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Series Expansion of Dirichlet Eta Function

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Series Expansion of Dirichlet Eta Function 08 Series Expansion of Dirichlet Eta Function 8.1 Dirichlet Eta Series Dirichlet eta function z is defined in Rez >0 by the following expression called Dirichlet eta series. s-1 -zlogs 1 1 1 1 (1.0) ()z = Σ()-1 e = z - z + z - z +- s=1 1 2 3 4 This is represented as follows for each real part and imaginary part. Formula 8.1.1 When the real and imaginary parts of Dirichlet eta function x,y are r , i respectively, the following expressions hold for x >0. cos()y log s s-1 r()x,y = Σ()-1 x s=1 s sin()y log s s-1 i()x,y = -Σ()-1 x s=1 s Proof When z = x +iy , from (1.0) ()x,y = Σ()-1 s-1e-x + iy logs = Σ()-1 s-1e-x log se-iy logs s=1 s=1 s-1 cos()y log s -i sin()y log s = Σ()-1 x s=1 s s-1 cos()y log s s-1 sin()y log s = Σ()-1 x - iΣ()-1 x s=1 s s=1 s From this, we obtain the desired expressions. These are drawn as follows. The left is the real part and the right is the imaginary part. In both figures, orange is the left side and blue is the right side. It is well known that x =0 is a line of convergence. - 1 - Formula 8.1.1 can be accelerated in convergence by applying the Knop transformation. Further, an analytic continuation can be expected. In addition, about Knopp transformation, see " 10 Convergence Acceleration & Summation Method by Double Series of Functions " ( A la carte ). Formula 8.1.1' ( Accelerated type ) When the real and imaginary parts of Dirichlet eta function x,y are r , i respectively, the following expressions hold for arbitrary positive number q . k k-s k q s-1 cosy log s r()x,y = Σ ()-1 Σ k+1 x k=1 s=1 q +1 s s k k-s k q s-1 siny log s i()x,y = - ()-1 ΣΣ k+1 x k=1 s=1 q +1 s s When q =1 , these are drawn as follows. The left is the real part and the right is the imaginary part. In both figures, orange is the left side and blue is the right side. We can see that the line of convergence x =0 has disappeared due to the effect of the analytic continuation. Non-Trivial Zeros of Riemann Zeta Theorem 8.1.2 The non-trivial zeros of Riemann zeta function are given by the real solutions of the following simultaneous equations. s-1 cos()y log s Σ()-1 x = 0 s=1 s Where, 0 < x < 1 s-1 sin()y log s -Σ()-1 x = 0 s=1 s And, Riemann hypothesis is that one of the solutions will always be x =1/2 . However, this simultaneous equation is a transcendental equation with respect to x and y , and the analysis is not easy. - 2 - 8.2 Power Series with Stieltjes Constants 8.2.0 Power of Complex Number Lemma 8.2.0 When x,y are real numbers, r is non-negative integer, the following holds. r -1 r -1 2 2 r s r r-2s 2s s r r-2s-1 2s+1 ()x +iy = Σ ()-1 x y + i Σ ()-1 x y (2.0) s=0 2s s=0 2s+1 0 Where, 0 = 1 , x is the ceiling function , x is the floor function. Proof 1 1 1-s s s ()x +iy = Σ1Ｃs x i y s=0 1-0 0 1-1 1 = 10Ｃ x y + i 11Ｃ x y 1-1 0 0 0 = s 1-2s 2s s 1-2s-1 2s+1 2 = 12Ｃ s 12Ｃ s+1 Σ()-1 x y + iΣ()-1 x y 1-1 s=0 s=0 0 = 2 2 2 2-s s s ()x +iy = Σ2Ｃs x i y s=0 2-0 0 2-2 2 2-1 1 = 20Ｃ x y - 22Ｃ x y + i 21Ｃ x y 2-1 1 0 1 = s 2-2s 2s s 2-2s-1 2s+1 2 = 22Ｃ s 22Ｃ s+1 Σ()-1 x y + iΣ()-1 x y 2-1 s=0 s=0 0 = 2 3 3 3-s s s ()x +iy = Σ3Ｃs x i y s=0 3-0 0 3-2 2 3-1 1 3-3 3 = 30Ｃ x y - 32Ｃ x y + i 31Ｃ x y - 33Ｃ x y 3-1 1 1 1 = s 3-2s 2s s 3-2s-1 2s+1 2 = 32Ｃ s 32Ｃ s+1 Σ()-1 x y + iΣ()-1 x y 3-1 s=0 s=0 1 = 2 4 4 4-s s s ()x +iy = Σ4Ｃs x i y s=0 4-0 0 4-2 2 4-4 4 4-1 1 4-3 3 = 40Ｃ x y - 42Ｃ x y - 44Ｃ x y + i 41Ｃ x y - 43Ｃ x y 4-1 2 1 2 = s 4-2s 2s s 4-2s-1 2s+1 2 = 42Ｃ s 42Ｃ s+1 Σ()-1 x y + iΣ()-1 x y 4-1 s=0 s=0 1 = 2 Hereafter, by induction, we obtain the desired expression. - 3 - 8.2.1 Power Series Expansion with Stieltjes Constants " 03 Complementary Series of Dirichlet Series " Formula 3.1.2 was as follows. r+1 r-1 r r r log 2 r-s z -1 ()z = log 2 + Σ()-1 - Σ s log2 r=1 r +1 s=0 s r ! s n ()log k s ()log n +1 Where, s = lim Σ - n k=1 k s+1 This is represented as follows for each real part and imaginary part. Formula 8.2.1 When the real and imaginary parts of Dirichlet eta function x,y are r , i respectively, the following expressions hold on the whole complex plane. r r+1 r-1 r ()-1 log 2 r-s rx,y = log 2 + Σ -Σ s log2 r=1 r ! r +1 s=0s r -1 2 r t Σ ()-1 t x -1 r-2t y 2 t=0 2t r r+1 r-1 r ()-1 log 2 r-s ix,y = Σ -Σ s log2 r=1 r ! r +1 s=0s r -1 2 r t Σ ()-1 t x -1 r-2t-1 y 2 +1 t=0 2t+1 0 Where, 0 = 1 , x is the ceiling function , x is the floor function. Proof " 03 Complementary Series of Dirichlet Series " Formula 3.1.2 was as follows. r r+1 r-1 r ()-1 log 2 r-s r ()z = log 2 + Σ -Σ s log2 z -1 r=1 r ! r +1 s=0s On the other hand, replacing x with x -1 in Lemma 8.2.0 , r -1 r 2 r t ()x-1+iy = Σ ()-1 t x -1 r-2t y 2 t=0 2t r -1 2 r t + i Σ ()-1 t x -1 r-2t-1 y 2 +1 t=0 2t+1 r Substituting this for z -1 and writing the real and imaginary parts as r ,i , we obtain the formula. These are drawn on the next page, The left is the real part and the right is the imaginary part. In both figures, orange is the left side and blue is the right side. We can see that this formula holds on the whole complex plane. - 4 - The convergence of Formula 8.2.1 is very fast. However, it contains two binomial coefficients and is not beautiful. So applying Formula 14.1.2 in " 14 Taylor Expansion by Real Part & Imaginary Part " (A la carte) to (1.2) , we obtain the follows. It is a beautiful two-variable Taylor series, and the convergence is fast. Formula 8.2.2 When the real and imaginary parts of Dirichlet eta function x,y are r , i respectively, the following expressions hold on the whole complex plane. s+1 s-1 s s s log2 s-t z -1 ()z = log 2 + Σ()-1 - Σ t log2 s=1 s +1 t=0 t s ! s+1 s-1 s s s log 2 s-t ()-1 x -1 r()x,y = log 2 + Σ - Σ t log2 s=1 s +1 t=0 t s ! 2r+s+1 2r+s-1 r s s s r 2r log2 2 + 2r+s-t ()-1 x -1 ()-1 y + ΣΣ - Σ t log2 r=1s=02r +s +1 t=0 t s ! 2r ! 2r+s+2 2r+s r s log 2 2 + +1 2r+s+1-t i()x,y = -ΣΣ - Σ t log2 r=0 s=0 2r +s +2 t=0 t r ()-1 sx -1 s ()-1 ry2 +1 s ! 2r +1 ! n t t+1 logk logn 0 Where, t = lim Σ - , 0 = 1 n k=1 k t +1 The verification results by numerical calculation of the real and imaginary parts are as follows. The upper part is the real part and the lower part is the imaginary part. Both sides are exactly the same. - 5 - 8.3 Taylor Series & Complementary Series 8.3.1 Taylor Series Formula 8.3.1 When the real and imaginary parts of Dirichlet eta function x,y are r , i respectively, the following expressions hold for x >0 and real constant c other than zero of .
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