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A new definition for the Riemann zeta function for all positive integernumber s > 1 is presented. We discover a most elegant expression and easymethod for calculating the Riemann zeta function for small even integer values.Through this new reformulation we provide a one-line proof of the value ofzeta(2) and demonstrate that zeta(2s) may be calculated by hand using only thecotangent function when the magnitude of the integer s is small.

Elegant expressions and generic formulas for the Riemann zeta function for integer arguments