Σk=1. g r 3/2 z. 2 3-z. g 3 ( 3/2 ) g r 2. = 1 r = 0. () z = ( a ) + Σ. c n () a = ( a) 3-z -a. 3-z. z - + Σ. z 3, 5, 7, z ! = !

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1 09 Maclauin Seies of Completed Riemann Zeta 9. Maclauin Seies of Lemma 9.. ( Maclauin seies of gamma function ) When is the gamma function, n is the polygamma function and B n,kf, f, ae Bell polynomials, Whee, g = () =0 = + g /! g ( / ) +! g ( / ) +!,, 7, g ( / ) +! = = 0 B,k 0,,, =,,, k= (.g) Poof Fomula.. in " Seies Expansion of Gamma Function & the Recipocal " (A la Cate) was as follows. Whenis gamma function, n Whee, () = ( a ) + n = c n () a = ( a) k= is polygamma function and B n,kf, f, is Bell polynomial, c n () a ( a) n a 0,,,, n! n Bn,k 0 () a, () a,, n () a Using this, / can be expanded in Taylo seies as follows. () a = ( a) k= Putting a =/, () Hee, putting Then, g () () a = ( a ) + =! B,k = + = = k= a 0 () a, () a,, () a B,k /! 0, n=,,, a,,7,,, =,,, (.w) =,,, = = 0 B,k 0,,, =,,, k= = g =0,,,

2 Substituting this fo (.w), i.e. = + = =0( ) = g () /! g /! Multiplying both sides by /, we obtain the desied expession. = Both sides of (.g) ae dawn as follows. The left side is blue and the ight side is oange. Thee ae singula points at =,,7,, so the convegence adius is. Lemma 9.. ( Maclauin seies of holomophied Riemann eta ) When is the Riemann eta function and is Stieltjes constant, the following expession holds on the whole complex plane. ( ) = c =0 (.) Whee, 0 = 4 0!!!! c = = 0 =,,,!

3 Poof When is Stieltjes constant, it is known that the following expession holds on the whole complex plane except =. () = + =0 Multiplying both sides by, ( ) () = + =0 Replacing with, i.e. () ( )! ( ) ( ) = + () = ( ) = = () ( ) + = + () ( )! = ( )! = + = ( ) ( )! ( ) ( )! ( )!! In this expession, howeve may be defined, cannot be included into. So, defining a new coefficient c including the denominato as in the poviso of the lemma, we obtain the desied expessions. Q.E.D. Both sides of (.) ae dawn as follows. The left side is blue and the ight side is oange. Both sides ae exactly ovelapped and the left side (blue) is invisible. Thee is no singulaity, so the convegence adius is.

4 Theoem 9.. ( Maclauin seies of ) Let completed Riemann eta be () = ( ) Then, the following expession holds on the whole complex plane. () = =0 s () (.) log s s s! () st g st ( /) ct (.) st s t! Whee, n is the polygamma function, B n,kf, f, is Bell polynomials, is Stieltjes constant, g = = 0 B,k 0,,, =,,, k= c = = 0 =,,,! Poof i.e. () = ( ) () = ( ) = ( ) () = = As fist, Maclauin seies of / is = =0 log! ( ) ( ) ( ) (.') Fom Lemma 9.. and Lemma 9.., the two subsequent functions ae as follows. ( ) = c =0 = () =0 g /! (.p) (.g) (.) Whee, g = = 0 B,k 0,,, =,,, k= 4

5 c = = 0 =,,,! Substituting these fo (.'), () = =0 log! () =0 g /! c =0 Accoding to Fomula.. in " 0 Powe of Infinite Seies " ( Infinitedegee Equation ), the poduct of the powe seies is expessed by the following equation. a b =0 =0 =0 c = =0 s a s b st c t So, putting log a =!, b = () g ( /)! we obtain Whee, () = =0 g s log s s s! () st g st ( /) ct (.) st s t! = = 0 B,k 0,,, =,,, k= c = = 0 =,,,! Q.E.D. The fist few of (.) ae () = + log g ( /) 0!! 0! + log g ( /) +!!! log g ( / ) g ( /) +!!! + log g ( /)!!! 0 log 0 0!! 0! log! log +! g ( /) log!! g ( /) log!! 0 0! +! g ( /)! g ( /)! 0 0!!

6 + log +! g ( /)! 0 0! Both sides of (.) ae dawn as follows. The left side is blue and the ight side is oange. Although the ight side is calculated up to 4, both sides ae exactly ovelapped and the left side (blue) is invisible. The singula points of the gamma function (.g) ae offset by the tivial eos of Riemann eta (.) and ae disappeaed. so the convegence adius is. Howeve, vey high calculation pecision is equied at a point fa fom the oigin. The following is the function value at the fist eo point / + i 4.47 of. It is calculated up to 6 at 64 digit pecision and is baely matched up to the 7th decimal place. 6

7 9. Maclauin Seies of Lemma 9.. ( Maclauin seies of gamma function ) When is the gamma function, n is the polygamma function and B n,kf, f, ae Bell polynomials, Whee, + 4 g 4 = 4 =0 g /4!, 9,, = = 0 B,k 0 4, 4,, 4 =,,, k= (.g) Poof Fomula.. in " Seies Expansion of Gamma Function & the Recipocal " (A la Cate) was as follows. When is gamma function, n Whee, () = ( a ) + n = c n () a = ( a) k= is polygamma function and B n,kf, f, is Bell polynomial, c n () a ( a) n a 0,,,, n! n Bn,k 0 () a, () a,, n () a n=,,, Using this, /4+ / can be expanded in Taylo seies as follows. + 4 () a = ( a) k= () a = ( a ) + =! B,k + a 4 0 () a, () a,, () a a, 9,, =,,, Putting a = /4, + 4 = 4 = 4 4 () Hee, putting Then, g 4 () 4 + = B,k k= ( /4)! 4 0, 4,, 4 =,,, = = 0 B,k 0 4, 4,, 4 =,,, k= Substituting this fo (.w), + 4 = 4 g =0,,, = 4 + = 4 g ( /4)! = 4 =0 g ( /4)! (.w) Thee ae singula points at = /,9/,/,, so the convegence adius is /. 7

8 Lemma 9.. ( Maclauin seies of holomophied Riemann eta ) When is the Riemann eta function and is Stieltjes constant, the following expession holds on the whole complex plane. Whee, Poof + c = = =0c = 0 ( / ) () s= Let holomophied Riemann eta be s s! s ( ) () = + () ( ) = ( )! Replacing with +/, + = + = ( ) s =,,, (.) ( )! (.t) 0 = + 0!! +! 4! This is a Taylo expansion aound =/. 4 + Since the function is holomophied, thee is no singulaity on the whole complex plane. So the convegence adius of the Taylo seie is. Hee, fom the binomial theoem, = Substituting this fo (.t), i.e. + + () s s = =0 = =0 ( ) = =0 s s () s s= ( ) s= ( )! This has to be equal to the following Maclauin seies. + = =0 a () s s s () s ( s )! s s ( s )! s s s s s Because if it is not so, it contadicts the uniqueness of the powe seies. Theefoe, a = s= ( ) s ( s )! s s (.m) 8

9 Fom (.t), the st tem /()! of a 0 has to be ead as. That is, a = s= () s= Futhemoe, fom (.m), Next, a 0 = s= s ( s )! Doubling both sides, + + Then, putting we obtain a c = ( / ) = + s s! s s! = =0 s = 0 s s = a = a 0 =0 = =0 s =,,, a = a 0 a ( / ) =0 a = 0 ( / ) () s= = =0c s s! s s =,,, Thee is no singulaity, so the convegence adius is. Theoem 9.. ( Maclauin Seies of ) Let completed Riemann eta be () = + Then, the following expession holds on the whole complex plane. 0 =0 () = ( ) s ( ) ( 0 ) = 4 / (.) s log s g st ( /4) c s ( s )! st t (.) ( st )! = Whee, n is the polygamma function, B n,kf, f, is Bell polynomials, is Stieltjes constant, g 4 = = 0 B,k 0 4, 4,, 4 =,,, k= 9

10 c = = 0 ( / ) () s= s s! s s =,,, Poof Let holomophied Riemann Zeta be () = + This can be tansfomed as follows. () = The st function can be expanded in Maclauin seies as follows. + = /4 =0( ) log! Fom Lemma 9.. and Lemma 9.., the two subsequent functions ae as follows. Whee, + 4 = 4 =0 + g 4 c = g /4! = =0c + (.) + (.'), 9,, = = 0 B,k 0 4, 4,, 4 =,,, k= = 0 ( / ) Substitutin these fo (.'), () = /4 () s= 4 s s! s () =0 s =,,, log! =0 g /4! c =0 Accoding to Fomula.. in " 0 Powe of Infinite Seies " ( Infinitedegee Equation ), the poduct of the powe seies is expessed by the following equation. a b =0 =0 =0 So, putting a = () =0 ( 0 ) = 4 /4 log! c = =0, b = =0 4 s a s b st c t g /4! = (.p) (.g) (.) 0

11 we obtain Whee, () = ( ) g 4 c = 0 =0 s log s s ( s )! s ( ) g st /4 st ( st )! c t (.) = = 0 B,k 0 4, 4,, 4 =,,, k= = 0 ( / ) () s= s s! s s =,,, Q.E.D. The fist few of (.) ae () = ( ) 0 + log g ( /4 ) + + c!! + log g + ( /4 ) log g ( + c /4 ) g + ( /4) c!!!!!! log c log g ( /4 ) log g ( /4 ) log g ( /4) c + + c + c!!!!!! + log! g ( /4 ) log g ( /4) log c + c!!!!! g ( /4 ) c We can see that the coefficients of the odd degee ae almost eo. Both sides of (.) ae dawn as follows. The left side is blue and the ight side is oange. Although the ight side is calculated up to 4, both sides ae ovelapped and the left side (blue) is almost invisible. The singula points of the gamma function (.g) ae offset by the tivial eos of Riemann eta (.) and ae disappeaed. so the convegence adius is.

12 Howeve, vey high calculation pecision is equied at a point fa fom the oigin. The following is the function value at the fist eo point i 4.47 of. It is calculated up to 6 at 6 digit pecision and is baely matched up to the 4th decimal place.

13 9. Appendix 9..0 Highe ode diffeential coefficient of Fomula 6..h in " 6 Highe and Supe Calculus of Zeta Function etc " ( SupeCalculus ) was as follows. When is Riemann eta function, () n is the lineal n th ode deivative and is Stieltjes constant, the following expession holds on the whole complex plane except =. n () = () n n! + () s ( ) ( ) n+ s +sn sn Given =a in this equation, the n th ode diffeential coefficient at a is obtained. That is n () n n! () a = ( a ) n+ + () s ( a ) sn s +sn (.a) () Highe ode diffeential coefficient at 0 In paticula, puttig a =0 in (.a), the n th ode diffeential coefficient at 0 is obtained. That is n () 0 = () n s sn! n! n =0,,, (.0) The fist few ae as follows. We can see that all the diffeential coefficients ae negative and the diffeential coefficient of the fist o highe ode is close to n!. () 0 () 0 = 0. () () 0 = ! () () 0 = ! () () 0 = ! () 4 () 0 = ! () () 0 = ! cf. (.0) is equivalent to the following equation. n () 0 = () n s+n n! n =0,,, s! Fom this, fomula of O.Maichev ( ) is obtained. That is s+n = () n n!+ n () 0 n =0,,, s! () Highe ode diffeential coefficient at / In paticula, when a =/ in (.a), the n th ode diffeential coefficient at / is obtained as follows.

14 i.e. n n ( ) () n n! = ( /) n+ + () s s ( /) sn +sn () n n! = (/) n+ + () s (/) sn s +sn n n! = () n () n+ + s n () s () sn s sn! / = n () n s s sn! n! (.h) The fist few ae as follows. We can see that all the diffeential coefficients ae negative and the diffeential coefficient of the fist o highe ode is close to n+ n!. () 0 ( / ) = () ( / ) = ! () ( / ) = ! () ( / ) = ! () 4 ( / ) = ! () ( / ) = ! 9.. Maclauin Seies of Riemann Zeta Using the highe ode diffeential coefficiets, the Maclauin seies of Riemann eta can be easily obtained. Fomula 9.. ( Maclauin seies of ) Whee, () 0 () = < (.) =0! () 0 = () s s!! =0,,, The fist few tems of (.) ae as follows. Because the absolute values of the numeato and denominato ae vey close, the coefficients of the th and highe ode ae almost. Moeove, D figue and D figue of the eal pat of (.) ae as follows. The left side is blue and the ight side 4

15 is oange. This seies (oange) can epesent only the inside of the cicle of D figue. Theefoe, this seies can not epesent the eos of (). This is the eason I was hesitating the Maclauin expansion of () until now. Fomula 9..' ( Maclauin seies of +/ ) Whee, + = =0 / <! / = () s s s!! =0,,, (.') The fist few tems of (.') ae as follows. Since the numeato is close to n+ n! and the denominato is n!, the coefficients of the th and highe degee ae almost n+. In addition, the convegence egion of the seies of (.') is within a cicle with a adius of / centeed on the oigin of the complex plane. Theefoe, this seies also can not epesent the eos of +/. 9.. Maclauin Seies of Holomophied Riemann Zeta These fomulas ae almost useless because the convegence ange is naow. The cause is the existence of a singula point. Because, the convegence adius of Maclauin seies is the distance fom the oigin to the singula point. If so, if the singulaity is emoved fom (), the convegence adius of the seies must be infinite. To do, and multiply +/ by /. so, we may simply multiply () by

16 Fomula 9.. ( Maclauin seies of () ) Whee, () = + = () 0 = () s s! () 0 () 0!! =0,,, (.) Poof Fom Fomula 9.. (.), () 0 () () = + () 0 =0! =0! = = = + = () ( )! 0 () 0 () 0 0 0! + () 0 =! () 0 () 0! (.) D figue and D figue of the eal pat of (.) ae as follows. The left side is blue and the ight side is oange. No convegence cicle can be found in both figues. It seems that (.) holds on the whole complex plane. In a simila way, the following is obtain fom (.'). This also holds on the whole complex plane. Fomula 9..' ( Maclauin seies of /+/ ) Whee, h = / + = + = () s s s! h h!! =0,,, (.') 6

17 9.. Maclauin Seies of Completed Riemann Zeta In a simila way to 9. and 9., the followings ae obtained using (.) and (.'). Theoem 9.. ( Maclauin seies of ) Let completed Riemann eta be () = ( ) () Then, the following expession holds on the whole complex plane. () = =0 s log s s s! s ( ) ( st ) () st s t! t ( t ) 0 () 0 t Whee, n is the polygamma function, B n,kf, f, is Bell polynomials, is Stieltjes constant, () = = 0 B,k 0 (), (),, () =,,, 0 = k= () 0 = s s!! = 0,,, t! cf. Theoem 9.. was as follows. Whee, () = =0 s log s s s! c = = 0 =,,,! This expession is a Maclauin expansion of () = ( ) = () st g st ( /) ct st s t! ( ) This is a ea attack. The weak point is a little had to ovelook the whole. The stoong point is that the diffeential coefficient of is obtained fom only one s. Theefoe, although high pecision is equied, it is possible to calculate these eos even on notebook computes. On the contay, Theoem 9.. is Maclauin expansion of () = + ( ) () This is a fontal attack. The stong point is easy to ovelook the whole. The weak point is that s has to be accumulated infinitely to obtain the diffeential coefficient of. Fo this pupose, high pecision and a lage calculation amount ae equied. Theefoe, it is almost impossible to calculate these eos on note 7

18 book computes accoding to Theoem 9... Theoem 9..' ( Maclauin seies of ) Let completed Riemann eta be () = + Then, the following expession holds on the whole complex plane. 0 =0 () = ( ) ( 0 ) = s log s s s! s ( ) 4 4 / g st h t th t st s t! t! = Whee, n is the polygamma function, B n,kf, f, is Bell polynomials, is Stieltjes constant, g 4 h = = = 0 B,k 0 4, 4,, 4 =,,, k= ( / ) 0 = () s s s!! = 0,,, cf. Theoem 9.. was as follows. Whee, 0 =0 () = ( ) c = s log s s ( s )! s ( ) g st /4 st ( st )! c t = 0 ( / ) () s= s s! s s =,,, This theoem and Theoem 9..' ae almost the same. Both ae othodox appoaches. The diffeence is the the fomula fo the diffeential coefficient of holomophied Riemann eta. Compaing c and h, we see that the fome is half of the latte at the amount of calculation. This diffeence is not small. When calculating eos by using a notebook compute, the calculation by Thoem 9.. is somehow possible, but the calculation by Thoem 9..' is almost impossible Alien's Mathematics Kano Kono 8

Solution to HW 3, Ma 1a Fall 2016

Solution to HW 3, Ma 1a Fall 2016 Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.