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:

    “Any relation to the land, the habit of tilling it, or mining it, or even hunting on it, generates the feeling of patriotism. He who keeps shop on it, or he who merely uses it as a support to his desk and ledger, or to his manufactory, values it less.”
    —Ralph Waldo Emerson (1803–1882)

    “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)

    “To criticize is to appreciate, to appropriate, to take intellectual possession, to establish in fine a relation with the criticized thing and to make it one’s own.”
    —Henry James (1843–1916)

    “Think of the tools in a tool-box: there is a hammer, pliers, a saw, a screwdriver, a rule, a glue-pot, nails and screws.—The function of words are as diverse as the functions of these objects.”
    —Ludwig Wittgenstein (1889–1951)