Liouville Function - Series

Series

The Dirichlet series for the Liouville function gives the Riemann zeta function as

The Lambert series for the Liouville function is

\sum_{n=1}^\infty \frac{\lambda(n)q^n}{1-q^n} = \sum_{n=1}^\infty q^{n^2} = \frac{1}{2}\left(\vartheta_3(q)-1\right),

where is the Jacobi theta function.

Read more about this topic:  Liouville Function

Famous quotes containing the word series:

    “History is nothing but a procession of false Absolutes, a series of temples raised to pretexts, a degradation of the mind before the Improbable.”
    —E.M. Cioran (b. 1911)

    “Personality is an unbroken series of successful gestures.”
    —F. Scott Fitzgerald (1896–1940)

    “Depression moods lead, almost invariably, to accidents. But, when they occur, our mood changes again, since the accident shows we can draw the world in our wake, and that we still retain some degree of power even when our spirits are low. A series of accidents creates a positively light-hearted state, out of consideration for this strange power.”
    —Jean Baudrillard (b. 1929)