2010 Mathematics Subject Classification: Primary 11M06; Secondary 11B65.

Transcription

1 New proof that the sum of atural umber is -1/ of zeta fuctio K. Sugiyama 1 Published 2014/03/30; revised 2015/02/15. Abstract We prove that the sum of atural umber is -1/ of the value of the zeta fuctio by the ew method. Abel calculated the sum of the diverget series by the Abel summatio method. However, we caot calculate the sum of atural umber by the method. I this paper, we calculate the sum of the atural umber by the exteded Abel summatio method Mathematics Subject Classificatio: Primary 11M06; Secodary 11B65. CONTENTS 1 Itroductio Issue Importace of the issue Research treds so far New derivatio method of this paper Old method by Abel summatio method New method New method by ew propositio Proof of the propositio Proof of the propositio Coclusio Future issues Supplemet Numerical calculatio Sigular equatio Geeral damped oscillatio summatio method Defiitio of the propositios by the (ε, δ)-defiitio of limit Relatio with q-aalog q-aalog Zeta fuctio Aalytic zeta fuctio Reflectio itegral equatio Appedix Hurwitz zeta fuctio Lerch zeta fuctio Goursat's theorem Bibliography Itroductio 1.1 Issue The itegral represetatio of the zeta fuctio coverges by the aalytic cotiuatio. O the other had, the zeta fuctio also has a series represetatio. Oe of the represetatio is the sum of all 1 / 36

2 atural umbers. The sum does ot coverge. It diverges. This paper coverges the sum by the exteded Abel summatio method. " " = 1 (1.1) 1.2 Importace of the issue It is desirable that both the itegral represetatio ad the series represetatio have a same value for mathematical cosistecy. Oe method to coverge the diverget series is Abel summatio method. The method coverges the diverget series by multiplyig covergece factor. However, the sum of all atural umbers does ot coverge by the method. Therefore, it is a importat issue to fid a ew summatio method. 1.3 Research treds so far Leohard Euler 2 suggested that the sum of all atural umbers is 1/ i Berhard Riema 3 showed that the itegral represetatio of the zeta fuctio is 1/ i Sriivasa Ramauja 4 proposed that the sum of all atural umbers is 1/ by Ramauja summatio method i Niels Abel 5 itroduced Abel summatio method i order to coverge the diverget series i about J. Satoh 6 costructed q-aalogue of zeta-fuctio i M. Kaeko, N. Kurokawa, ad M. Wakayama 7 derived the sum of the double quoted atural umbers by the q-aalogue of zetafuctio i New derivatio method of this paper We defie a ew sum of all atural umbers by a ew summatio method, damped oscillatio summatio method. The method coverges the diverget series by multiplyig covergece factor, which is damped ad oscillatig very slowly. The traditioal sum diverges for the ifiite terms. O the other had, the ew sum is equal to the traditioal sum for the fiite term. I additio, the sum coverges o 1/ for the ifiite term. (Damped oscillatio summatio method for the sum of all atural umbers) lim x 0+ k exp( kx) cos (kx) = 1 (1.2) 1.5 Old method by Abel summatio method Niels Abel itroduced Abel summatio method i about We cosider the followig sum of the series. S = a k (1.3) 2 / 36

3 The we defie the followig fuctio. (Abel summatio method) F(x) = a k exp( kx) (1.4) 0 < x (1.5) We defie Abel sum as follows. S A = lim x 0+ F(x) (1.6) We cosider the followig diverget series. S = ( 1) k 1 k (1.7) The we defie the followig fuctio. G(x) = ( 1) k 1 k exp( kx) (1.8) 0 < x (1.9) Here, we will use the followig formula. (Formula of the geometric series that has coefficiets of atural umbers) kr k = r (1 r) 2 (1.10) We obtai the followig equatio by the above formula. F(x) = We calculate the Abel sum as follows. e x (1 + e x ) 2 (1.11) S A = lim x 0+ F(x) = e 0 (1 + e 0 ) 2 = 1 4 (1.) Abel calculated the sum of the diverget series by the Abel summatio method as above stated. However, we caot calculate the sum of atural umber by the method. We cosider the followig sum of all atural umbers. 3 / 36

4 S = k (1.13) The we defie the followig fuctio. G(x) = k exp( kx) (1.14) 0 < x (1.15) Here, we will use the followig formula. (Formula of the geometric series that has coefficiets of atural umbers) kr k r = (1 r) 2 (1.16) We obtai the followig equatio by the above formula. G(x) = We calculate the Abel sum as follows. e x (1 e x ) 2 (1.17) S A = lim G(x) = e 0 x 0+ (1 e 0 ) 2 = (1.18) The Abel sum diverges as above stated. Therefore, we caot caluculate the sum of the atural umber by the method. I order to examie the mechaism of the divergece, we use the followig defiitioal formula of Beroulli umbers. (Defiitioal formula of Beroulli umbers) ze z e z 1 = B! z =0 (1.19) I this paper, Beroulli umbers B is Beroulli polyomial B (1). We will square the both sides of the above formula. I additio, we will divide the both sides by z 2. The we obtai the followig equatio. e 2z (1 e z ) 2 = 1 z 2 ( B! z =0 ) 2 (1.20) 4 / 36

5 We will multiply the left side of the above formula by e z. I additio, we will multiply the right side by Maclauri series of e z. The we obtai the followig equatio. e z (1 e z ) 2 = 1 z 2 ( B! z =0 ) 2 1! ( z) =0 (1.21) Therefore, we express the fuctio G (x) as follows. G(x) = 1 x 2 ( B 2! ( x) ) 1! x =0 =0 (1.22) We obtai the followig equatio by calculatig the above formula. G(x) = 1 x 2 (1 x x2 + O(x3 )) (1 + x + x2 2 + O(x3 )) (1.23) G(x) = 1 5 (1 x + x2 x2 + O(x 3 )) (1 + x + x2 2 + O(x3 )) (1.24) G(x) = 1 x 2 (1 + (1 1)x + ( ) x2 + O(x 3 )) (1.25) G(x) = 1 x 2 (1 1 x2 + O(x 3 )) (1.26) G(x) = 1 x O(x) (1.27) Here the symbols O (x) are Ladau symbols. The symbols mea that the error has the order of the variable x. The first term diverges. The first term is called sigular term. Therefore, the followig Abel sum diverges. It is the purpose of this paper to remove this divergece. S A = lim x 0+ G(x) = (1.28) 2 New method 2.1 New method by ew propositio We have the followig propositio. We will prove the propositio i the ext sectios. 5 / 36

6 Propositio 1. lim x 0+ k exp( kx) cos(kx) = k (2.1) We have the followig propositio. We will prove the propositio i the ext sectios, too. Propositio 2. lim x 0+ k exp( kx) cos(kx) = 1 (2.2) We defie the fuctio S ad H (x) as follows. S = k H (x) = k exp( kx) cos(kx) (2.3) (2.4) The we defie the followig symbols = S (2.5) = lim S (2.6) " " = lim x 0+ H (x) (2.7) " " = lim x 0+ ( lim H (x)) (2.8) The we have the followig propositios = (2.9) " " = (2.10) " " = 1 (2.11) The double quotes mea the aalytic cotiuatio of the sum of atural umbers. 6 / 36

7 The traditioal sum diverges for the ifiite terms. O the other had, the ew sum is equal to the traditioal sum for the fiite term. I additio, the sum coverges o 1/ for the ifiite term. 2.2 Proof of the propositio 1 Propositio 1. lim x 0+ k exp( kx) cos(kx) = k (2.) Proof. We defie the fuctio H (x) as follows. H (x) = k exp( kx) cos(kx) (2.13) O the other had, we have the followig equatio. lim H (x) = k exp( k0) cos(k0) = k x 0+ Therefore, we have the followig equatio. lim x 0+ k exp( kx) cos(kx) = k (2.14) (2.15) This completes the proof. 2.3 Proof of the propositio 2 Propositio 2. lim x 0+ k exp( kx) cos(kx) = 1 (2.16) Proof. We cosider the followig sum of all atural umbers. S = k (2.17) The we defie the followig fuctio. 7 / 36

8 H(x) = k exp( kx) cos(kx) (2.18) Here, we use the followig formula. (Euler's formula) We obtai the followig formula from the above formula. 0 < x (2.19) exp(iθ) = cos(θ) + i si(θ) (2.20) cos(θ) = 1 (exp(iθ) + exp( iθ)) (2.21) 2 We obtai the followig formula from the above formula by puttig θ = k x. cos(kx) = 1 (exp(ikx) + exp( ikx)) (2.22) 2 Hece, we express the fuctio H (x) as follows. H(x) = k exp( kx) 1 (exp(ikx) + exp( ikx)) 2 H(x) = 1 k exp( kx) exp(ikx) k exp( kx) exp( ikx) 2 H(x) = 1 k exp( kx + ikx) + 1 k exp( kx ikx) 2 2 H(x) = 1 k exp( k(x ix)) k exp( k(x + ix)) 2 (2.23) (2.24) (2.25) (2.26) I additio, we defie the followig fuctio. G(z) = k exp( kz) (2.27) z C (2.28) We express the fuctio G (z) from the formula (1.27). 8 / 36

9 G(z) = 1 z O(z) (2.29) Hece, we express the fuctio G (x ix) as follows. G(x ix) = k exp( k(x ix)) (2.30) G(x ix) = The, we express the fuctio H (x). 1 (x ix) O(x) (2.31) H(x) = 1 2 G(x ix) G(x + ix) (2.32) H(x) = 1 2 ( 1 (x ix) O(x)) ( 1 (x + ix) O(x)) (2.33) H(x) = 1 2(x ix) (x + ix) O(x) (2.34) The first term is show below. 1 2(x ix) 2 = 1 2(x 2 2ix 2 x 2 ) = 1 4ix 2 (2.35) O the other had, the secod term is show below. 1 2(x + ix) 2 = 1 2(x 2 + 2ix 2 x 2 ) = 1 4ix 2 (2.36) Therefore, the sum of the first term ad the secod term vaishes. 1 2(x ix) (x + ix) 2 = 0 (2.37) Sice the sigular term vaished, we have the followig equatio. H(x) = 1 + O(x) (2.38) We express the fuctio H (x) as follows. H (x) = k exp( kx) cos(kx) (2.39) O the other had, from the formula (2.38) we have the followig equatio. 9 / 36

10 lim H (x) = k exp( kx) cos(kx) Therefore, we have the followig equatio. This completes the proof. = H(x) = 1 + O(x) (2.40) lim ( lim H (x)) = lim ( O(x)) = x 0+ x 0+ (2.41) 3 Coclusio We obtaied the followig results i this paper. We proved that the sum of atural umber is -1/ of the value of the zeta fuctio by the ew method. 4 Future issues The future issues are show below. To study the relatio betwee the damped oscillatio summatio ad the q-aalog. 5 Supplemet 5.1 Numerical calculatio We will calculate the followig value umerically H = k exp( k0.01) cos (k0.01) (5.1) The result is show below. H = (5.2) This value is very close to the followig 1/. 1 = (5.3) The graph is show i the Figure / 36

11 Figure 5.1: Damped oscillatio of atural umber Here, we will double the atteuatio factor as follows. We do ot chage the vibratio period H = k exp( k0.02) cos (k0.01) (5.4) The result is show below. H = (5.5) Therefore, the series does ot coverge if the atteuatio factor ad the vibratio period do ot have a special relatio. We will cosider the relatio i the ext sectio. 5.2 Sigular equatio We cosider the followig series. S = a k (5.6) The we defie the followig fuctio. 11 / 36

12 H(x) = a k exp( kφ(x)) cos(kx) (5.7) 0 < x (5.8) 0 < φ(x) (5.9) I additio, we defie the followig fuctio. G(z) = k exp( kz) (5.10) z C (5.11) We express the fuctio G (z) by usig sigular term A (z), ad the costat C. G(z) = A(z) + C + O(z) (5.) Here, we use the followig formula. (Euler's formula) exp(iθ) = cos(θ) + i si(θ) (5.13) We express the fuctio H (x) as follows by usig the above formula. H(x) = 1 2 (G(z) + G(z )) (5.14) z = φ(x) + ix (5.15) z = φ(x) ix (5.16) We obtai the followig equatio by usig the formula (5.) of the fuctio G (z). H(x) = 1 (A(z) + A(z )) + C + O(z) (5.17) 2 We determie the fuctio ϕ (x) by the followig sigular equatio i order to remove the sigular term of the above equatio. (Sigular equatio) 1 2 (A(z) + A(z )) = O(z) (5.18) The series a k ad the fuctio G (z) ad ϕ (x) are show below. / 36

13 Series a k Fuctio G(z) Fuctio φ(x) 1 G(z) = 1 z O(z) φ(x) = x ta(π/2) + O(x3 ) k G(z) = 1 z O(z) x φ(x) = ta(π/4) + O(x4 ) k 2 G(z) = 2 z 3 + O(z) φ(x) = x ta(π/6) + O(x5 ) k 3 G(z) = 6 z O(z) x φ(x) = ta(π/8) + O(x6 ) The series ad the example of the damped oscillatio summatio method are show below. Series a k Example of the damped oscillatio summatio method 1 H(x) = 1 exp( kx 3 ) cos (kx) k H(x) = k exp( kx) cos (kx) k 2 H(x) = k 2 exp( kx 3 ) cos (kx) k 3 H(x) = k 3 exp ( kx(1 + 2 )) cos (kx) Therefore, we have the followig equatios. 13 / 36

14 " " = 1 2 (5.19) " " = 1 (5.20) " " = 0 (5.21) " " = 1 0 (5.22) We calculate the above equatios as follows. lim 1 x 0+ exp( kx3 ) cos(kx) = 1 2 (5.23) lim x 0+ k exp( kx) cos (kx) = 1 (5.24) lim x 0+ k2 exp( kx 3 ) cos (kx) = 0 (5.25) lim x 0+ k3 exp ( kx(1 + 2 )) cos (kx) = 1 0 (5.26) 5.3 Geeral damped oscillatio summatio method We cosider the followig sum of all atural umbers. The we defie the followig fuctio. S = k H(x) = k exp( kx) cos(kx) G(z) = k exp( kz) (5.27) (5.28) (5.29) We express the fuctio H (x) as follows. 14 / 36

15 H(x) = 1 (G(x ix) + G(x + ix)) (5.30) 2 We iterpret the above fuctio as a sum of the fuctio G (z1) ad G (z2) like the followig Figure 5.2. Im(z) G(z1) O Re(z) G(z2) Figure 5.2: Special damped oscillatio summatio method I the above Figure 5.2, the two fuctios G (z1) ad G (z2) approach to the origi O at the two agles. This is special coditio. Is it possible to make the coditio more geeral? The we cosider the case that the may fuctios G (z) approach to the origi from the all poits o the circle of the radius x like the followig Figure / 36

16 Im(z) G(z) G(z) G(z) G(z) G(z) G(z) O G(z) x Re(z) G(z) G(z) G(z) G(z) G(z) Figure 5.3: Geeral damped oscillatio summatio method We express the above cosideratio i the followig equatios. H(x) = 1 2 (G(z 1) + G(z 2 )) (5.31) H(x) = 1 3 (G(z 1) + G(z 2 ) + G(z 3 )) (5.32) H(x) = 1 4 (G(z 1) + G(z 2 ) + G(z 3 ) + G(z 4 )) (5.33) H(x) = 1 (G(z 1) + G(z 2 ) + G(z 3 ) + + G(z )) (5.34) H(x) = 1 G(z k) (5.35) The fuctio H (x) is the mea value of the fuctio G (z). We suppose that the circumferece of the circle of the radius x is L. We express the circumferece L of the circle as follows. 16 / 36

17 L = 2πx (5.36) O the other had, we express the circumferece L as follows. L = z k+1 z k (5.37) Therefore, we have the followig equatio. 2πx = z k+1 z k (5.38) The, we express the ormalized costat 1/ as follows. 1 = z k+1 z k 2π z k (5.39) z k = x (5.40) O the other had, the directio of i (zk+1 zk) is same as the directio of zk as show i the Figure 5.4 Im(z) zk +1 zk +1 - zk -i -i (zk +1 - zk) zk O x Re(z) Figure 5.4: Geeral damped oscillatio summatio method Therefore, we express 1/ as follows. 17 / 36

18 1 = z k+1 z k = i(z k+1 z k ) = iδz k (5.41) 2π z k 2πz k 2πz k δz k = z k+1 z k (5.42) Therefore, we express the fuctio H (x) as follows. H(x) = iδz k G(z 2πz k ) k (5.43) We express the above sum as the followig itegratio. H(x) = H(x) = z =x z =x idz 2πz G(z) dz G(z) 2πiz (5.44) (5.45) The above result is equal to the residue theorem. We ca iterpret the residue theorem the mea value theorem of cotour itegratio. We obtai the followig result by the residue theorem. H(x) = dz z =x 2πiz ( 1 z O(z)) = 1 (5.46) Therefore, we obtai the followig ew geeral summatio method. (Geeral damped oscillatio summatio method) S = a k G(z) = a k exp( kz) H(x) = z =x dz G(z) 2πiz (5.47) (5.48) (5.49) S H = lim x 0+ H(x) (5.50) This summatio method ca sum ay series by the residue theorem. We call the damped oscillatio summatio method with the residue theorem geeral damped oscillatio summatio method. 18 / 36

19 We call the damped oscillatio summatio method without the residue theorem special damped oscillatio summatio method. 19 / 36

20 5.4 Defiitio of the propositios by the (ε, δ)-defiitio of limit The value of the limit depeds o the order of the limit as follows. lim ( lim x 0 ke kx cos(kx) ) = 1 (5.51) lim (lim x 0 ke kx cos(kx) ) = (5.52) I this paper, we defie the order of the limit as follows. lim f k(x) x 0 : = lim x 0 ( lim f k (x)) (5.53) f k (x) = ke kx cos(kx) (5.54) This order of the limit meas the followig relatio. 0 1 x (5.55) We cofirm the above relatio by the (ε, δ)-defiitio of limit. I this sectio, we defie the followig propositio by the (ε, δ)-defiitio of limit. We defie the fuctio S ad H (x), ad the costat α as follows. S = k H (x) = k exp( kx) cos(kx) α = 1 (5.56) (5.57) (5.58) We have the followig propositio. lim S = (5.59) We ca express the above propositio by (R, N)-defiitio of the limit as follows. Give ay umber R > 0, there exists a atural umber N such that for all satisfyig 20 / 36

21 N <, (5.60) we have the followig iequatio. R < S (5.61) We have the followig propositio. Propositio 1. lim x 0+ H (x) = S (5.62) We ca express the above propositio by (ε, δ)-defiitio of the limit as follows. Give ay umber x > 0 ad ay atural umber, there exists a umber δ > 0 such that for all x > 0 satisfyig x < δ, (5.63) we have the followig iequatio. H (x) S < ε (5.64) We ca summarize the above defiitio as follows. ε > 0, N, δ > 0 s. t. x > 0, x < δ H (x) S < ε (5.65) We have the followig propositio. Propositio 2. lim ( lim H (x)) = α (5.66) x (5.67) x We ca express the above propositio by (ε, δ)-defiitio of the limit as follows. Give ay umber ε > 0, there exist a atural umber N ad a umber δ > 0 such that for all satisfyig N <, (5.68) we have the followig iequatio. H (δ) α < ε (5.69) We ca summarize the above defiitio as follows. 21 / 36

22 ε > 0, N N, δ > 0 s. t. N, N < H (δ) α < ε (5.70) The propositio 1 is the regio of ad the propositio 2 is the regio 1/ i the followig figure. 1/x ε=0.01 1/ 1/δ O N Figure 5.5: (ε, δ)-defiitio of limit 22 / 36

23 5.5 Relatio with q-aalog q-aalog F. H. Jackso 8 defied the followig quatized ew atural umber, q-umber i (q-umber) [] q = 1 + q + q q 1 (5.71) [] q = q k 1 (5.72) [] q = 1 q 1 q (5.73) We call the quatized mathematical object q-aalog geerally. The q-aalog of the atural umber is q-umber []q. We express the atural umber by the classical limit of the q-umber. = lim q 1 [] q (5.74) The zeta fuctio is show below. (Zeta fuctio) ζ(s) = 1 s =1 (5.75) The q-aalog of the zeta fuctio is q-zeta fuctio. The fuctio is show below. 7 (q-zeta fuctio) ζ q (s) = 1 s [] q(s 1) q =1 (5.76) We express the zeta fuctio by the classical limit of the q-zeta fuctio. ζ(s) = lim q 1 ζ q(s) (5.77) ζ(s) = 1 s [] 1(s 1) =1 1 ζ(s) = 1 s =1 (5.78) (5.79) 1 < Re(s) (5.80) 23 / 36

24 5.5.2 Zeta fuctio This paper defies the followig waved (quatized) ew atural umber, atural fuctio. (Natural fuctio) ``z = e z + e 2z + e 3z + + e z (5.81) ``z = e kz (5.82) 1 ez z ``z = e (5.83) 1 e z ν z : = `ν `z: = ``z (5.84) The relatio betwee the atural fuctio ad q-umber as follows. ν z = q[] q (5.85) q = e z (5.86) We differetiate the atural fuctio by the variable z. ν z = e kz d dz ν z = ke kz d 2 dz 2 ν z = k 2 e kz d 3 dz 3 ν z = k 3 e kz (5.87) (5.88) (5.89) (5.90) Therefore, we obtai the followig fuctio by differetiatig the atural fuctio by the variable z m times. d m dz m ν z = k m e kz (5.91) 24 / 36

25 We defie the followig fuctio, atural derivative by replacig the atural umber m by the complex s-1. (Natural derivative) `ν `z(s): = k s 1 e kz (5.92) ν z (s): = `ν `z(s) (5.93) O the other had, we defie the atural umber as follows. (Natural umber) `` = (5.94) `` = 1 (5.95) = `` (5.96) We express the atural umber by a limit of the atural derivative. = lim s 1 (lim z 0 ν z (s)) (5.97) = 1 k 0 ek0 = 1 (5.98) (5.99) We defie the zeta fuctio as follows. (Zeta fuctio) `ζ`(s) = 1 1 s s s + (5.100) `ζ`(s) = 1 k s (5.101) ζ(s): = `ζ`(s) (5.102) We express the zeta fuctio by a limit of the atural derivative. 25 / 36

26 ζ(s) = lim z 0 (lim 0 ν z (1 s)) (5.103) ζ(s) = 1 k s ek0 ζ(s) = 1 k s (5.104) (5.105) We defie the partitio fuctio as follows. (Partitio fuctio) `Z`(z): = e z + e 2z + e 3z + (5.106) `Z`(z) = e kz (5.107) Z(z): = `Z`(z) (5.108) We express the partitio fuctio by a limit of the atural derivative. Z(z) = lim s 1 ( lim ν z (s)) (5.109) Z(z) = k 1 1 e kz Z(z) = e kz (5.110) (5.111) 26 / 36

27 We summarize the above results as follows. The atural umber is a limit of the atural derivative. The zeta fuctio is a limit of the atural derivative. The partitio fuctio is a limit of the atural derivative. We express these relatios i the followig figure. Zeta fuctio (s) Limit Natural derivative z(s) Limit Natural umber Limit Partitio fuctio Z (z) Figure 5.6: Zeta fuctio Aalytic zeta fuctio We defie the harmoic atural derivative as follows by averagig (ucertaiizig) the atural derivative. (Harmoic atural derivative) "" ε (s) = z =ε dz 2πiz ν z(s) (5.1) ν ε (s): = " ν " ε (s): = "" ε (s) (5.113) We defie the aalytic zeta fuctio as follows. (Aalytic zeta fuctio) 27 / 36

28 "ζ"(s) = 1 Γ(s) 0 xs 1 e x dx (5.114) 1 ζ(s) = "ζ"(s) (5.115) The aalytic zeta fuctio is the aalytic cotiuatio of the zeta fuctio. We express the aalytic zeta fuctio by a limit of the harmoic atural derivative. ζ(s) = lim z 0 ( lim ν ε (1 s)) (5.116) ζ(s) = lim x 0 ζ(s) = lim dz z =ε 2πiz ν ε(1 s) ε 0 z =ε ζ(s) = lim ε 0 z =ε dz 2πiz 1 k s ekz dz 2πiz Li s(e z ) Here, the fuctio Lis (w) is the followig polylogarithm fuctio. (Polylogarithm fuctio) Li s (w) = 1 k s wk (5.117) (5.118) (5.119) (5.0) We obtai the followig equatio by usig Taylor expasio of the Lis (e z ). Li s (e z ) = O ( 1 z 1 s) + ζ(s) + ζ(s 1)z + O(z2 ) (5.1) The costat term of the Taylor expasio is the aalytic zeta fuctio. Therefore, we have the followig equatio by usig the residue theorem. ζ(s) = lim ε 0 z =ε We express these relatios i the followig figure. dz ζ(s) 2πiz (5.2) 28 / 36

29 Zeta fuctio (s) Natural derivative z(s) Limit Aalytic cotiuatio Average Aalytic zeta fuctio (s) Limit Harmoic atural derivative (s) Figure 5.7: Aalytic zeta fuctio 29 / 36

30 5.6 Reflectio itegral equatio We shoe the reflectio itegral equatio of the atural derivative. We defie the factorial fuctio by the gamma fuctio as follows. (Factorial fuctio) We have the followig equatio for the atural umber. Π(s) = Γ(s)s (5.3) Π() =! (5.4) We defie the fallig factorial as follows. (Fallig factorial) s t = Π(s) Π(s t) (5.5) We have the followig equatio for the atural umber. 1 = ( 0) (5.6) 2 = ( 0)( 1) (5.7) 3 = ( 0)( 1)( 2) (5.8) We defie the combiatio fuctio by the beta fuctio as follows. (Combiatio fuctio) C(s, t) = 1 s + t B(s, t) st (5.9) We have the followig equatio for the complex s, t. C(s, t) = Π(s + t) Π(s)Π(t) (5.130) We have the followig equatio for the atural umber m,. C(m, ) = (m + )! m!! m + = ( ) (5.131) We show the reflectio itegral equatio of the complex as follows. 9 (Reflectio itegral equatio of the complex) ν z (s + 1) = idt 2π B(s, t)ν z( t) (5.132) S 1 We express the equatio by usig the combiatio fuctio as follows. 30 / 36

31 We deform the equatio as follows. We replace the variable s+1 to s. ν z (s + 1) = idt 1 s + t ν 2π C(s, t) st z ( t) (5.133) S 1 ν z (s + 1) = idt ν z ( t) (5.134) S 1 2π C(s + 1, t)t ν z (s) = idt ν z ( t) (5.135) S 1 2π C(s, t)t We show the reflectio itegral equatio of the quaterio as follows. 10 (Reflectio itegral equatio of the quaterio) ν z (s + 1) = dt3 B(s, t + 2) 2π 2 (t + 1)t ν z( t) (5.136) S 3 We express the equatio by usig the combiatio fuctio as follows. ν z (s + 1) = dt3 1 s + t + 2 ν z ( t) 2π 2 C(s, t + 2) s(t + 2) (t + 1)t (5.137) S 3 We deform the equatio as follows. We replace the variable t to t-2. We replace the variable s+1 to s. ν z (s + 1) = dt3 ν z ( t) S 3 2π 2 C(s + 1, t + 2)(t + 2) 3 (5.138) ν z (s + 1) = dt3 ν z (2 t) S 3 2π 2 C(s + 1, t)t 3 (5.139) ν z (s) = dt3 ν z (2 t) S 3 2π 2 C(s, t)t 3 (5.140) We have the reflectio itegral equatio of the complex ad the quaterio as follows. (Reflectio itegral equatios) ν z (s) = idt ν z ( t) (5.141) S 1 2πt C(s, t) ν z (s) = dt3 ν z (2 t) 2π 2 t 3 (5.142) S 3 C(s, t) 31 / 36

32 6 Appedix 6.1 Hurwitz zeta fuctio We calculate the sum of atural umbers by the Hurwitz zeta fuctio. Adolf Hurwitz 11 published the followig zeta fuctio i (Defiitioal series of Hurwitz zeta fuctio) ζ(s, q) = 1 (q + 0) s + 1 (q + 1) s (6.1) (q + 2) s ζ(s, q) = 1 (q + k) s k=0 (6.2) Hurwitz zeta fuctio ad Riema zeta fuctio have the followig relatioship. (Relatio betwee Hurwitz zeta fuctio ad Riema zeta fuctio) ζ(s) = { 1 1 s s (q 1) s} + { 1 (q + 0) s } (6.3) (q + 1) s q 1 ζ(s) = 1 k s + ζ(s, q) (6.4) Therefore, we ca observe the value of the sum to the middle of the sum of the atural umbers. We calculate the Hurwitz zeta fuctio by the followig asymptotic expasio. Here, Bk is the Beroulli umbers. (Asymptotic expasio of the Hurwitz zeta fuctio) r ζ(s, q) = B k k! k=0 Γ(s + k 1) (6.5) Γ(s)qs+k 1 For example, we thik the sum of oe util three ad the sum of four util ifiity. 4 1 ζ( 1) = 1 k 1 We ca express the value of the Hurwitz zeta fuctio as follows. r + ζ( 1,4) (6.6) ζ( 1,4) = B k Γ(k 2) (6.7) k! Γ( 1)4k 2 Here, we use the followig reflectio formula of the gamma fuctio. k=0 32 / 36

33 π 1 Γ(1 s) = si(πs) Γ(s) We ca modified the Hurwitz zeta fuctio as follows by the above formula. r ζ( 1,4) = B k Γ(2) si(2π) k! Γ(3 k)4 k 2 si((3 k)π) k=0 (6.8) (6.9) We have the followig equatio for iteger k. si(2π) si((3 k)π) = ( 1)k+1 (6.10) Therefore, we ca obtai the followig equatio. r ζ( 1,4) = B k Γ(2) k! Γ(3 k)4 k 2 ( 1)k+1 (6.11) k=0 We have the followig equatio for iteger k 3. 1 Γ(3 k) = 0 (6.) Therefore, we ca obtai the followig equatio. ζ( 1,4) = B 01! ( 1) 0+1 0! (2 0)! B 11! ( 1) 1+1 1! (2 1)! B 21! ( 1) 2+1 (6.13) 2! (2 2)! 42 2 ζ( 1,4) = ( ) + ( ) + ( 1 40 ) (6.14) ζ( 1,4) = ζ( 1,4) = 6 1 (6.15) (6.16) Therefore, we ca obtai the followig result. "( ) + ( )" = 6 + ( 6 1 ) = 1 (6.17) We ca see that the sum of the atural umbers is icreasig certaily halfway accordig the above equatio. The, o the way leadig to the ifiity, it reduced. The series of the zeta fuctio seems to correct for egative itegers. However, why will it decrease o the way leadig to the ifiity? 33 / 36

34 I thik that it oscillates i a ifiite time o the way to ifiity. I thik that it is damped expoetially o the way to ifiity. This idea became the origi of the damped oscillatio summatio method. 6.2 Lerch zeta fuctio This paper proposed that we might iterpret Riema zeta fuctio as the Lerch zeta fuctio i the case of the variable λ is approximately zero. The Lerch zeta fuctio was published by Lerch i (Defiitioal series of Lerch zeta fuctio) L(λ, α, s) = exp(2πiλ0) (0 + α) s + exp(2πiλ1) (1 + α) s + exp(2πiλ2) (2 + α) s + (6.18) L(λ, α, s) = exp(2πiλk) (k + α) s k=0 (6.19) 6.3 Goursat's theorem We derive the followig theorem from the Cauchy's itegral formula. (Goursat's theorem of complex) f (m) (a) = m! 2πi Here, we cosider the followig partitio fuctio. (Partitio fuctio) C Z(z) = e kz We express the above formla by Goursat's theorem. Z (m) (0) = m! 2πi f(z) (z a) m+1 dz (6.20) = ez (6.21) 1 ez Z(z) z m+1 The, we replace the atural umber m to the complex -s. Z ( s) (0) = Γ(1 s) 2πi C dz (6.22) z s 1 Z(z)dz (6.23) C This formula is same as the cotour itegral defiitio of the zeta fuctio. (Cotour itegral defiitio of the zeta fuctio) 34 / 36

35 ζ(s) = Γ(1 s) 2πi z s 1 C The cotour itegral defiitio of the atural derivative is show below. (Cotour itegral defiitio of the atural derivative) ν ε (s) = Γ(s + 1) 2πi z =ε e z 1 e z dz (6.24) z s 1 e z e (+1)z 1 e z dz (6.25) O the other had, Goursat's theorem of quaterio is show below. (Goursat's theorem of quaterio) f (m) (a) = m! 2π 2 Here, we cosider the followig partitio fuctio. (Partitio fuctio) C Z(z) = e kz We express the above formla by Goursat's theorem. Z (m) (0) = m! 2π 2 f(z) (z a) m+3 dz 3 (6.26) = ez (6.27) 1 ez C Z(z) dz (6.28) z m+3 The, we replace the atural umber m to the complex -s. Z ( s) (0) = Γ(1 s) 2π 2 z s 3 Z(z)dz (6.29) C This formula is same as the cotour itegral defiitio of the zeta fuctio. (Cotour itegral defiitio of the zeta fuctio) ζ(s) = Γ(1 s) 2π 2 z s 3 C The cotour itegral defiitio of the atural derivative is show below. (Cotour itegral defiitio of the atural derivative) ν ε (s) = e z 1 e z dz (6.30) Γ(s + 1) 2π 2 z s 3 e z e (+1)z z =ε 1 e z dz (6.31) 7 Bibliography / 36

36 1 Mail: Site: ( 2 Leohard Euler (1768), Remarques sur u beau rapport etre les series des puissaces tat directes que reciproques (Remarks o a beautiful relatio betwee direct as well as reciprocal power series), Memoires de l'academie des scieces de Berli, 17, , Opera Omia, 1 (15), 70-90, 3 Berhard Riema (1859), Über die Azahl der Primzahle uter eier gegebee Grösse (O the Number of Primes Less Tha a Give Magitude), Moatsberichte der Berlier Akademie, Bruce C. Berdt (1939), Ramauja's Notebooks, Ramauja's Theory of Diverget Series, Spriger-Verlag (ed.), Chapter 6, G. H. Hardy (1949), Diverget Series, Oxford: Claredo Press, Juya Satoh (1989), q - aalogue of Riema's ζ - fuctio ad q - Euler umbers, J. Number Theory, 31, , X(89) M. Kaeko, N. Kurokawa, ad M. Wakayama (2003), A variatio of Euler's approach to values of the Riema zeta fuctio, Kyushu J. Math. 57 (1), , math/ F. H. Jackso, (1905), The Basic Gamma-Fuctio ad the Elliptic Fuctios, Proceedigs of the Royal Society of Lodo. Series A, Cotaiig Papers of a Mathematical ad Physical Character (The Royal Society), 76 (508), 7 144, 9 K. Sugiyama (2014), Derivatio of the reflectio itegral equatio of the zeta fuctio by the complex aalysis, 10 K. Sugiyama (2014), Derivatio of the reflectio itegral equatio of the zeta fuctio by the quaterioic aalysis, 11 Adolf Hurwitz, Zeitschrift fur Mathematik ud Physik vol. 27 (1882) p. 95. Lerch, Mathias, Note sur la foctio K (w, x, s) = k=0 e 2kπix (w + k) -s, Acta Mathematica (i Frech) 11 (1887) (1 4): (Blak space) (Blak space) 36 / 36