Theta Function - Relation To The Riemann Zeta Function

Relation To The Riemann Zeta Function

The relation

was used by Riemann to prove the functional equation for the Riemann zeta function, by means of the integral

\Gamma\left(\frac{s}{2}\right) \pi^{-s/2} \zeta(s) = \frac{1}{2}\int_0^\infty\left t^{s/2}\frac{dt}{t}

which can be shown to be invariant under substitution of s by 1 − s. The corresponding integral for z not zero is given in the article on the Hurwitz zeta function.

Read more about this topic:  Theta Function

Famous quotes containing the words relation to the, relation to, relation and/or function:

    “Much poetry seems to be aware of its situation in time and of its relation to the metronome, the clock, and the calendar. ... The season or month is there to be felt; the day is there to be seized. Poems beginning “When” are much more numerous than those beginning “Where” of “If.” As the meter is running, the recurrent message tapped out by the passing of measured time is mortality.”
    —William Harmon (b. 1938)

    “You must realize that I was suffering from love and I knew him as intimately as I knew my own image in a mirror. In other words, I knew him only in relation to myself.”
    —Angela Carter (1940–1992)

    “It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
    —René Descartes (1596–1650)

    “The function of muscle is to pull and not to push, except in the case of the genitals and the tongue.”
    —Leonardo Da Vinci (1425–1519)