Riemann Xi function

In mathematics, the Riemann Xi - function is a transform of the Riemann zeta function. Your zeros correspond exclusively to the non-trivial zeros of the zeta function, and in contrast to this, the Xi - function is holomorphic on the whole complex plane. Furthermore, it satisfies a particularly simple functional equation. Bernhard Riemann introduced in 1859 in the same paper on the distribution of primes, in which he also formulated the Riemann conjecture named after him later.

Definition

The Riemann Xi - function (" small xi " ) is defined as

Where the Riemann function and the gamma function called. The product term on the right side in front of the Riemann function eliminates exactly the negative zeros and the singularities of the zeta function at the point. The only zeros of are therefore precisely the non-trivial zeros of the function.

A variant of the Xi - function is usually ( " large Xi " ) designated and goes through the transformation of variables (ie ) out:

The Riemann conjecture is equivalent to the statement that all zeros of are real.

Remarkably, Riemann himself used the letters identifying those function that is now known ( Landau ) with; the cause of these initially confusing symbolism is in manifest error Riemann, but has no effect on the statements of his article.

Properties

Special values

The following applies:

For even natural numbers is valid:

Where the- th Bernoulli number called. From this representation, among other things, the values ​​are obtained:

Functional equation

The Xi - function satisfies the functional equation ( " reflection formula " )

Or equivalently for the function:

Is thus an even function.

Product representation

Which runs in the product formula over all zeros of.

Relationship with the Riemann - Siegel'schen Z function

It is

Asymptotic behavior

For real values ​​of s

So

(where the Landau symbol indicates ). Accordingly, valid for real values ​​of t

Li coefficients

The Xi - function has a close relationship to the so - called Li- coefficient

Because there are relations

And

The metallic criterion is the property for all positive. It is equivalent to the Riemann Hypothesis.