SEVERAL RESULTS ASSOCIATED WITH THE RIEMANN ZETA FUNCTION

Abstract

In 1859, Bernhard Riemann, in his epoch-making memoir, extended the Euler zeta function ζ(s) (s > 1, s ∈ R) to the Riemann zeta function ζ(s) ((cid:60)(s) > 1, s ∈ C) to investigate the pattern of the primes. Sine the time of Euler and then Riemann, the Riemann zeta function ζ(s) has involved and appeared in a variety of mathematical research subjects as well as the function itself has been being broadly and deeply researched. Among those things, we choose to make a further investigation of the following subjects: Evaluation of ζ(2k) (k ∈ N); Approximate functional equations for ζ(s); Series involving the Riemann zeta function.

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