New proof that the sum of natural numbers is -1/12 of the zeta function. Home > Quantum mechanics > Zeta function and Bernoulli numbers

Transcription

1 New proof that the sum of atural umbers is -/2 of the zeta fuctio Home > Quatum mechaics > Zeta fuctio ad Beroulli umbers 206/07/09 Published 204/3/30 K. Sugiyama We prove that the sum of atural umbers is -/2 of the value of the zeta fuctio by the ew method Figure 5.: Damped oscillatio of atural umber Abel calculated the sum of the diverget series by the Abel summatio method. However, we caot calculate the sum of atural umbers by the method. I this paper, we calculate the sum of the atural umbers by the exteded Abel summatio method. 200 Mathematics Subject Classificatio: Primary M06; Secodary B65. CONTENTS Itroductio Issue Importace of the issue Research treds so far New derivatio method of this paper Old method by the Abel summatio method New method New proof for the summatio formula of atural umbers Proof of the propositio Proof of the propositio / 36

2 3 Coclusio Future issues Supplemet Numerical calculatio of the oscillatig Abel summatio method Geeral formula of the oscillatig Abel summatio method The mea Abel summatio method by residue theorem Defiitios of the propositios by the (ε, δ)-defiitio of limit Appedix Proof of the summatio formula of the powers of the atural umbers Cauchy s itegral formula ad mea formula Lauret series ad the pole of the order zero The proof of the summatio formula of the atural umbers by harmoic derivative The proof of the summatio formula of the powers of the atural umbers by harmoic overtoe derivative The secod proof Riema zeta fuctio Jacobi s theta fuctio Relatio with the q-aalog The q-aalog Ackowledgmet Bibliography Itroductio. 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 atural umbers. The sum does ot coverge. It diverges. This paper coverges the sum by the exteded Abel summatio method. " " = 2 (.).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 the 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..3 Research treds so far Leohard Euler 2 suggested that the sum of all atural umbers is -/2 i 749. Berhard Riema 3 showed that the itegral represetatio of the zeta fuctio is -/2 i 859. Sriivasa Ramauja 4 proposed that the sum of all atural umbers is -/2 by Ramauja summatio method i 93. Niels Abel 5 itroduced the Abel summatio method i order to coverge the diverget series i about 829. J. Satoh 6 costructed q-aalogue of zeta-fuctio i 989. M. Kaeko, N. Kurokawa, ad M. Wakayama 7 derived the sum of the double quoted atural umbers by the q-aalogue of zeta-fuctio i / 36

3 .4 New derivatio method of this paper We defie a ew sum of all atural umbers by a ew summatio method, oscillatig Abel 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 -/2 for the ifiite term. (Summatio formula of atural umbers) lim x 0+ k exp( kx) cos (kx) = 2 (.2).5 Old method by the Abel summatio method Niels Abel itroduced the Abel summatio method i about 829. We cosider the followig sum of the series. S = a k (.3) The we defie the followig fuctio. (The Abel summatio method) F(x) = a k exp( kx) (.4) 0 < x (.5) We defie Abel sum as follows. S A = lim x 0+ F(x) (.6) We cosider the followig diverget series. S = ( ) k k (.7) The we defie the followig fuctio. 3 / 36

4 G(x) = ( ) k k exp( kx) (.8) 0 < x (.9) Here, we will use the followig formula. (Formula of the geometric series that has coefficiets of atural umbers) kr k = r ( r) 2 (.0) We obtai the followig equatio by the above formula. F(x) = We calculate the Abel sum as follows. e x ( + e x ) 2 (.) S A = lim x 0+ F(x) = e 0 ( + e 0 ) 2 = 4 (.2) Abel calculated the sum of the diverget series by the Abel summatio method as above stated. However, we caot calculate the sum of atural umbers by the method. We explai the reaso as follows. We cosider the followig sum of all atural umbers. S = k (.3) The we defie the followig fuctio. G(x) = k exp( kx) (.4) 0 < x (.5) Here, we will use the followig formula. (Formula of the geometric series that has coefficiets of atural umbers) kr k = r ( r) 2 (.6) We obtai the followig equatio by the above formula. 4 / 36

5 G(x) = We calculate the Abel sum as follows. e x ( e x ) 2 (.7) S A = lim G(x) = e 0 x 0+ ( e 0 ) 2 = (.8) The Abel sum diverges as above stated. Therefore, we caot calculate the sum of the atural umbers 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 = B! z =0 (.9) I this paper, Beroulli umbers B is Beroulli polyomial B (). 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 ( e z ) 2 = z 2 ( B! z =0 ) 2 (.20) 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 ( e z ) 2 = z 2 ( B! z =0 ) 2! ( z) =0 (.2) Therefore, we express the fuctio G (x) as follows. G(x) = x 2 ( B 2! ( x) )! x =0 =0 (.22) We obtai the followig equatio by calculatig the above formula. G(x) = x 2 ( x x2 2 + O(x3 )) ( + x + x2 2 + O(x3 )) (.23) G(x) = 5 ( x + x2 2 x2 + O(x 3 )) ( + x + x2 2 + O(x3 )) (.24) 5 / 36

6 G(x) = x 2 ( + ( )x + ( ) x2 + O(x 3 )) (.25) G(x) = x 2 ( 2 x2 + O(x 3 )) (.26) G(x) = x 2 + O(x) (.27) 2 Here the symbol O (x) is Ladau symbols. The symbol meas that the error has the order of the variable x. The first term diverges. The first term is called sigular term. Therefore, the followig the Abel sum diverges. It is the purpose of this paper to remove this divergece. S A = lim x 0+ G(x) = (.28) 2 New method 2. New proof for the summatio formula of atural umbers We explai the ew proof for the summatio formula of atural umbers. We have the followig propositio. We will prove the propositio i the ext sectios. Propositio. lim x 0+ k exp( kx) cos(kx) = k (2.) 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) = 2 (2.2) We defie the fuctio S ad H (x) as follows. 6 / 36

7 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) " " = (2.0) " " = 2 (2.) The double quotes mea the aalytic cotiuatio of the sum of atural umbers. 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 -/2 for the ifiite term. 2.2 Proof of the propositio Propositio. lim x 0+ k exp( kx) cos(kx) = k (2.2) Proof. We defie the fuctio H (x) as follows. H (x) = k exp( kx) cos(kx) (2.3) O the other had, we have the followig equatio. 7 / 36

8 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 (2.4) (2.5) This completes the proof. 2.3 Proof of the propositio 2 Propositio 2. lim x 0+ k exp( kx) cos(kx) = 2 (2.6) Proof. We cosider the followig sum of all atural umbers. S = k (2.7) The we defie the followig fuctio. H(x) = k exp( kx) cos(kx) (2.8) Here, we use the followig formula. (Euler's formula) We obtai the followig formula from the above formula. 0 < x (2.9) exp(iθ) = cos(θ) + i si(θ) (2.20) cos(θ) = 2 (exp(iθ) + exp( iθ)) (2.2) We obtai the followig formula from the above formula by puttig θ = k x. cos(kx) = (exp(ikx) + exp( ikx)) (2.22) 2 Hece, we express the fuctio H (x) as follows. 8 / 36

9 H(x) = k exp( kx) (exp(ikx) + exp( ikx)) 2 H(x) = k exp( kx) exp(ikx) 2 + k exp( kx) exp( ikx) 2 H(x) = k exp( kx + ikx) + k exp( kx ikx) 2 2 H(x) = k exp( k(x ix)) 2 + 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 (.27). G(z) = z 2 + O(z) (2.29) 2 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). (x ix) 2 + O(x) (2.3) 2 H(x) = 2 G(x ix) + G(x + ix) (2.32) 2 H(x) = 2 ( (x ix) O(x)) + 2 ( (x + ix) 2 + O(x)) (2.33) 2 H(x) = 2(x ix) 2 + 2(x + ix) 2 + O(x) (2.34) 2 9 / 36

10 The first term is show below. 2(x ix) 2 = 2(x 2 2ix 2 x 2 ) = 4ix 2 (2.35) O the other had, the secod term is show below. 2(x + ix) 2 = 2(x 2 + 2ix 2 x 2 ) = 4ix 2 (2.36) Therefore, the sum of the first term ad the secod term vaishes. 2(x ix) 2 + 2(x + ix) 2 = 0 (2.37) Sice the sigular term vaished, we have the followig equatio. H(x) = 2 + 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. lim H (x) = k exp( kx) cos(kx) Therefore, we have the followig equatio. This completes the proof. = H(x) = + O(x) (2.40) 2 lim ( lim H (x)) = lim ( + O(x)) = x 0+ x (2.4) 3 Coclusio We obtaied the followig results i this paper. - We proved that the sum of atural umbers is -/2 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 oscillatig Abel summatio method ad the q-aalog. 0 / 36

11 5 Supplemet 5. Numerical calculatio of the oscillatig Abel summatio method We will calculate the followig value umerically H = k exp( k0.0) cos (k0.0) (5.) The result is show below. H = (5.2) This value is very close to the followig -/2. = (5.3) 2 The graph is show i the Figure Figure 5.: 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 (k0.0) (5.4) The result is show below. H = (5.5) / 36

12 Therefore, the series does ot coverge if the atteuatio factor ad the vibratio period do ot have a special relatioship. We will cosider the relatio i the ext sectio. 5.2 Geeral formula of the oscillatig Abel summatio method We cosider the followig series. S = a k (5.6) The we defie the followig fuctio. 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) = a k exp( kz) (5.0) z C (5.) We express the fuctio G (z) by usig the sigular term A (z), ad the costat C. G(z) = A(z) + C + O(z) (5.2) Here, we use the followig formula. (Euler's formula) exp(iθ) = cos(θ) + i si(θ) (5.3) We express the fuctio H (x) as follows by usig the above formula. H(x) = (G(z) + G(z )) (5.4) 2 z = φ(x) + ix (5.5) z = φ(x) ix (5.6) We obtai the followig equatio by usig the formula (5.2) of the fuctio G (z). 2 / 36

13 H(x) = (A(z) + A(z )) + C + O(z) (5.7) 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) 2 (A(z) + A(z )) = O(z) (5.8) The series a k ad the fuctio G (z) ad φ(x) are show below. Series Fuctio Fuctio G(z) = z 2 + O(z) φ(x) = cot ( π 2 ) x + O(x3 ) k G(z) = z O(z) φ(x) = cot ( π 4 ) x + O(x4 ) k 2 G(z) = 2 z 3 + O(z) φ(x) = cot ( π 6 ) x + O(x5 ) k 3 G(z) = 6 z O(z) φ(x) = cot ( π 8 ) x + O(x6 ) k G(z) = A z + + ζ( ) + O(z) π φ(x) = cot ( ) x + O(x+3 ) The series ad the example of the damped oscillatio summatio method are show below. 3 / 36

14 Series Example of the damped oscillatio summatio method H(x) = 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( + 2 )) cos (kx) k H(x) = k π exp ( kx cot ( )) cos (kx) Therefore, we have the followig equatios. " " = 2 (5.9) " " = 2 (5.20) " " = 0 (5.2) " " = 20 (5.22) We calculate the above equatios as follows. 4 / 36

15 lim x 0+ exp( kx3 ) cos(kx) = 2 (5.23) lim x 0+ k exp( kx) cos (kx) = (5.24) 2 lim x 0+ k2 exp( kx 3 ) cos (kx) = 0 (5.25) lim x 0+ k3 exp ( kx( + 2 )) cos (kx) = 20 (5.26) Geeral formula of the oscillatig Abel summatio method is show below. " " = ζ( ) (5.27) lim π x 0+ k exp ( kx cot ( )) cos (kx) = ζ( ) (5.28) The mea Abel summatio method by residue theorem 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) (5.29) (5.30) (5.3) We express the fuctio H (x) as follows. H(x) = (G(x ix) + G(x + ix)) (5.32) 2 We iterpret the above fuctio as a sum of the fuctio G (z) ad G (z2) like the followig Figure / 36

16 Im(z) G(z) O Re(z) G(z2) Figure 5.2: The oscillatig Abel summatio method I the above Figure 5.2, the two fuctios G (z) ad G (z2) approach to the origi O at the two agles. This is a 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

17 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: The mea Abel summatio method We express the above cosideratio i the followig equatios. H(x) = 2 (G(z ) + G(z 2 )) (5.33) H(x) = 3 (G(z ) + G(z 2 ) + G(z 3 )) (5.34) H(x) = 4 (G(z ) + G(z 2 ) + G(z 3 ) + G(z 4 )) (5.35) H(x) = (G(z ) + G(z 2 ) + G(z 3 ) + + G(z )) (5.36) H(x) = G(z k) (5.37) 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. 7 / 36

18 L = 2πx (5.38) O the other had, we approximately express the circumferece L as follows for sufficietly large. Therefore, we have the followig equatio. L = z k+ z k (5.39) 2πx = z k+ z k (5.40) The, we express the ormalized costat / as follows. = z k+ z k 2π z k (5.4) z k = x (5.42) O the other had, the directio of -i (zk+ - zk) is same as the directio of zk as show i the Figure 5.4 Im(z) zk + zk + - zk -i -i (zk + - zk) zk O x Re(z) Figure 5.4: Geeral damped oscillatio summatio method Therefore, we express / as follows. 8 / 36

19 = z k+ z k = i(z k+ z k ) = iδz k (5.43) 2π z k 2πz k 2πz k δz k = z k+ z k (5.44) Therefore, we express the fuctio H (x) as follows. H(x) = iδz k G(z 2πz k ) k (5.45) 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 (5.46) (5.47) 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 ( z O(z)) = 2 (5.48) Therefore, we obtai the followig ew geeral summatio method. (The mea Abel summatio method) S = a k G(z) = a k exp( kz) H(x) = z =x dz G(z) 2πiz (5.49) (5.50) (5.5) S H = lim x 0+ H(x) (5.52) This summatio method ca sum ay series by the residue theorem. 5.4 Defiitios of the propositios by the (ε, δ)-defiitio of limit The value of the limit depeds o the order of the limit as follows. 9 / 36

20 lim ( lim x 0 k exp( kx) cos(kx) ) = 2 (5.53) lim (lim k exp( kx) cos(kx) ) = (5.54) x 0 I this paper, we defie the order of the limit as follows. lim f k(x) : = lim ( lim f k (x)) (5.55) x 0 x 0 f k (x) = k exp( kx) cos(kx) (5.56) This order of the limit meas the followig relatio. 0 (5.57) x 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 = 2 (5.58) (5.59) (5.60) We have the followig propositio. lim S = (5.6) 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 N <, (5.62) we have the followig iequatio. R < S (5.63) 20 / 36

21 We have the followig propositio. Propositio. lim x 0+ H (x) = S (5.64) 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, (5.65) we have the followig iequatio. H (x) S < ε (5.66) We ca summarize the above defiitio as follows. ε > 0, N, δ > 0 s. t. x > 0, x < δ H (x) S < ε (5.67) We have the followig propositio. Propositio 2. lim ( lim H (x)) = α (5.68) x 0+ 0 (5.69) 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 <, (5.70) we have the followig iequatio. H (δ) α < ε (5.7) We ca summarize the above defiitio as follows. ε > 0, N N, δ > 0 s. t. N, N < H (δ) α < ε (5.72) The propositio is the regio of ad the propositio 2 is the regio -/2 i the followig figure. 2 / 36

22 /x ε=0.0 /2 /δ O N Figure 5.5: (ε, δ)-defiitio of limit 6 Appedix 6. Proof of the summatio formula of the powers of the atural umbers The author foud the followig formula ad proved it i March 204. (Summatio formula of the atural umbers) ζ( ) = lim x 0+ k exp( kx) cos (kx) (6.) N.S foud the followig formula by geeralizig the above formula ad published i March 205. (Summatio formula of the powers of the atural umbers) ζ( 2t) = lim x 0+ k2t exp( k t x) cos (k t x) I this sectio, we prove the above formula i the followig coditio. (6.2) t (6.3) 6.. Cauchy s itegral formula ad mea formula We express the fuctio F(z) which is differetiated s-times as follows. 22 / 36

23 F (s) (z) = ds dz s F(z) (6.4) We call the differetiated fuctio F (s) (z) the derivative. We call the fuctio we differetiate the geeratig fuctio. O the other had, Cauchy s itegral formula is show below. (Cauchy s itegral formula) F(a) = C dz 2πi(z a) F(z) (6.5) The fuctio F(z) is holomorphic i the Cauchy s itegral formula. If the fuctio F(z) is ot holomorphic, the itegrated fuctio is differet from the origial fuctio by residue theorem. I this paper, we call the formula with the o-holomorphic fuctio F(z) Cauchy s mea formula i order to distiguish the Cauchy s itegral formula. (Cauchy s mea formula) f(a) = C dz 2πi(z a) F(z) (6.6) We iterpret the fuctio f(a) as the mea value of the values o the cotour path C of the fuctio F(z). Therefore, we iterpret the Cauchy s mea formula as the mea value theorem. We iterpret the fuctio f (a) is the aalytic cotiuatio of the fuctio f (z). We call the aalytic cotiuatio of the fuctio the mea fuctio. We use differet fuctio ame for the fuctio f(a) sice the mea fuctio is differet from the origial fuctio geerally. Cauchy s mea differetiatio formula is show below. (Cauchy s mea differetiatio formula) f (s) (a) = Γ(s + ) C dz F(z) 2πi(z a) (z a) s (6.7) We call the differetiated fuctio f (s) (z) the mea derivative. We iterpret that the Cauchy s mea differetiatio formula is the formula that averages ad differetiates the fuctio. We iterpret that the mea derivative f (s) (z) is the fuctio we average ad differetiate s-times. We express the mea derivative by from derivative by Cauchy s mea formula. (Cauchy s mea formula) f (s) (a) = C dz 2πi(z a) F(s) (z) (6.8) We express the geeratig fuctio by from mea derivative by Lauret series. (Lauret series) F(z) = s= f(s) (a) (z a)s Γ(s + ) (6.9) The relatio of the fuctios i this sectio is show below. 23 / 36

24 Geeratig fuctio F (z) Derived fuctio ( ) F s ( z) Differetiatio Average Average Mea geeratig fuctio f (a) Differetiatio Mea derived fuctio ( ) f s ( a) Figure 6.: Mea geeratig fuctio ad mea derivative 6..2 Lauret series ad the pole of the order zero Geerally, the order of the pole is a positive iteger. (The pole of the order ) (6.0) z Z (6.) Though the order of the pole is greater or equal to oe geerally, we itroduce the followig the pole of the order zero by usig fiite positive real umber h i this paper. (The pole of the order zero) hz h (6.2) 0 < h (6.3) The pole of the order zero is diverget at z=0. The differetiatio of the above pole is show below. d dz ( hz h) = z O the other had, the differetiatio of the atural logarithm is show below. (6.4) 24 / 36

25 d dz (log ( z )) = z (6.5) Therefore, we iterpret the atural logarithm as the pole of the order zero. We derive the costat term from the coefficiet of the argumet of the atural logarithm. log ( C z ) = log ( z ) + log(c) = log ( ) + γ (6.6) z Therefore, the ew iterpretatio that we regard the term of the order zero as the pole of the order zero is cosistet with the old iterpretatio that we regard the term of the order zero as the costat term. We express ay fuctio by the ew Lauret series with atural logarithm. (Lauret series with atural logarithm) F(z) = f(k) (0) Γ(k + ) zk h k= (6.7) f (k) (0) = Γ(k + ) C dz 2πiz F(z) (6.8) zk We express the fuctio F(z) as follows if the fuctio has the pole of the order zero at most by usig the costat A ad γ. F(z) = A log ( z ) + γ + f(k) (0) Γ(k + ) zk (6.9) 6..3 The proof of the summatio formula of the atural umbers by harmoic derivative Harmoic umber is show below. (Harmoic umber) H = k We defie the harmoic geeratig fuctio as follows. (Harmoic geeratig fuctio) (6.20) H (z): = exp(kz) (6.2) k We obtai the harmoic derivative by differetiatig the harmoic geeratig fuctio s-times. (Harmoic derivative) 25 / 36

26 H (s) (z) = ks exp(kz) (6.22) k We itroduce the followig symbol. The, we itroduce the followig symbol. H(z) = lim H (0) (z) = Li (e z ) (6.23) H (s) (z) = lim H (s) (z) = Li ( s) (e z ) (6.24) We defie the mea harmoic derivative by Cauchy s mea formula. (Mea harmoic derivative) ν (s) (0): = dz C 2πiz H (s) (z) (6.25) We call the above fuctio the atural derivative. We call the atural derivative the atural fuctio shortly. We itroduce the followig symbol. The, we itroduce the followig symbol. ν (s) (0) = lim ν (s) (0) = ζ( s) (6.26) ν (s) = ν (s) (0) (6.27) We obtai the atural fuctio by Cauchy s mea differetiatio formula. (The atural fuctio) ν (s) (0) = Γ(s + ) dz C 2πiz H(z) zs (6.28) We choose the followig coditio. Re(z) < 0 (6.29) O the other had, the harmoic geeratig fuctio is show below. H(z) = log( e z ) (6.30) We calculate the order of the pole of the harmoic geeratig fuctio at z=0 by usig Ladau s symbol as follows. H(z) = log( e z ) = log( ( + O(z))) = O (log ( ) ) (6.3) z Sice the harmoic geeratig fuctio has the pole of the order zero at z=0, we express the fuctio as the Lauret series by usig the costat A ad γ. (Lauret series of the harmoic geeratig fuctio) 26 / 36

27 H(z) = A log ( z ) + γ + ν(k) (0) Γ(k + ) zk (6.32) We differetiate the above fuctio two-times. H (2) (z) = A z 2 + ν(2) (0) + O(z) (6.33) We prove the summatio formula of the atural umbers as follows. S = lim S = lim x 0+ S = lim x 0+ x 0+ k exp( kx) cos(kx) (6.34) 2 (H(2) ( x + ix) + H (2) ( x ix)) (6.35) 2 ( A ( x + ix) 2 + A ( x ix) 2) + ν(2) (0) + O(x) (6.36) S = ν (2) (0) = ζ( ) = 2 (6.37) (Q.E.D.) 6..4 The proof of the summatio formula of the powers of the atural umbers by harmoic overtoe derivative We defie the harmoic overtoe geeratig fuctio as follows. (Harmoic overtoe geeratig fuctio) H t (z): = exp(k t z) (6.38) k We obtai the harmoic overtoe derivative by differetiatig the harmoic overtoe geeratig fuctio s-times. (Harmoic overtoe derivative) H t(s) (z) = kst exp(k t z) (6.39) k We itroduce the followig symbol. The, we itroduce the followig symbol. H t (z) = lim H t(0) (z) (6.40) H t(s) (z) = lim H t(s) (z) (6.4) 27 / 36

28 We defie the mea harmoic overtoe derivative by Cauchy s mea formula. (Mea harmoic overtoe derivative) ν t(s) (0): = dz C 2πiz Ht(s) (z) (6.42) We call the above fuctio the atural overtoe derivative. We call the atural overtoe derivative the atural overtoe fuctio shortly. We itroduce the followig symbol. The, we itroduce the followig symbol. ν t(s) (0) = lim ν t(s) (0) = ζ( st) (6.43) ν t(s) = ν t(s) (0) (6.44) We obtai the atural overtoe fuctio by Cauchy s mea differetiatio formula. (The atural overtoe fuctio) ν t(s) (0) = Γ(s + ) dz C 2πiz z s Ht (z) (6.45) We choose the followig coditio. t (6.46) The, we have the followig equatio for the positive k. I additio, we choose the followig coditio. The, we have the followig equatio for the positive k. k k t (6.47) Re(z) < 0 (6.48) exp(k t z) exp(kz) (6.49) Therefore, we have the followig equatio by choosig the absolute value of the z is very small. H t (z) = exp(kt z) exp(kz ) = H(z) (6.50) k k From the above result, we foud that the harmoic overtoe geeratig fuctio does ot have the pole of higher order tha the pole of the harmoic geeratig fuctio. We calculate the order of the pole of the harmoic geeratig fuctio at z=0 by usig Ladau s symbol as follows. H(z) = log( e z ) = log( ( + O(z))) = O (log ( )) (6.5) z Sice the harmoic overtoe geeratig fuctio has the pole of the order zero at z=0, we express the fuctio as the Lauret series by usig the fuctio A(t) ad γ(t). (Lauret series of the harmoic overtoe geeratig fuctio) 28 / 36

29 H t (z) = A(t) log ( z ) + γ(t) + νt(k) (0) Γ(k + ) zk (6.52) We differetiate the above fuctio two-times. H t(2) (z) = A(t) z 2 + νt(2) (0) + O(z) (6.53) We prove the summatio formula of the powers of the atural umbers as follows. (Q.E.D.) S = lim S = lim x 0+ k2t exp( k t x) cos(k t x) (6.54) S = lim x 0+ x 0+ 2 (Ht(2) ( x + ix) + H t(2) ( x ix)) (6.55) 2 ( A(t) ( x + ix) 2 + A(t) ( x ix) 2) + νt(2) (0) + O(x) (6.56) S = ν t(2) (0) = ζ( 2t) (6.57) 6..5 The secod proof I the previous sectio, we prove the summatio formula of the powers of the atural umbers o the followig coditio. t (6.58) I this sectio, we prove the formula o the followig coditio. 0 < t (6.59) O the above coditio, we have the followig formula for the atural umber k. k t < (k + ) t (6.60) Therefore, we have the followig formula. exp(k t ) < exp((k + ) t ) (6.6) I additio, we choose the followig coditio. Re(z) < 0 (6.62) The, we have the followig formula for the atural umber k ad very small z. exp(k t z) > exp((k + ) t z) (6.63) I the same way, we have the followig formula for the atural umber k ad very small z. 29 / 36

30 exp(kt z) k > exp((k + )t z) (6.64) k + Therefore, the absolute value of the followig fuctio mootoically decreases. f(x) = exp(xt z) x (6.65) As show i the followig figure, the first term ad the itegratio of the fuctio f(x) from oe to ifiity is greater tha the harmoic overtoe geeratig fuctio Figure 6.2: Harmoic overtoe geeratig fuctio Therefore, we have the followig formula for very small z. H t (z) = exp(kt z) < e z + exp(xt z) dx (6.66) k x Here, we trasform the variable as follows. y = x t (6.67) We differetiate the both sides. dy = tx t dx (6.68) dx t dy = tx x dy = ty dx x dy t y = dx x (6.69) (6.70) (6.7) Therefore, we trasform the variable of the itegratio as follows. 30 / 36

31 e z + exp(xt z) dx x = e z + t exp(yz) dy y Here, we cosider the followig fuctio. g(y) = exp(yz) y (6.72) (6.73) As show i the followig figure, the harmoic geeratig fuctio is greater tha the first term ad the itegratio of the fuctio g(y) from oe to ifiity Figure 6.3: Harmoic geeratig fuctio Therefore, we have the followig formula for very small z. e z + t exp(yz) dy < e z + y t exp(kz) (6.74) k As show i the previous sectio, the right side of the followig formula has the pole of zero-th at z=0. H t (z) = exp(kt z) < e z + k t exp(kz) (6.75) k Therefore, as show i the previous sectio, we have the followig formula. ζ( 2t) = lim x 0+ k2t exp( k t x) cos(k t x) (6.76) 3 / 36

32 6.2 Riema zeta fuctio Harmoic derivative is show below. (Harmoic derivative) H (s) (z) = ks exp(kz) (6.77) k We express the harmoic umber by the limit of the harmoic derivative. H = lim z 0 (lim s 0 H (s) (z)) (6.78) We defie the partitio fuctio as follows. (Partitio fuctio) Z(z) = exp(kz) (6.79) We express the partitio fuctio by the limit of the harmoic derivative. Z(z) = lim s ( lim H (s) (z)) (6.80) Riema defied the zeta fuctio. (Zeta fuctio) ζ(s) = s = (6.8) < Re(s) (6.82) We express the zeta fuctio by the limit of the harmoic derivative. ζ(s) = lim z 0 ( lim H ( s) (z)) (6.83) We summarize the above results as follows. - The harmoic umber is the limit of the harmoic derivative. - The partitio fuctio is the limit of the harmoic derivative. - The zeta fuctio is the limit of the harmoic derivative. We express these relatios i the followig figure. 32 / 36

33 Zeta fuctio (s) Limit Harmoic derivative ( ) H s ( z) Limit Harmoic umber H Limit Partitio fuctio Z (z) The atural fuctio is show below. (The atural fuctio) ν (s) (0) = dz Figure 6.4: Zeta fuctio C 2πiz H (s) (z) (6.84) We obtai the followig formula by itegratig the above formula by parts (s-)-times. We replace the variable s by -s. ν (s) (0) = Γ(s) dz C 2πiz z s H () (z) (6.85) ν ( s) (0) = Γ( s) dz O the other had, Riema defied the aalytic zeta fuctio. (The aalytic zeta fuctio) C 2πiz zs H () (z) (6.86) ζ(s) = Γ( s) dz ez 2πiz zs ez (6.87) We express the aalytic zeta fuctio by the limit of the atural fuctio. We express these relatios i the followig figure. C ζ(s) = lim ν ( s) (0) (6.88) 33 / 36

34 Zeta fuctio (s) Limit Harmoic derivative ( ) H s ( z) Aalytic cotiuatio Average Aalytic zeta fuctio (s) Limit Natural fuctio (s) (0) Figure 6.5: Aalytic zeta fuctio 6.3 Jacobi s theta fuctio Riema expressed the zeta fuctio by the Jacobi s theta fuctio. (The express of the zeta fuctio by the Jacobi s theta fuctio) ζ( 2s) = Γ( s) 0 dx x x s ψ(x) πs (6.89) ψ(x) = exp( k 2 πx) (6.90) (Jacobi s theta fuctio) θ(z, τ) = exp(πik 2 τ + 2πikz) (6.9) k= θ(0, ix) = + 2ψ(x) (6.92) O the other had, we have the mea harmoic overtoe derivative as follows. (Mea harmoic overtoe derivative) ν t(s) (0) = dz C 2πiz Ht(s) (z) (6.93) We obtai the followig formula by itegratig the above formula by parts s-times. 34 / 36

35 ν t(s) (0) = Γ(s + ) dz C 2πiz z s Ht(0) (z) (6.94) We express the zeta fuctio by the harmoic overtoe geeratig fuctio. (The express of the zeta fuctio by the harmoic overtoe geeratig fuctio) (Harmoic overtoe geeratig fuctio) ζ( st) = Γ(s + ) dz C 2πiz z s Ht(0) (z) (6.95) H t(0) (z) = exp(k t z) (6.96) k 6.4 Relatio with the q-aalog 6.4. The q-aalog F. H. Jackso 8 defied the followig quatized ew atural umber, the q-umber i 904. (The q-umber) [] q = + q + q q (6.97) [] q = q k (6.98) [] q = q q (6.99) We call the quatized mathematical object the q-aalog geerally. The q-aalog of the atural umber is the q-umber []q. We express the atural umber by the classical limit of the q- umber. = lim q [] q (6.00) The zeta fuctio is show below. (Zeta fuctio) ζ(s) = s = (6.0) The q-aalog of the zeta fuctio is the q-zeta fuctio. The fuctio is show below. 7 (The q-zeta fuctio) ζ q (s) = s [] q(s ) q = We express the zeta fuctio by the classical limit of the q-zeta fuctio. (6.02) 35 / 36

36 ζ(s) = lim q ζ q(s) (6.03) ζ(s) = s [] (s ) = ζ(s) = s = (6.04) (6.05) < Re(s) (6.06) The relatio with the harmoic derivative is show below. [] q = exp(z) H () (z) (6.07) q = exp(z) (6.08) 7 Ackowledgmet I writig this paper, I thak from my heart to NS who gave valuable advice to me. 8 Bibliography Mail: sugiyama_xs@yahoo.co.jp 2 Leohard Euler (768), 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, 7, 83-06, Opera Omia, (5), Berhard Riema (859), Uber die Azahl der Primzahle uter eier gegebee Grosse (O the Number of Primes Less Tha a Give Magitude), Moatsberichte der Berlier Akademie, Bruce C. Berdt (939), Ramauja's Notebooks, Ramauja's Theory of Diverget Series, Spriger-Verlag (ed.), Chapter 6, G. H. Hardy (949), Diverget Series, Oxford: Claredo Press, Juya Satoh (989), q - aalogue of Riema's ζ - fuctio ad q - Euler umbers, J. Number Theory, 3, , 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 (), 75-92, math/ F. H. Jackso, (905), 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), 27-44, 36 / 36