6 Global definition of Riemann Zeta, and generalization of related coefficients. p + p >1 (1.1)

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1 6 Global definition of Riemann Zeta, and generalization of related coefficient 6. Patchy definition of Riemann Zeta Let' review the definition of Riemann Zeta. 6.. The definition by Euler The very beginning definition of the Riemann Zeta wa a follow. ( p ) p + p + 4 p + p > (.) Serie on the right-hand ide i a generalized harmonic erie, and eem to be called p-erie. Euler obtained the following even number zeta baed on thi expreion. n ( n ) ( ) - n+ B n ( n )! However, thi erie wa divergence at p. Therefore, ( / ), etc. were not able to be treated. Then, Euler derived the following expreion. ( p ) - -p - p - 4 p +- p 0, p (.) p The erie in thi parenthei i called Dirichlet Eta. Riemann zeta can be defined on the half-plane Gauian if uing thi formula. In addition, the function on p > may be (.). A well known, thi derived a follow. - -

2 From thi, + p p + 4 p + p - p - 4 p + p + 4 p + - -p p p - p - 4 p +- + p p 4 p + 6 p + + p p + - p p - 4 p +- Thi i a really myteriou equation. When 0 p <, though the left ide i a divergent erie, the right ide i a convergent erie. Subtituting p 0 for thi, Euler obtained the following trange reult. ( 0 ) " " - Furthermore, Euler found out the following equation. B n+ ( -n) - n,,3,4, (.3) n + Where, B n i Bernouilli number.t. B 0, B /6, B 4 -/30, B -/, B 3 B 5 B 7 0 Thi i alo myteriou equation. Although the left ide mut be infinite, the right ide i a poitive or negative rational number. ( -) " " - ( -) " " 0 ( -3) " " 0 ( -4) " " 0 Here, p -,-4,-6, are called the trivial zero of Riemann Zeta. In addition, (.3) i obtained alo from Euler-Maclaurin ummation formula ( See " 05 Generalized Bernoulli Polynomial and Number " ) - -

3 6.. Riemann' functional equation Riemann dicovered the following functional equation. ( p ) ( -p) in p ( -p) p 0, (.3') -p Thi equation hold on the whole complex plane except p 0,. Applying thi at p <0, we can moothly connect the point of (.3). At that time, -p of the right ide have to be (.). By thi equation, the following calculation wa enabled. ( -/) " " Patchy definition of Riemann Zeta or Thu, if Riemann Zeta i defined on the complex plane uing the above expreion, it i a follow. ( p ) ( p ) r r p - -p - -p r r p ( -p) in p -p r Re( p ) > (-) r- 0 Re( p) p r p ( -p) in p -p - p r (-) r- Re( p) 0 p 0 r -p (-) r- Re( p) 0 p - p r (-) r- Re( p) 0 p 0 r -p (.4) (.4') Formula manipulation oftware eaily draw the graph uch a the above. Becaue, the function uch a (.4) or (.4') i implemented in the oftware. In fact, the graph in thi ection i drawn by (.4). Such a graph can never be drawn only by (.) or (.). Moreover, we can never find out the zero of p baed on the definition (.). It i becaue the zero doe not exit on the plane defined by the expreion

4 The definition of Riemann zeta function uing the Dirichlet erie and the functional equation i o complicated. However, uing a double erie, we can define Riemann zeta function very imply. Beide, the double erie can generalize Bernoulli Number, Euler Numver, tangent number, etc. In the following ection, I mention them

5 6. Global definition of Riemann Zeta 6.. Global definition of Riemann Zeta Definition 6.. We difine the Riemann Zeta Function on the complex plane a follow. ( p ) - -p r r r+ (-) - r -p p (.4p) It i aid that it i proved by Hae around 930 that thi formula hold on the whole complex plane except p. We will make ure of it by example. Example ( -6) ( -) Example ( -/ ),( 0 ),( 3) Example3 Non-trivial zero 6.. Derivation of the definition So neceary for the next ection, I derive the definition along the line of Mr.Sugimoto. () Baic equence When m,n are natural number, the following expreion hold. ( See " 07 New Formula for Power Sum " ) Where, n m n r0 nb r m C r r nb r ( -) r- r n r0,,,,n (.0) Thi generate the following value

6 n \r : : : : : Here, what hould be paid attention i that all value are 0 for r > n. Thi i a property reulting from a general binomial coefficient. Thereby, at any time, we can replace the upper limit r by and can change n to the real number. Thi well behaved property i maintained all the time hereafter. Since the alternate erie i required in order to obtain Riemann Zeta, we tranform (.0) a follow. r nb r ( -) - n r,,3,,n (.) n \r : : : : : () Tangent number Multiplying (.) by n-r and umming up them, we obtain a tangent number. T n r n r n-r (-) - r n n,,3, (.n) n \r Total : : : : : In addition, although original tangent number i a non-negative integer, in thi chapter, we call the ighed one. alo tangent number. There i the following relation between T n- and the original tangent number T n

7 T n- (-) n- T n- n,,3, (3) Bernoulli number Multiplying (.n) by i.e. n + n+ n+ -, we obtain a Bernoulli number B n+. n + n + B n+ T n+ n+ n - n+ n+ - n + B n+ n+ - r n r r+ n T n B n n r n-r r ( ) - - r (-) - r n n,,3, (.3 n ) n (4) Riemann Zeta Finally, ubtituting (.3 n ) for the well known B n+ ( -n) - n + n,,3, we obtain Riemann Zeta at negative integer. n r ( -n) - n+ r r+ (-) - r n n,,3, (.4 -n ) Since all value are 0 for r > n. replacing the upper limit r by we can extend the natural number n to the complex number p. ( -p) - +p r r r+ Since p i a complex number, we can revere the ighn. ( p ) - -p r r r+ (-) - r p p - (.4 -p ) (-) - r -p p (.4 p ) - 7 -

$r n r,,3,,n n \r 3 4 5 : 0 0 0 0 : - 0 0 0 3: -3 0 0 4: -7 6-0 5: -5 5-0 Since all value are 0 for r > n, we can replace the upper limit r by.$

8 6.3 Generalization of related coefficient 6.3. Generalized Stirling Number of the nd kind Dividing (.) of the previou ection by r!, we obtain Stirling Number of the nd kind. r S ( n,r ) (-) - r! r n r,,3,,n n \r : : : : : Since all value are 0 for r > n, we can replace the upper limit r by. Then, ince n doe not need to be a natural number any longer, it i extenible to a more global number. Definition 6.3. (Generalized Stirling Number of the nd kind) S ( p,r ) ( -) r! - r p r,,3, (3.) Example S 3, S 3,5 and S 3.0, S 3.0,5 Although original Stirling Number i a non-negative integer, in thi chapter, we call the ighed one alo Stirling Number. There i the following relation between S p,r and the original Stirling Number S p,r. S ( p,r ) (-) r+ S ( p,r) r0,,, (3.') - 8 -

The following formula i known for a natural number n. ns n- -, n S 3 3 n - 3 n + 3 6 (3.') atifie the following formula alo for a complex number p.

9 The following formula i known for a natural number n. ns n- -, n S 3 3 n - 3 n (3.') atifie the following formula alo for a complex number p. S p, p- -, S p,3 3 p Example S +i, and S +i, Generalized Tangent Number From (.n) in the previou ection, T n r n r n-r (-) - r n n,,3, Since all value are 0 for r > n. we can replace the upper limit r by and extend the natural number n to the complex number p. r T p p-r r ( -) - r p p 0 (3.) Adding the point p 0 to thi, we define a follow. Definition 6.3. (Generalized Tangent Number) T p 0 p 0 r p-r r ( -) - r p p 0 (3.) - 9 -

Example T T 5 and T.0 T 5.0 Example Non-trivial zero 6.3.3 Generalized Bernoulli Number From (.3 n ) in the previou ection, n + B n+ n+ - n r r r+ (-) - r n n,,3, Since all value are 0 for r > n.

10 Example T T 5 and T.0 T 5.0 Example Non-trivial zero Generalized Bernoulli Number From (.3 n ) in the previou ection, n + B n+ n+ - n r r r+ (-) - r n n,,3, Since all value are 0 for r > n. we can replace the upper limit r by and extend the natural number n to the complex number p. p + B p+ p+ p+ - Replacing p with p -, p B p p - r r r+ r p-r r Adding two point p 0, to thi, we define a follow. (-) - r p p - (-) - r p- p (3.3p) Definition (Generalized Bernoulli Number) B p - p p - r r r+ p (-) - r p- p (3.3) - 0 -

11 Example B B 6 and B.0 B 6.0 Example B -4.0 B -4.9 Example3 Non-trivial zero cf. The value of Example are exactly conitent with the value of the table of 5. in " 5 Generalized Bernoulli Polynomial and Number ". Functional equation The following functional equation hold. B -p -p p ( -p) p in Bp -p p 0,,,3, Notation of Riemann Zeta by the variou coefficient Uing thee variou coefficient, we can define Riemann Zeta Function a follow. Formula ( p ) - -p ( p ) - -p r -p r! r+ S (-p,r) (3.4) T -p p (3.5) B - p ( p ) - p (3.6) -p - -

12 6.4 Dirichlet Eeta Function Looking back at the reult of the previou ection, the element of all the coefficient wa the following erie. r r r+ (-) - r -p here, the global definition of p and the related expreion were ( p ) - -p r ( p ) ( p) - -p r r+ (-) - r -p p Comparing thee, we notice the following immediately Global definition of Dirichlet Eeta Function Definition 6.4. We difine the Dirichlet Eeta Function on the complex plane a follow. ( p ) r r r+ (-) - r -p (4.) Thi definition reult in the uual definition of p in a half-plane. Some preparation i required in order to how thi. Lemma r Cr r for,,3, Proof C C0 0 C C3 C C 0 C 0 C C0 0 C 0 C C0 0 C 0 C

13 Next, C C3 C4 C C C C3 C C C C Here, Then, C C 3 C 4 C C C C C C C C C C C C C C C C C C Next, C C3 C4 C C3 3 3 C4 3 C5 3 C C 3 C3 C4 C C C3 C C C3 C Here, C 3 C3 C C C3 C C C3 C

14 Then, C 4 C 5 C3 3 3 C3 C C3 C C C C3 C C3 C C4 3 C5 3 C Hereafter, by induction, we obtain the diired expreion. Theorem 6.4. When p i Dirichlet Eta function, if Re p ( p ) r r r+ (-) - r -p r 0, the following expreion hold. (-) r- r p Proof ( p ) r r r+ (-) - r -p -p -p - -p 3 -p - 3 -p p 4 -p - 4 -p p p -p p 3 -p p

15 ( ) - - According to the above Lemma, Cr r r Uing thi -p r r r for,,3, (The order of the alternating erie are unchanged.) ( p ) (-) - -p When p 0 ( 0 ) (-) When Re( p ) >0, let p x+iy. Then ( x+iy ) (-) - -x- iy x >0 Uing exponential function, ( x+iy ) (-) - -x log -i y log e Uing trigonometric function, ( x+iy ) (-) - e -x log co( y log ) - iin( y log ) When i a natural number and y i a real number, 0 < co( y log ) - iin( y log ) < Therefore, thi erie diverge at x <0 and converge at x >0. Q.E.D Notation of Dirichlet Eeta by the variou coefficient Uing thee variou coefficient, we can define Dirichlet Eeta Function a follow. Formula ( p ) r r! r+ S (-p,r) (4.) T -p ( p ) -p (4.3) B ( p ) -- -p - p p (4.4) -p Alien' Mathematic K. Kono - 5 -

Lecture 3. January 9, 2018

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