Conversion from special function to continued fraction

There is another expansion in infinite product. That is called Euler’s product.

   for x>1

where p={2,3,5,7,11,13,…} that are primes.

We use the following formula

     (regular expression)

　in special form in 0220.

Thus

 in special form

There are a few more expansions for Riemann Zeta function.

Euler’s product is equivalent to the following

using primes in order.

We can use the following formula

   (regular expression)

   in special form

in 0218.

   in special form

There is another version of expansion of Riemann Zeta function. That is,

Using the following formula of series

  (regular expression)

   in special

Applying gamma-shaped multipliers and rule of signs repeatedly, we finally get

   in special form