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