The Riemann Zeta and the Calculus

(Under construction: Reprising investigations over several years.)

By virtue of the relation between the values of the Riemann zeta function at the negative integers, \zeta(-n<1), and the Bernoulli numbers and between the Bernoulli polynomials and the partial sums of the powers of the natural numbers and derivatives of analytic functions, the Riemann zeta can be related to the integration and differentiation of analytic functions.

Through the relation between the values of the Riemann zeta function at the positive natural numbers greater than one, \zeta(n>1), and a series expansion of the digamma function and between a digamma differential operator and the infinigen (infinitesimal generator) of a fractional calculus, the Riemann zeta can be related to the fractional calculus-the calculus of fractional integral and differential operators acting on real functions analytic on the positive real axis. .

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