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.

Bernoulli numbers and Euler numbers are as follows.

Harmonic number is

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.

Where, are the Eulderian Numbers and are the tangent numbers. These are defined as follows respectively.

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.

And Harmonic number is

For ,

For ,

Especially,

Example

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.

And Harmonic number is

For ,

For ,

Especially,

Example

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.

For ,

For ,

Especially,

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.

And gamma function and incomplete gamma function are

And is a complex number such that .

For ,

For ,

Especially,

2012.12.05

K. Kono