is defined in the half plane Re ( z ) >0 as follows.

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0) r- r z (.) (.0) and (.) are the ame in Re( z ) >0. Although (.0) can not epre the left ide of line the of convergence, (.) can epre alo the left ide of thi.

1 0 Abolute Value of Dirichlet Eta Function. Dirichlet Eta Function.. Definition Dirichlet Eta Function ( z) i defined in the half plane Re( z ) >0 a follow. () z = r- r z Thi erie i analytically continued to the whole comple plane by applying ome kind of acceleration method. The eaiet of thee i the Euler tranformation a follow. () z = k= k k k r (.0) r- r z (.) (.0) and (.) are the ame in Re( z ) >0. Although (.0) can not epre the left ide of line the of convergence, (.) can epre alo the left ide of thi. Therefore, we can define Dirichlet Eta Function ( z) by (.)... Overview The D figure of the real part and the imaginary part of Dirichlet Eta Function ( iy ) are a follow. Further, the D figure of the abolute value i a follow. In the left figure, trivial zero of ( z ) are oberved along the ai. The right figure i a view of the left figure from the bottom. We can ee that zero of ( z) are located in two line along =/ and =. - -

2 A een in thee figure, Dirichlet Eta Function ( z) () Trivial zero -,-,-6,-8, ha three kind of zero a follow. () Non-trivial zero / i.7, / i.00, / i 5.008, () ( z) pecific zero i /log, i /log, 6 i /log, Among thee, () and () are common to the zero of Riemann zeta function. Further, it i well known that non-trivial zero () eit in 0< < called Critical Strip. Moreover, it i proved that they have to eit ymmetrically with repect to =/. And, fortunately, thi critical trip i included within the convergence range of the erie (.0). - -

. Squared Abolute Value of Dirichlet Eta Squared abolute value of Dirichlet eta function i ( ) f(,y ) =,y Thi i a real-valued function with two variable. And it i hown in the figure a follow. (.0) In the left figure, dent are oberved along =/ and =.

3 . Squared Abolute Value of Dirichlet Eta Squared abolute value of Dirichlet eta function i ( ) f(,y ) =,y Thi i a real-valued function with two variable. And it i hown in the figure a follow. (.0) In the left figure, dent are oberved along =/ and =. The right figure i a view of the left figure from the bottom. We can ee that zero of ( z) are located in two line along =/ and =. The zero on the =/ correpond to the zero of ( z) function and the zero on the = are ( z) pecific zero. On the other hand, there i no zero on the =0. Feature in 0 / Let u focu on pace 0 /. The figure of ection in =0, /, / are drawn a follow. Looking at thi, it look like ( 0,y ) > ( /,y) > ( /,y) in.6 y 5. It i the ame alo in 00 y 5. Below, we oberve thi in more detail. () 0 y.58 The front view of D in thi interval i the left figure. The cutaway view at y =0,.77,.58 of thi i the right figure. In thi interval, (,y) eem to be monotonically increaing with repect to. - -

().58 < y <.6 The front view of D in thi interval i the left figure. The cutaway view at y =.6,.6055,.58 of thi i the right figure.

58 look like monotonically increaing, it i decreaing at the left end when it i een enlarged. Although the curve of y =.

The cutaway view at y =0,.6, 8, of thi i the right figure. In thi interval, (,y) eem to be monotonically decreaing with repect to.

4 ().58 < y <.6 The front view of D in thi interval i the left figure. The cutaway view at y =.6,.6055,.58 of thi i the right figure. In thi interval, (,y) i not monotonic with repect to. In the right figure, although the curve of y =.58 look like monotonically increaing, it i decreaing at the left end when it i een enlarged. Although the curve of y =.6 look like monotonically decreaing, it i increaing at the right end when it i een enlarged. () y.6 The front view of D in thi interval i the left figure. The cutaway view at y =0,.6, 8, of thi i the right figure. In thi interval, (,y) eem to be monotonically decreaing with repect to. Baed on the obervation above, I preent the net hypothei equivalent to the Riemannian hypothei. Hypothei.. When (,y) i the Dirichlet eta function on the comple plane, the quared abolute value (,y) i a monotonically decreaing function in the region 0 < < /, y. Remark The zero common to the Riemann zeta function eit in 0< < called critical trip. Moreover, it i - -

5 proved that they have to eit ymmetrically with repect to =/. So, if (,y) i monotonically decreaing with repect to in the region 0 < < /, y, zero do not eit in the region and the oppoite region / < <, y. Thi i equivalent to the Riemann hypothei. Incidentally, in the oppoite region / < <, y, (,y) function with repect to. i not necearily a monotone - 5 -

6 . Epreion of Squared Abolute Value by Serie.. Epreion of Dirichlet Eta Function by Serie A een in.., Dirichlet Eta Function ( z) wa defined a follow. () z = r- r z Re() z > 0 When Re( z ) >0, let z = iy. Then, (,y ) = r- r -- iy >0 If thi i repreented by an eponential function, (,y ) = r- e -( iy) logr = r- -logr-i ylog r e If thi i repreented by a trigonometric function, (,y ) = r- co( y log r) r - i ( ) - r- in( y log r) r (.).. Epreion of by Double Serie Squared abolute value of Dirichlet eta function (,y) (,y) co( y log r) = ( - ) r- r i epreed uing (.) a follow. in( y log r) r- r Although it look like a very complicated, it become an unepectedly imple epreion when it i epanded and organized. Formula.. When (,y) i the Dirichlet Eta Function, (,y) = r co y log r (.) Proof Let r- co( y log r ) = C r. Then, Thi quare i r- co( y log r) r = C C C C C 5 5 C 6 6 C C C 5 5 C

7 = C C C C C C C C C C C C C C i.e. = C r C r r C r- co( y log r) C C r = r = Let r- in( y log r ) = S r. Then, in a imilar way, we obtain Then, (,y) r- in( y log r) r S r = r = r- co( y log r) r = Returning to the original ymbol, i.e. Here, (,y) = (,y) = C r C S r S = C r C S r S ( - - ) r- in( y log r ) r ( -) r- co( y log r ) - co( y log ) r r- in( y log r ) - in( y log ) co( y log r) co( y log ) in( y log r) in( y log ) co( y log r) co( y log ) in( y log r) in( y log ) = coy log r Uing thi, (,y) = r co y log r (.) The firt few line are a follow

Then, we attach the parallel accelerator ( See " Convergence Acceleration of Multiple Serie " (A la carte)

8 ,y = ( ) co ylog - ( ) coylog ( ) coylog - ( ) coylog ( ) coylog - ( ) coylog ( ) coylog - ( ) coylog ( ) coylog When both ide of (.) are illutrated, it i a follow. The right figure i drawn with the upper limit of a each 00, but it doe not coincide much with the left figure near =0. Then, we attach the parallel accelerator ( See " Convergence Acceleration of Multiple Serie " (A la carte) ) to the right ide of (.). f r (,y,q ) = k= k k q k-r- k ( q ) k r coy log r r When thi i illutrated at q =/ and m=0, it i a follow. (m i the upper limit of. Same a below.) Both ide overlap eactly and look like pot. (.') - 8 -

9 Incidently, when (.') i calculated at =-, y =0, q =/, m=0, it i a follow. Since =-, y =0 i a trivial zero of (,y ), thi mean that the right ide of (.) i analytically continued in the negative direction beyond the line of convergence =0. Uing Formula.., a hypothei equivalent to the Riemann hypothei can alo be decribed a follow. Hypothei.. When (,y) (,y) = Remark i the Dirichlet eta function on the comple plane, the following inequality hold. From the definition of the abolute value, r co y log r > 0 for 0 < < / (.) (,y) 0 i obviou. It i the Riemann hypothei that the equality hold only when =/. If (.) hold, there are no zero on both ide of =/, o, (.) i equivalent to the Riemann hypothei. Uing (.'), when q =/, m=0, the figure of ection in =0, /, / are drawn a follow. Thi i eactly the ame a the figure in the previou ection. Thi figure i certainly drawn a a hypothei. however, it ha to be proved analytically. I worked on the proof for year, but I wa defeated adly

10 . Partial Derivative of Squared Abolute Value Abolute value of Dirichlet eta function (,y ) i a dicontinuou function with repect to and y. But the quare (,y) i a continuou function with repect to and y. That i, (,y) i differentiable with repect to and y. Uing the erie in the previou ection and differential method, the hypothee mentioned in Section can be decribed a follow. Hypothei.. r log co y log r > 0 for 0 < < / y (.) Remark (,y) i monotonically decreaing with repect to, if the partial derivative i negative within a certain domain. That i, ( ),y = - r Revering thi ign, we obtain the deired inequality. log co y log r < 0 for 0 < < / y The proof of (.) i more difficult than the proof of (.) in the previou ection. So, I recently came up with the net inequality. Hypothei.. In 0 < < /, y 7, the following inequality hold. ( ) - r log( ) r co y log r > ( ) - r co y log r (.) Well, although the hypothei i preented, i thi really true? So, let u verify thi graphically. Since honet drawing i not obtained by (.), we accelerate thi. Since it take time in the parallel acceleration method, the erie acceleration method i adopted thi time. At firt, we convert both ide to half double erie. Then, replacing r with r -, Right: f(,y ) = Left: g(,y ) = r ( ) - r r ( ) - r ( r-) co y log r- log( ) ( r-) r- co y log r- Applying Knopp tranformation ( See " 0 Convergence Acceleration & Summation Method " (A la carte) ) to thee, Right: f(,y,q ) = k= Left: g(,y,q ) = k= k r k r q k-r ( q ) k q k-r ( q ) k k ( ) r k ( ) r - r ( r -) co y log r - - r log( r -) ( r -) co y log r - When both ide are illutrated together at q =/ and m=0, it i a follow

Furthermore, the figure of ection in =/, / are drawn a follow. The left i =/ and the right i =/. From the right figure, the lower limit of y at which thi hypothei hold eem to be eactly 6.78.

11 Furthermore, the figure of ection in =/, / are drawn a follow. The left i =/ and the right i =/. From the right figure, the lower limit of y at which thi hypothei hold eem to be eactly A far a thee figure are oberved, the hypothei eem to hold. The quetion i whether thi can be proved analytically. Hypothei.. trie to prov e Hypothei.. with (,y) 0 a a tepping tone. Roughly peaking, the difference between both ide of (.) i only amplitude. If o, the proof of thi hypothei eem to be a bit eaier than the proof of the previou two hypothee. Can omehow prove it? - -

12 .5 Appendi Formula.. wa a follow.,y = ( ) co ylog - ( ) coylog ( ) coylog - ( ) coylog ( ) coylog - ( ) coylog ( ) coylog - ( ) coylog ( ) coylog Interetingly, at the zero of, each of thee row have to be all 0. Below, thi i decribed a a theorem. Theorem.5. When (,y) i Dirichlet Eta Function, if (,y ) =0, Proof Let r r co y log r be the r th row of thi double erie. Then, r = = r co y log r r r = 0 for,,, (5.) = r r co( y log - y log r) co y log r Here, co( y log -y log r ) = co( y log r) co( y log ) in( y log r) in( y log ) Uing thi, i.e. r = r = At the zero of, Therefore, r r r ( ) r - co( y log r) co( y log ) in( y log r) in( y log ) ( ) - r co( y logr) co( y log ) in( y logr) in( y log) co( y log ) = 0, = 0 for r =,,,. in( y log ) = 0 From thi, the following corollary follow. Corollary.5.' When (,y) i Dirichlet Eta Function, if (,y ) =0, - -

13 i.e. co y log r co ylog co ylog co ylog co = 0 for,,, (5.') ylog co ylog co ylog co ylog co ylog co ylog - co ylog - = 0 - co ylog - = 0 - co ylog - = 0 Putting = y log r in thi corollary, we obtain the following. Corollary.5." When (,y) i Dirichlet Eta Function, if (,y ) =0, co( y log - ) = 0 for real number (5.") Since the convergence of thi left ide i low and it i difficult to obtain an accurate value, we apply Knopp tranformation ( See " 0 Convergence Acceleration & Summation Method " (A la carte) ) to thi a follow. k q k- h(, y,, q ) = k= ( q ) k k The left figure i drawn at q =/, m=5, y =, and the right figure i drawn at q =/, m=5, y =.7. co( y log -) In the left figure there i no contour line to be the zero of (,y) =/. anywhere, but in the right figure it eit at Net, The left figure i drawn at q =/, m=5, =0.6, and the right figure i drawn at q =/, m=5, =

14 In the left figure there i no contour line to be the zero of (,y) anywhere, but in the right figure it eit at y =.7. In the firt place, It i wonder that traight contour line appear in uch a twited figure even if it i irregular Alien' Mathematic Kano Kono - -

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

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