Summary
1 Zeta Generating Functions
Both of hyperbolic functions and trigonometric functions can be expanded to Fourier series and Taylor series. And if the
termwise higher order integration of these is carried out, Riemann Zeta Functions are obtained.
Where, these are automorphisms which are expressed by lower zetas. However, in this chapter, we stop those so far.
The work that obtain the non-automorphism formulas by removing lower zetas from these are performed subsequent to
Chapter2 .
In this chapter, we obtain the following polynomials from the zeta generating functions
Where, Riemann Zeta, Dirichlet Eta and Dirichlet Lambda are as follows.
math0001.png
Bernoulli numbers and Euler numbers are as follows.
math0002.png
math0003.png
Harmonic number is math0004.png
math0005.png
math0006.png
math0007.png
math0008.png
math0009.png
math0010.png
math0011.png
math0012.png
math0013.png
math0014.png
math0015.png
math0016.png
math0017.png
math0018.png
math0019.png
math0020.png
Furthermore, if the termwise higher order differentiation of the Fourier series of each family of tanh, cot and tan are carried
out, the following expressions are obtained.
math0021.png
math0022.png
math0023.png
Where, math0024.png are the Eulderian Numbers and math0025.png are the tangent numbers. These are defined as follows respectively.
math0026.png
By-products
math0027.png
2 Formulas for Riemann Zeta at natural number
In this chapter, removing the lower zetas from automorphism formulas in the previous chapter, we obtain
non-automorphism formulas for Riemann Zeta at natural number.
Where, Bernoulli numbers and Euler numbers are as follows.
math0028.png
math0029.png
And Harmonic number is math0030.png
For math0031.png,
math0032.png
math0033.png
math0034.png
For math0035.png,
math0036.png
math0037.png
math0038.png
math0039.png
math0040.png
Especially,
math0041.png
math0042.png
math0043.png
math0044.png
math0045.png
math0046.png
math0047.png
math0048.png
math0049.png
math0050.png
Example
math0051.png
math0052.png
math0053.png
math0054.png
math0055.png
math0056.png
math0057.png
3 Formulas for Riemann Zeta at odd number
In this chapter, we obtain non-automorphism formulas for Riemann Zeta at odd number.
Where, Bernoulli numbers, Euler numbers and tangent numbers are as follows.
math0058.png
math0059.png
math0060.png
And Harmonic number is math0061.png
For math0062.png,
math0063.png
math0064.png
math0065.png
math0066.png
For math0067.png,
math0068.png
math0069.png
math0070.png
math0071.png
math0072.png
math0073.png
Especially,
math0074.png
math0075.png
math0076.png
Example
math0077.png
math0078.png
math0079.png
4 Formulas for Riemann Zeta at even number
In this chapter, we obtain non-automorphism formulas for Riemann Zeta at even number.
Where, Bernoulli numbers and Euler numbers are as follows.
math0080.png
math0081.png
For math0082.png,
math0083.png
math0084.png
For math0085.png,
math0086.png
math0087.png
math0088.png
math0089.png
math0090.png
Especially,
math0091.png
math0092.png
By-products
math0093.png
math0094.png
5 Formulas for Riemann Zeta at complex number
In this chapter, we obtain the formulas for Riemann Zeta at a complex number by processing " 2 Formulas for Riemann
Zeta at natural number "
Where, Bernoulli numbers are as follows.
math0095.png
And gamma function and incomplete gamma function are
math0096.png
And math0097.pngis a complex number such that math0098.png.
For math0099.png,
math0100.png
math0101.png
For math0102.png,
math0103.png
math0104.png
math0105.png
math0106.png
math0107.png
Especially,
math0108.png
math0109.png
math0110.png
math0111.png
math0112.png
math0113.png
2012.12.05
K. Kono
Alien's Mathematics