet erie

.3 Dirichlet Serie & Logarithmic Power Serie.3. Relatio betwee Dirichlet Serie & Logarithmic Power Serie Dirichlet erie a a a a 3 3 a 4 4 - Im() < (.

8 .3 Dirichlet Serie & Logarithmic Power Serie.3. Relatio betwee Dirichlet Serie & Logarithmic Power Serie Dirichlet erie a a a a 3 3 a Im() < (.0) i traferred to the followig erie by coverio e - z. a z log a z log a z log a 3 z log3 (.) 0 Derivatio a a e -log a (e - ) log a z log Sice, the coverio z e - i may-to-oe fuctio, the rage from which (.0) i correctly traferred to (.) i limited to - Im <. Neverthele, we ca ue (.) a aalyi tool of Dirichlet erie (.0) i the rage. We will call the erie (.) Logarithmic Power Serie..3. Circle of covergece & Lie of covergece A ee above, the relatio betwee the variable z of logarithmic power erie ad the variable of Dirichlet erie wa a follow.. -log z -log z - i arg z Here, let it, z xiy. The, it -log x i y - i arg( x i y) Although coordiate x -y are traferred to coordiate -t by the coverio, the coordiate -t are imilar to pola coordiate r -. The differece amog both i oly the exitece of log i the real part. From thi, -log x y t -arg( x i y ) (.t) The real part (.) i draw o the left figure. (.) - 8 -

9 All the horizotal ectio of the real part (.) are circle cetered at the origi. Radiu of each circle i give by x y. That i, r x y e -. Therefore, the covergece radiu R of (.) i give by the lie of covergece c of (.0), a follow. R e -c. For example, if the lie of covergece i c /, the covergece radiu i R e -/ If the left figure i horizotally cut at the height /, it i the right figure. Radiu of the bae i ad the upper part i a covergece regio of (.)

.4 p - erie & Logarithmic Geometric Serie Riema zeta fuctio () () e -log i expaded i a Dirichlet eri at Re() > a follow. f() The right-had ide Dirichlet erie i particularly called p- erie.

I both figure, the left-had ide i orage, ad the right-had ide i blue. I both figure, we caee that the lie of covergece of the right-had ide i.

10 .4 p - erie & Logarithmic Geometric Serie Riema zeta fuctio () () e -log i expaded i a Dirichlet eri at Re() > a follow. f() The right-had ide Dirichlet erie i particularly called p- erie. Whe it, the real part ad the imagiary part of both ide are illutrated repectively a follow. The left figure i a real part, the right figure i a imagiary part. I both figure, the left-had ide i orage, ad the right-had ide i blue. I both figure, we caee that the lie of covergece of the right-had ide i. Further, for compario with latter figure, appropriate 3 poit o the circumferece i marked i white. (.f) Subtitutig e - z ( i.e. -log z ) for (.f ), ( -log z ) z log g() z (.g) Though the right ide g() z i a kid of Logarithmic power erie, we will call g() z Logarithmic Geometric Serie i particular. Whe z x iy, the real part ad the imagiary part of both ide are illutrated repectively a follow. The left figure i a real part, the right figure i a imagiary part. I both figure, the left-had ide i orage, ad the right-had ide i blue. I both figure, we caee that the radiu R of covergece of the right ide i approximately Further, we caee that the 3 poit i the former figure are traferred to the red poit i both figure by z e

11 Note I thi covergece circle, it.t. or t ca ot be expreed. Further, ice the lie of covergece of f() i c, the radiu of covergece of g() z i R e Expreio by ummatio formula Applyig Euler-Maclauriummatio formula to the logarithmic geometric erie g() z, we obtai the approximate value eaily. Theorem.4. Let B m be Beroulli umber, B m () t be Beroulli polyomial, S m, z e -. kid. The the followig expreio hold for a complex umber z.t. z log k k z log - z log m r B r r- ( r )! be Stirlig umber of the t z log r- - S ( r-, ) log z R m (.) m R m - ( m )! S ( m,) log z Bm t- t Epecially, whe z < e z log k - k - log z - log z - log z -log z B t- t t m (.r) t (.) (.') Proof Whe f x i a fuctio of cla C m o a cloed iterval a,b, t i the floor fuctio, B r i Beroulli umber ad B m () So, let b f k a bf t dt ka R m - ( m )! x i Beroulli polyomial, Euler-Maclauriummatio formula wa a follow. f b f a b Bm a S z log, z log, z log3,, z log The, f() t z log t f() t dt f m () t t z log t log z m r t- t z log - z log t m S ( m, ) log z t m l - - B r r! f ( r-) b - f ( r-) a R m f m () t dt

12 f r- () z log t r- t t r- S ( r-, ) log z Subtitutig thee for the above formula, z log k k z log - z log m r R m - ( m )! Subtitutig m for (.) ad (.r), z log k k B r r- ( r )! z log r- - S ( r-, ) log z R m (.) Bm t- t z log t m S ( m, ) log zdt t m m - ( m )! S ( m,) log z Bm t- t z log - z log B z log - z log R - S ( ), log z B t- t log z - log z z log t B t- t dt t t m z log - S (, ) log z R z log - log z R t Whe 0< z < e -, lim zlog 0, lim zlog 0. The, z log k log z - - R k R log z - log z z log t B t- t dt t Subtitutig the latter for the former, we obtai (.). 00 log Example i k, m k (.r) - -

13 Example i k log k Example 4.3 k z log k, z 0., 0.,

.5 - erie & Logarithmic Geometric Serie Dirichlet eta fuctio () i expaded i a Dirichlet eri at Re() >0 a follow. () (-) - e -log f() (.f) The right-had ide erie i particularly called Dirichlet - erie.

14 .5 - erie & Logarithmic Geometric Serie Dirichlet eta fuctio () i expaded i a Dirichlet eri at Re() >0 a follow. () (-) - e -log f() (.f) The right-had ide erie i particularly called Dirichlet - erie. Whe it, the real part ad the imagiary part of both ide are illutrated repectively a follow. The left figure i a real part, the right figure i a imagiary part. I both figure, the left-had ide i orage, ad the right-had ide i blue. I both figure, we caee that the lie of covergece of the right-had ide i 0. Further, for compario with latter figure, appropriate 3 poit o the circumferece i marked i white. Subtitutig e - z ( i.e. -log z ) for (.f ), ( -log z ) (-) - z log g() z (.g) Whe z x iy, the real part ad the imagiary part of both ide are illutrated repectively a follow. The left figure i a real part, the right figure i a imagiary part. I both figure, the left-had ide i orage, ad the right-had ide i blue. I both figure, we caee that the radiu R of covergece of the right ide i approximately. Further, we caee that the 3 poit i the former figure are traferred to the red poit i both figure by z e -. Note I thi covergece circle, it.t. 0 or t ca ot be expreed. Further, ice the lie of covergece i c 0, the radiu of covergece of g() z i R e

15 Expreio by ummatio formula Applyig Euler-Maclauriummatio formula to the logarithmic geometric erie g() z, we obtai the approximate value eaily. Theorem.5. Let B m be Beroulli umber, B m () t be Beroulli polyomial, S m, kid. The the followig expreio hold for a complex umber z.t. z e -. (-) k- z log k k - log( ) -z log ez m r ( r )! - z log be Stirlig umber of the t B r - r z log - -z log ( ) r- m R m - ( m )! S ( m,) log z z log t B m t - t dt-z log t m Epecially, whe z < e -0 (-) k- z log k k r- S ( r-,) log z R m (.) - log( ) z log t B m t - t dt -z log - log z ez t m (.r) -z log log z-log z z log t B t- t dt (.) -z log - - log z t (.') Proof z log k S k ev k S z logk z log z log k z log S k S - (-) k- z log k ev S - S k i.e. S - (-) k- z log k S -z log S k Applyig Theorem.4. i previou ectio to S, S, z log k k z log - z log m r B r ( ) r- ( r )! z log r- - S ( r-, ) log z R m - 5 -

16 Therefore, - ( m )! R m z log k k ( ) k A firt, A ecod, Lat, R m m S ( ) z log - z log m r - ( m )! ( r )! m, log z B m t- t B r r- t m z log r- - S ( r-, ) log z R m m S ( ) m, log z Bm t- t t m - k- z log k z log - z log -z log z log - z log m r m r B r z log - ( ) r- m B r ( r )! ( ) r- ( r )! - z log r R m - z log R m r- S ( r-, ) log z z log r- - S ( r-, ) log z z log -z log z log - z log z log - - z log z log( ez ) log - z log - r ( r )! - log( ) -z log ez - z log B r z log r- - ( ) S ( ) r- m B r z ( r )! log - r- B r - ( r )! r z log - ( ) r- - z log r m R m -z log R m - ( m )! r-, log z r- S ( r-, ) log z -z log r- S ( r-,) log z m S ( ) m, log z B m t- t z log m ( m )! S ( m,) log z Bm t- t t m t m - 6 -

17 m - ( m )! S ( m,) log z B m t- t z log t dt-z log Bm t m t- t Subtitutig thee for the above, weobtai (.) ad (.r). Whe 0< z e - <, lim z log 0. The, (-) k- z log k - k Givig m, -z log - -z log m r B r ( r )! t m r- S ( r-, ) log z R m -z log m R m - ( m )! S ( m,) log z Bm t- t (-) k- z log k k - log( ) -z log ez B - -z log S (, ) log z R t m - log( ) -z log ez - -z log log z R R - -z log S ( ) Subtitutig the latter for the former, we obtai (.). 50 Example 5. k, log z Bm t- t t -z log log z-log z z log t B t- t dt t (-) k-.3 log k, m - 7 -

18 Example 5. (-) k- 0.9 log k k Example 5.3 (-) k- z log k, z 0.6, 0.7, 0.8 k Acceleratio of Logarithmic Power Serie (Aalytic Cotiuatio) Applyig Kopp Traformatio to Logarithmic Power Serie (-) k- x iy log k, k q j-k m j g( x,y,q,m ) k j ( q ) j j k (-) k- ( xiy) log k (3.) - 8 -

Sice q may arbitrary poitive umber, we give q, m30. Whe (3.) ad -log z are draw together, it i a above. The left figure i a real part, the right figure i a imagiary part.

19 Sice q may arbitrary poitive umber, we give q, m30. Whe (3.) ad -log z are draw together, it i a above. The left figure i a real part, the right figure i a imagiary part. I both figure, orage i -log z ad blue i (3.). Both are coitet well ad are ee ipot. Comparig thee with the figure of (.g), we ee that logarithmic geometric erie i aalytically cotiued from the circle to the quare. Virtual zero of Dirichlet Eta Fuctio Eve if Kopp Traformatio i applied, the area traferred to both figure from it i limited to >0, t <. Sice all kow zero of () are out of thi rage, the zero of () are ot cotaied all over thee figure. Regrettably, logarithmic geometric erie ca ot alo be ued a aalyi tool of Dirichlet erie. However, It i poible to draw the virtual image of the zero of() o thee figure. For example, by e -, /i i reduced 4 i ad i equated with /i So, if 00 zero /it,,,50 of () are mechaically chaged ito z ad (3.) i draw, it i a follow. Blue i -t ad red i t. Although thee are virtual zero, we caee that thee are ditributed o the circumferece of radiu e -/ Alie' Mathematic Kao. Koo - 9 -

Math 210A Homework 1

Math 210A Homework 1 Math 0A Homework Edward Burkard Exercise. a) State the defiitio of a aalytic fuctio. b) What are the relatioships betwee aalytic fuctios ad the Cauchy-Riema equatios? Solutio. a) A fuctio f : G C is called