01 General Dirichlet Series & Power Series

1.1 Definition & Theorems

Definition 1.1.1 (General Dirichlet Series)

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

When s = +it and an are complex numbers, we call the following series General Dirichlet Series.  -n s Σan e n =1

Note

Especially when n = log n in General Dirichlet Seris, it is called Ordinary Dirichlet Series. This is discussed in another chapter.

Theorem 1.1.2  -n s Let Dirichlet serie f()s =Σan e . And assume f()s is convergent at s =sc =c + tc i . Then n =1  1. f()s converges uniformly in Args -s   < . c 2

2. f()s converges for any s = + t i s.t.  > c .

This c is called the line of convergence. By convention, c = if f()s converges nowhere and

c =- if f()s converges everywhere on the complex plane.

How to calculate c n 1. When Σak is divergent, k=1

loga1 + a2 + + an c = limsup n  n n 2. When Σak is convergent, k=1

logan+1 + an+2 + an+3  c = limsup n  n

Absolute Convergence   -n s -n s A Dirichlet series f()s =Σan e is absolutely convergent if the series Σan e is convergent. n =1 n =1

There exist a a  c such that f()s converges absolutely for  > a and converges non-absolutely for  < a . This a is called the line of absolute convergence.

- 1 - How to calculate a n 1. When Σak is divergent, k=1

loga1 +a2 + +an a = limsup n  n n 2. When Σak is convergent, k=1

logan+1 +an+2 +an+3  a = limsup n  n

Uniform Convergence  -n s As seen in Theorem 1.1.2 , Dirichlet serie f()s =Σan e converges uniformly in a certain domain. n =1 If we see it a little in detail, it is as follows.

There exist a u c  u  a such that f()s converges uniformly for  > u and converges non-uniformly for  < u . This u is called the line of uniform convergence.

How to calculate u

log Tx u = limsup x log x where

-i n y Tx = sup Σ an e ||y <[]x n

Theorem 1.1.3 (Holomorphy)  -n s If Dirichlet serie f()s =Σan e s = + t i converges for  > c , n =1

f()s is holomophic at  > c . And the derivative of f()s is given as follows.  k k k -n s f ()s = ()-1 Σn an e n =1

Theorem 1.1.4 (Uniqueness) Let two Dirichlet series are as follows.   -n s -n s f()s =Σan e , g()s =Σbn e n =1 n =1

If both are convergent in a certain domain and f()s = g()s holds at there, an = bn for n =1,2,3, .

Bibliography 「数論入門」D.B.ザギヤー著、片山考次訳岩波 1990年, etc.

- 2 - 1.2 General Dirichlet Series & Power Series General Dirichlet Series can contain Power Series other than Ordinary Dirichlet Series. In the following section, we will consider this.

1.2.1 Relation between Power Series and General Dirichlet Series   n -n s Relations between power series Σcn z -z0 and general Dirichlet series Σan e are as follows. n =1 n =1 -s cn = an , n = n , z -z0 = e (2.1) Especially, when the power series starts from n =0 , -s cn-1 = an , n-1= n , z -z0 = e (2.0)

Calculation     n n-1 -s n -n s Σcn z -z0 = Σcn-1 z -z0 = Σan(e ) = Σan e n =0 n =1 n =1 n =1     -n s -s n n-1 n Σan e = Σcn-1(e ) = Σcn-1z -z0 = Σcn z -z0 n =1 n =1 n =1 n =0

Note When a function f()z is expressed with a power series, Dirichlet series equivalent to this does not express -s -s f()s . Since z -z0 = e , it is fz = f (e +z0) . That is, Dirichlet series equivalent to the power series express the composition f .

1.2.2 Circle of convergence & Line of convergence

As seen above, the relation between the variable z -z0 of power series and the variable s of Dirichlet series was as follows.. -s z -z0 = e That is,

s = -logz -z0 = -logz -z0 - i argz -z0 Here, let s =  +i t , z = x +iy . Then

 +it = -logx -x0 + i y -y0 - i argx -x0 + i y -y0 Although coordinates x -y are transferred to coordinates -t by the conversion, the coordinates -t are similar to pola coordinates r - . The difference among both is only the existence of log in the real part. From this, 2 2  = -log x -x0 + y -y0 (2.)

t = -argx -x0 + i y -y0 (2.t)

When z0 = x0+iy0 =1+i1 , the real part (2.) is drawn on the left figure. All the horizontal sections of the real part (2.) are circles centered on the point 1,1 . Radius of each circle is given by x -1 2+ y -1 2

- 3 - For example, if the radius is 1 ,  =-log1=0 from (2.) . If the left figure is horizontally cut at this height, it is the right figure.

When the radius is 1 , the power series converges in the circle. This means that the corresponding Dirichlet series should converge at the upper than the circle i.e.  >0. The height c =-log R which provides the convergence circle with the radius R is the line of convergence.

So far, we have seen the relationship between circle of convergence and line of convergence in the figure. It is as follows analytically. -s using z -z0 = e

-c -s - - i t - e  R > z -z0 = e = e = e i.e. e- < e-c = R Taking the logarithm of both sides, we obtain

 > c = -logR (2.2) c.f.

According to the definition in 1.1 , the convergence line c is calculated as follows.   n When Σcn or Σcn is convergent, Σck is also convergent. Then n =1 n =0 k=1

logan+1 + an+2 + an+3  logcn+1 + cn+2 + cn+3  c = limsup = limsup n  n n  n The calculation by this formula is difficult in many cases. It is better to use the above (2.2) .

Byproduct -1/n From the above and R = lim infcn , c = -logR , we obtain the following n  1/n 1/n limsupcn + cn + cn  = limsupcn () = 1/R (2.3) n  +1 +2 +3 n 

- 4 - 1.3 Dirichlet series expression of primary functions

1.3.1 Dirichlet series expression of logarithmic function Function log z is expanded in power series as follows.

 n-1 ()-1 n log z = Σ ()z-1 = f()z (1.f) n =1 n When z = x +iy , the real part and the imaginary part of both sides are illustrated respectively as follows. The left figure is a real part, the right figure is an imaginary part. In both figures, the left-hand side is orange, and the right-hand side is blue. In both figures, we can see that the radius of convergence of the right-hand side is 1. Further, for comparison with latter figures, appropriate 5 points on the circumference is marked in red.

-s Substituting z -1 = e for (1.f) ,  n-1 -s ()-1 -ns log1+e = Σ e = g()s (1.g) n =1 n n-1  ()-1 -n s Let an = , n = n . Then the right-hand side becomes Σan e , which is general Dirichlet n n =1 -s series. Where, we should note that the left-hand side is not log s but log(1+e ) . When s = +it , the real part and the imaginary part of both sides are illustrated respectively as follows.

The left figure is a real part, the right figure is an imaginary part. In both figures, the left-hand side is orange, and the right-hand side is blue. In both figures, we can see that the line of convergence of the right-hand side

- 5 - is 0. Further, we can see that 5 points in the former figures are transferred from the circle of convergence to the line of convergence by s =-log(z -1) . ( White points.)

In addition, the line of convergence c is calculated as follows using 1.2.2 (2.2) .

c =-logR =-log 1 = 0

1.3.2 Dirichlet series expression of tangent function

Function tan z is expanded in power series as follows. Where, Bk is Bernouilli number.

 2n 2n n-1 2 2 -1 2n-1 tan z = Σ()-1 B2n z = f()z (2.f) n =1 2n ()2n -1 ! When z = x +iy , the real part and the imaginary part of both sides are illustrated respectively as follows. The left figure is a real part, the right figure is an imaginary part. In both figures, the left-hand side is orange, and the right-hand side is blue. In both figures, the radius of convergence of the right-hand side looks like /2 Further, for comparison with latter figures, appropriate 4 points on the circumference is marked in red.

-s Substituting z = e for (2.f) ,  2n 2n -s n-1 2 2 -1 -2n-1 s tan e = Σ()-1 B2n e = g()s (2.g) n =1 2n ()2n -1 ! 2n 2n  n-1 2 2 -1 -n s Let an = ()-1 B2n , n = 2n -1 , Then the right-hand side becomes Σan e , 2n 2n -1 ! n =1 -s which is general Dirichlet series. Where, we should note that the left-hand side is not tan s but tan e . When s = +it , the real part and the imaginary part of both sides are illustrated respectively as follows.

- 6 - The left figure is a real part, the right figure is an imaginary part. In both figures, the left-hand side is orange, and the right-hand side is blue. In both figures, we can see that the line of convergence of the right-hand side is near -0.5 . Further, we can see that 4 points in the former figures are transferred from the circle of convergence to the line of convergence by s =-log z . ( White points.)

In addition, the line of convergence c is calculated as follows using 1.2.2 (2.2) .   =-logR =-log = -0.451852 c 2

1.3.3 Dirichlet series expression of exponential function z Function e is expanded in power series as follows.

 n z z e = Σ = f()z (3.f) n =0 n ! When z = x +iy , the real part and the imaginary part of both sides are illustrated respectively as follows. The left figure is a real part, the right figure is an imaginary part. In both figures, the left-hand side is orange, and the right-hand side is blue.

In both figures, something like a circle of convergence is found nowhere. Both sides are completely overlap. For comparison with latter figures, appropriate 5 points is marked in red. -s Substituting z = e for (3.f) ,

 -n-1 s e-s e e = Σ = g()s (3.g) n =1 ()n-1 !  1 -n s Let an = , n = n -1 , Then the right-hand side becomes Σan e , which is general Dirichlet n -1 ! n =1 s e-s seies. Where, we should note that the left-hand side is not e but e .

When s = +it , the real part and the imaginary part of both sides are illustrated respectively on the following page. The left figure is a real part, the right figure is an imaginary part. In both figures, the left-hand side is orange, and the right-hand side is blue. In both figures, something like a line of convergence is found nowhere. Further, we can see that 5 points in the former figures are transferred to white points by s =-log z .

- 7 - In addition, the line of convergence c is calculated as follows using 1.2.2 (2.2) .

c =-logR =-log  = -

2016.08.05 Kano. Kono Alien's Mathematics

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