Liouville function

The Liouville function, named after Joseph Liouville, is a multiplicative number-theoretic function. It is designated by the Greek letter, and is defined as follows:

It denotes the order of, meaning the number of (not necessarily distinct) prime factors.

It also defines and.

The first values ​​are (starting at )

Properties

It is

The Liouville function is related to the Möbius function by

String

The Dirichlet series of the Liouville function can be expressed by the Riemann zeta function:

Your Lambert series is given by

Where the Jacobi theta function called.

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It should be

The Pólya conjecture states that it is - how can the graphics suspect right - always

This assumption has now been refuted; the smallest counterexample is. It is, however, so far not known whether its sign infinitely often changes.

A related sum is

For this it was assumed she was always positive for sufficiently large; This was disproved in 1958 by the English mathematician Colin Brian Haselgrove, where he showed that infinitely often assumes negative values ​​. A proof of the conjecture would have the accuracy of Riemann conjecture result.