10 Dirichlet Series & Taylor Series

Abstract (1) Dirichlet series can be converted to Taylor series within its convergence area. (2) If the coefficients of the Dirichlet series and the center of the Taylor expansion are limited to real numbers, the Taylor series for each real part and imaginary part can be obtained. (3) A finite or infinite Dirichlet series cannot represent a polynomial ( more than one term ) .

10.1 General Dirichlet Series & Taylor Series

Definition 10.1.0 (General Dirichlet Series)

Let R be a real number set. And let x , y  R and n  R , n < n+1 n =1,2,3, .

When z =x +iy and an are complex numbers, we call the following series General Dirichlet Series.  - t z f()z = Σat e t=1

General Dirichlet series can be converted to Taylor series within its convergence area. The following shows the conversion formula.

Formula 10.1.1 When a function fz that is holomorphic on domain D is expanded into a general Dirichlet series, the following expression holds for arbitrary complex numbers z,c belonging to the convergence area.

   s - t z -c t s ()z-c f()z = Σat e = ΣΣat e -t (1.1) t=1 s=0 t=1 s!

Proof  - t z f()z = Σat e t=1

- t z Expanding e into Taylor series with respect to z around c ,

 s - t z -c t s ()z-c e = Σe -t s=0 s! Substituting this for the right-hand side and swapping  , we obtain the desired expression.

Example 1 coth z - 1 coth z is expanded into Fourier series for Rez >0 as follows. ez + e-z 1+ e-2z coth z = = = 1+ e-2z 1 + e-2z + e-4z + e-6z + ez - e-z 1- e-2z -2z -4z -6z -8z = 1 + 2e + 2e + 2e + 2e +- = 1 + 2cos2iz + cos4iz + cos6iz + cos8iz +  + 2isin 2iz + sin 4iz + sin 6iz + sin 8iz + 

- 1 - From this,  coth z -1 = Σ2 e-2t z Re()z > 0 t=1

Put at = 2 , t = 2t in the right hand. Then t < t+1 This is a general Dirichlet series. So, from the formula,    ()z-c s coth z -1 = Σ2 e-2t z =ΣΣ2e-2t c()-2t s t=1 s=0 t=1 s! The function values of the three parties at z =2+1.5i are calculated as follows. The center is the general Dirichlet series and the right end is the Taylor series at c =1+i. All three are exactly the same.

In a similar way, the following examples are obtained.

Example 2 tanh z - 1    ()z-c s tanh z -1 = Σ2()-1 t e-2t z = ΣΣ2()-1 te-2t c()-2t s t=1 s=0 t=1 s!

Example 3 csch z s    s ()z-c csch z = Σ2e-2t-1 z = ΣΣ2e-c 2t-1 -()2t-1 t=1 s=0 t=1 s!

Example 4 sech z s    s ()z-c sech z = Σ2()-1 t-1e-2t-1 z = ΣΣ2()-1 t-1e-c 2t-1 -()2t-1 t=1 s=0 t=1 s!

If coefficients of the Dirichlet series at and the center c of the Taylor expansion are limited to real numbers, the Taylor series for each real part and imaginary part can be obtained.

Formula 10.1.2  - t z When f()z =Σat e () z = x+iy is general Dirichlet serie, u ,v are real and imaginary parts t=1 of fz and c , at t=1,2,3, are arbitrary real numbers, the following expressions hold in the convergence area.   s -c t s ()z-c f()z = ΣΣat e -t s=0 t=1 s!    s r 2r -c t 2r+s ()x-c ()-1 y u()x,y = ΣΣΣat e -t r=0 s=0 t=1 s! ()2r !    s r 2r+1 -c t 2r+s+1 ()x-c ()-1 y v()x,y = ΣΣΣat e -t r=0 s=0 t=1 s! ()2r+1 !

- 2 - 0 Where, 0 = 1

Proof  - t z f()z = Σat e t=1   - t x +iy - t x f()x,y = Σat e = Σat e cost y - i cost y t=1 t=1 i.e.   - t x - t x f()x,y = Σat e cost y - iΣat e sint y t=1 t=1 From this,  - t x u()x,y = Σat e cost y t=1  - t x v()x,y = -Σat e sint y t=1

Expanding coss y , sins y into Maclaurin series with respect to y respectively,  r 2r 2r ()-1 y cost y = Σt r=0 ()2r !  r 2r+1 2r+1 ()-1 y sins y = Σt r=0 ()2r+1 !

-t x Next, expanding e into Taylor series with respect to x around real number c ,

 s -t x -c t s s ()x-c e = Σe ()-1 t s=0 s! Substituting these for the above,   s  r 2r -c t s s ()x-c 2r ()-1 y u()x,y = Σat Σe ()-1 t Σt t=1 s=0 s! r=0 ()2r !   s  r 2r+1 -c t s s ()x-c 2r+1 ()-1 y v()x,y = -Σat Σe ()-1 t Σt t=1 s=0 s! r=0 ()2r+1 ! Rearranging  ,    s r 2r -c t s 2r+s ()x-c ()-1 y u()x,y = ΣΣΣat e ()-1 t r=0 s=0 t=1 s! ()2r !

   s r 2r+1 -c t -ct s+1 2r+s+1 ()x-c ()-1 y v()x,y = ΣΣΣat e e ()-1 t r=0 s=0 t=1 s! ()2r+1 ! Here, let  -c t s 2r+s 2r+s Σat e ()-1 t := f ()c t=1  -c t -ct s+1 2r+s+1 2r+s+1 Σat e e ()-1 t := f ()c t=1 Then,  -c t s s s Σat e ()-1 t := f ()c t=1

- 3 - Therefore, according to Taylor's theorem

 s   s ()s ()z-c -c t s s ()z-c f()z = Σf c = ΣΣat e ()-1 t s=0 s ! s=0 t=1 s!   s -c t s ()z-c = ΣΣat e -t s=0 t=1 s! And

   s r 2r -c t s 2r+s ()x-c ()-1 y u()x,y = ΣΣΣat e ()-1 t r=0 s=0 t=1 s! ()2r !    s r 2r -c t 2r+s ()x-c ()-1 y = ΣΣΣat e -t r=0 s=0 t=1 s! ()2r !    s r 2r+1 -c t s+1 2r+s+1 ()x-c ()-1 y v()x,y = ΣΣΣat e ()-1 t r=0 s=0 t=1 s! ()2r+1 !

   s r 2r+1 -c t 2r+s+1 ()x-c ()-1 y = ΣΣΣat e -t r=0 s=0 t=1 s! ()2r+1 !

t-1 t Example at = ()-1 , t = 1.1 General Dirichlet series is

 t f()z = Σ()-1 t-1 e- 1.1 z t=1 Taylor series are s   t s ()z-c f()z = ΣΣ()-1 t-1 e- c 1.1 -1.1t s=0 t=1 s! s r 2r    t 2r+s ()x-c ()-1 y u()x,y = ΣΣΣ()-1 t-1 e- c 1.1 -1.1 t r=0 s=0 t=1 s! ()2r ! s r 2r+1    t 2r+s+1 ()x-c ()-1 y v()x,y = ΣΣΣ()-1 t-1 e- c 1.1 -1.1t r=0 s=0 t=1 s! ()2r+1 !

When c=5 , Dirichlet series and Taylor series are drawn in the 2D figure as follows. Orange is Dirichlet series and blue is Taylor series. The red dot is the center of the expansion. Although the abscissa of convergence is observed at x =0 , both are exactly overlapped and Dirichlet series (orange) is almost invisible.

- 4 - When c=5 , the real part and the imaginary part are drawn in the 3D figure as follows. The left is the real part and the right is the imaginary part. In both figures, orange is Dirichlet series and blue is Taylor series.

The convergence area of Taylor series is a square inscribed in a circle with a radius of 5 . And the right outside is asymptotic expansion.

Further, the following theorem is derived from Formula 10.1.1 .

Theorem 10.1.3 General Dirichlet series cannot represent finite Taylor series with more than one term ( i.e. polynomial ) .

Proof From Formula 10.1.1 ,    s - t z -c t s ()z-c f()z = Σat e = ΣΣat e -t (1.1) t=1 s=0 t=1 s! Now, assume that the general Dirichlet series is equal to the finite Taylor series with more than one term. Then, there exists a natural number m that satisfies the following equation.

 m  s - t z -c t s ()z-c f()z = Σat e = ΣΣat e -t (1.m) t=1 s=0 t=1 s! In order for this to hold  s -c t s f ()c = Σat e -t = 0 s = m+1, m+2, m+3, (1.s') t=1

-c t Here, e  0 . And, since t < t+1 t=1,2,3, , there is at most one t such that t =0.

Let it be t =k i.e. k =0 . Then, in order for (1.s') to hold, at = 0 for t  k . Then,  - t z - k z Σat e = ak e = ak t=1 m  s m s -c t s ()z-c -c k s ()z-c 0 ΣΣat e -t = Σak e -k = ak  0 =1 s=0 t=1 s! s=0 s! Thus, (1.m) becomes

f()z = ak = ak This is contrary to the assumption. Q.E.D.

- 5 - 10.2 Finite General Dirichlet Series & Taylor Series

The formulas and theorem in the previous section also hold for the finite general Dirichlet series.

Formula 10.2.1 When a function fz that is holomorphic on domain D is expanded into a finite general Dirichlet series, the following expression holds for arbitrary complex numbers z,c

n  n s - t z -c t s ()z-c f()z = Σat e = ΣΣat e -t (2.1) t=1 s=0 t=1 s!

t-1 Example at = 2()-1 , t = 2t -1 s n  n s ()z-c f()z = Σ2()-1 t-1e-2t-1 z = ΣΣ2()-1 t-1e-c 2t-1 -()2t-1 t=1 s=0 t=1 s! When n =4, c =2+3i , finite Dirichlet series and Taylor series are drawn in the 2D figure as follows. Orange is finite Dirichlet series and blue is Taylor series. Both are exactly overlapped and finite Dirichlet series (orange) can not be seen. In addition, since an abscissa of convergence does not exist in the former, a circle of convergence does not exist in the latter also.

Formula 10.2.2 n - t z When f()z =Σat e () z = x+iy is finite general Dirichlet serie, u ,v are real and imaginary t=1 parts of fz and c , at t=1,2,3, are arbitrary real numbers, the following expressions hold.  n s -c t s ()z-c f()z = ΣΣat e -t s=0 t=1 s!   n s r 2r -c t 2r+s ()x-c ()-1 y u()x,y = ΣΣΣat e -t r=0 s=0 t=1 s! ()2r !   n s r 2r+1 -c t 2r+s+1 ()x-c ()-1 y v()x,y = ΣΣΣat e -t r=0 s=0 t=1 s! ()2r+1 ! 0 Where, 0 = 1

- 6 - t-1 t Example at = ()-1 , t = 1.1 Finite general Dirichlet series is

n t f()z =Σ()-1 t-1 e-1.1 z t=1 Taylor series are s  n t s ()z-c f()z = ΣΣ()-1 t-1 e- c 1.1 -1.1t s=0 t=1 s! s r 2r   n t 2r+s ()x-c ()-1 y u()x,y = ΣΣΣ()-1 t-1 e- c 1.1 -1.1 t r=0 s=0 t=1 s! ()2r ! s r 2r+1   n t 2r+s+1 ()x-c ()-1 y v()x,y = ΣΣΣ()-1 t-1 e- c 1.1 -1.1t r=0 s=0 t=1 s! ()2r+1 ! When n =4 , c=5 , these are drawn in the 3D figure as follows. The left is the real par and the right is the imaginary part. In both figures, orange is Dirichlet series and blue is Taylor series. The red dot is the center of the expansion.

The following theorem is obtained as in the previous section.

Theorem 10.2.3 Finite general Dirichlet series cannot represent finite Taylor series with more than one term ( i.e. polynomial ) .

- 7 - 10.3 Ordinary Dirichlet Series & Taylor Series

Ordinary Dirichlet Series is what is put t = log t at General Dirichlet Series, and is defined as follows.

Definition 10.3.0 (Ordinary Dirichlet Series)

When z,an n =1,2,3, are complex numbers, we call the following Ordinary Dirichlet Series.  a a a a a t 1 2 3 4  f()z = Σ z = z + z + z + z + t=1 t 1 2 3 4

Ordinary Dirichlet series can be converted to Taylor series within its convergence area.

Tthe conversion formula is obtained immediately by putting t = log t in Formula 10.1.1 ..

Formula 10.3.1 When a function fz that is holomorphic on domain D is expanded into a ordinary Dirichlet series, the following expression holds for arbitrary complex numbers z,c belonging to the convergence area.

   s at at s ()z-c (3.1) f()z =Σ z = ΣΣ c ()-log t t=1 t s=0 t=1 t s!

Example Dirichlet Eta Series Dirichlet eta series is  ()-1 t  ()z = Σ z t=1 t t Since at = ()-1 , according to the formula, s  ()-1 t   ()-1 t ()z-c  s ()z = Σ z = ΣΣ c ()-log t t=1 t s=0 t=1 t s! When c =1+i , the function values of Dirichlet series and Taylor series at z =3+4i are as follows. Both are almost equal.

If coefficients of the Dirichlet series at and the center c of the Taylor expansion are limited to real numbers,

Taylor series for each real part and imaginary part can be obtained. This is also obtained by putting t = log t in Formula 10.1.2 ..

Formula 10.3.2  z When fz =Σat /t  z = x +iy is ordinary Dirichlet serie, u ,v are real and imaginary parts t=1 of fz and c , at t =1,2,3, are arbitrary real numbers, the following expressions hold in the convergence area.

- 8 -   s at s ()z-c f()z = ΣΣ c ()-log t s=0 t=1 t s!

   s r 2r at 2r+s ()x-c ()-1 y u()x,y = ΣΣΣ c ()-log t r=0 s=0 t=1 t s! ()2r ! s r 2r+1    at ()x-c ()-1 y 2r+s+1 v()x,y = ΣΣΣ c ()-log t r=0 s=0 t=1 t s! ()2r+1 ! 0 Where, 0 = 1

Example Dirichlet Beta Series When Dirichlet character is m, j, t , Dirichlet Beta Series is expressed  ()4,2,t  ()-1 t-1  ()z =Σ z = Σ z t=1 t t=1 ()2t-1

Since at = 4,2,t , according to the formula, s   ()4,2,t ()x-c  s ()z = ΣΣ c ()-log t s=0 t=1 t s!

   s r 2r ()4,2,t 2r+s ()x-c ()-1 y u()x,y = ΣΣΣ c ()-log t r=0 s=0 t=1 t s! ()2r ! s r r    ()4,2,t ()x-c ()-1 y2 +1 2r+s+1 v()x,y = ΣΣΣ c ()-log t r=0 s=0 t=1 t s! ()2r+1 !

Though there is no problem as it is, since 4,2,t t=1,2,3, is 1 , 0 , -1 , 0 ,  , these can be further rewritten as follows.   ()-1 t-1 z-c s ()z = ΣΣ -log2t -1 s s=0 t=1 2t -1 c s !

   t-1 s r 2r ()-1 r s x -c ()-1 y u x,y = -log2t -1 2 + ()ΣΣΣ c r=0 s=0 t=1 2t -1 s! ()2r !

   t-1 s r 2r+1 ()-1 r s x -c ()-1 y v x,y = -log2t -1 2 + +1 ()ΣΣΣ c r=0 s=0 t=1 2t -1 s! ()2r+1 ! 0 Where, 0 = 1

When c=0.5 , Dirichlet series and Taylor series are drawn in the 2D figure on the next page. Orange is the Dirichletseries and blue is the Taylor series. The red dot is the center of the expansion. Although abscissa of convergence is observed at x =0 , both are exactly overlapped and Dirichlet series (orange) is almost invisible.

- 9 - Further, when c = 0.5 , the function values of Dirichlet series and Taylor series at z =2+3i are as follows. Both are exactly equal.

Further, the following theorem is derived from Formula 10.3.1 .

Theorem 10.3.3 Ordinary Dirichlet series cannot represent finite Taylor series with more than one term ( i.e. polynomial ) .

Proof From Formula 10.3.1 ,

   s at at s ()z-c (3.1) f()z = Σ z = ΣΣ c ()-log t t=1 t s=0 t=1 t s! Now, assume that the ordinary Dirichlet series is equal to the finite Taylor series with more than one term. Then, there exists a natural number m that satisfies the following equation.

 m  s at at s ()z-c (3.m) f()z = Σ z = ΣΣ c ()-log t t=1 t s=0 t=1 t s! In order for this to hold  s at s  (3.s') f ()c = Σ c ()-log t = 0 s = m+1, m+2, m+3, t=1 t c Here, 1/t  0 . And, -log 1=0 , -log t  0 t=2,3,4, .

Then, in order for (3.s') to hold, at =0 t=2,3,4, . Then,

 at a1 Σ z = z = a1 t=1 t 1 s s m  at ()z-c m a1 ()z-c s s  0 ΣΣ c ()-log t = Σ c ()-log 1 = a1 0 =1 s=0 t=1 t s! s=0 1 s! Thus, (3.m) becomes

f()z = a1 = a1 This is contrary to the assumption. Q.E.D.

- 10 - 10.4 Finite Ordinary Dirichlet Series & Taylor Series The formulas and theorem in the previous section also hold for the finite ordinary Dirichlet series.

Formula 10.4.1 When a function fz that is holomorphic on domain D is expanded into a ordinary Dirichlet series, the following expression holds for arbitrary complex numbers z,c belonging to the convergence area.

n  n s at at s ()z-c (4.1) f()z = Σ z = ΣΣ c ()-log t t=1 t s=0 t=1 t s!

t-1 Example at = ()-1 , n=6

6 t-1  6 t-1 s ()-1 ()-1 s ()z-c f()z = Σ z = ΣΣ c ()-log t t=1 t s=0 t=1 t s!

These are the first 6 terms of the Dirichlet eta series and its Taylor series. When c=1/2+15i , the values near these zeros are calculated as follows. The top two lines are the finite Dirichlet eta series and the bottom two lines are the Taylor series. Both are almost 0 .

Formula 10.4.2 n z When f()z =Σat /t () z = x+iy is finite ordinary Dirichlet serie, u ,v are real and imaginary t=1 parts of fz and c , at t =1,2,3, are arbitrary real numbers, the following expressions hold.

 n s at s ()z-c f()z = ΣΣ c ()-log t s=0 t=1 t s!   n s r 2r at 2r+s ()x-c ()-1 y u()x,y = ΣΣΣ c ()-log t r=0 s=0 t=1 t s! ()2r ! s r 2r+1   n at ()x-c ()-1 y 2r+s+1 v()x,y = ΣΣΣ c ()-log t r=0 s=0 t=1 t s! ()2r+1 !

Where, 00 = 1

t-1 Example at = ()-1 t , n = 2 Finite ordinary Dirichlet series is

2 t-1 ()-1 t 1 2 1-z f()z =Σ z = z - z = 1-2 t=1 t 1 2 Taylor series are

- 11 - t  2 -1 s ()-1 t s ()z-c f()z = ΣΣ c ()-log t s=0 t=1 t s! t   2 -1 s r 2r ()-1 t 2r+s ()x-c ()-1 y u()x,y = ΣΣΣ c ()-log t r=0 s=0 t=1 t s! ()2r ! t-1 s r r   2 ()-1 t ()x-c ()-1 y2 +1 2r+s+1 v()x,y = ΣΣΣ c ()-log t r=0 s=0 t=1 t s! ()2r+1 ! When c=3 , these are drawn in the 3D figure as follows. The left is the real part and the right is the imaginary part. In both figures, orange is finite ordinary Dirichlet series and blue is Taylor series. Red dot is the center of the of the expansion.

This finite ordinary Dirichlet series has zeros on x =1 , In fact, the real and imaginary parts on this line are drawn in the 2D figure as follows. The purple dots are the zeros.

z These zeros can be easily obtained from 1-21- =0 to x=1 , y=2n/log 2 n=0,1,2, . These are the zeros specific to the Dirichlet eta function z . Because, there is the following relationship between Riemann zeta function z and Dirichlet eta function z . 1 2 1 1 1 1 1 1 1 1 - + + + + = - + - +- 1z 2z 1z 2z 3z 4z 1z 2z 3z 4z

- 12 - The following theorem is obtained as in the previous section.

Theorem 10.4.3 Finite ordinary Dirichlet series cannot represent finite Taylor series with more than one term ( i.e. polynomial ) .

2020.10.03 Kano. Kono Alien's Mathematics

- 13 -