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# Euler Riemann Zeta Function Calculator _ Online Computing Tools

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### App description

Riemann function is a special function, which was discovered and put forward by German mathematician Riemann. Riemann function is defined on [0,1], and its basic definition is as follows:

R(x)=1/q，当x=p/q(p, q are positive integers, p/q is a reduced true fraction); R(x)=0, when x=0，1 and the irrational number in (0,1). The definition of Riemann Zata function (s) is as follows: let a complex number s whose real part is greater than 1 and:

Function R (x) defined on interval [0,1]:

When x=p/q (p, q are positive integers, p/q is a reduced true fraction), R(x) = 1/q;

When x = 0,1 and irrational number, R(x) = 0 is called Riemann function.

According to the definition of Riemann function, it has the following characteristics:

(1) Riemann function is a bounded function on the interval [0,1], whose supremum is 1/2 and its infimum is 0. Its range of values is only one, and the accumulation point is 0. It is also the limit value of sequence {1/q}, where q is a natural number.

(2) The image of Riemann function at rational point is symmetric about the line x0 = 1/2.

(3) For, there are only a finite number of rational numbers in the interval (0,1) such that R(x)=R（p/q）=1/q>ε）.

### Usage example:

Parameters of Riemann zeta function = 55

Click "calculate" to output the result.

The approximate Zeta function (55) = 1, n=20000