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function useful in number theory for investigating properties of prime numbers. Written as zeta(x), it was originally defined as the infinite series zeta(x) = 1 + 2x + 3x + 4x +.When x = 1, this series is called the harmonic series, which increases without bound-i.e., its sum is infinite. For values of x larger than 1, the series converges to a finite number as successive terms are added. If x is less than 1, the sum is again infinite. The zeta function was known to the Swiss mathematician Leonhard Euler in 1737, but it was first studied extensively by the German mathematician Bernhard Riemann.