RIEMANN ZETA FUNCTION

function useful in number theory for investigating properties of primes. Written as z(x), it was originally defined as the infinite sum z(x) = 1 + (1/2)x + (1/3)x> + (1/4)x + . . . etc. When x = 1, this series is called the harmonic series, which has no finite sum. For values of x larger than 1, the series adds up to a finite number. If x is less than 1, the sum is again infinite. The zeta function was known to the Swiss mathematician Leonhard Euler in 1737 and was studied more extensively by the German mathematician Bernhard Riemann (for whom it was named) in 1859. A more complicated function can be defined (by a continuation process) that equals this series for values of x greater than 1, but has finite values for any real or complex number the real part of which is different from 1. This is the zeta function that is studied and discussed in the literature. One of the first questions asked about a function by mathematicians is for what values it is equal to zero, and this question is still not answered for the zeta function. It is known that the function equals zero when x is -2, -4, -6, . . . , and that it has an infinite number of zero values for the set of complex numbers the real part of which is between zero and one, but it is not known exactly for what complex numbers these zeros occur. Riemann conjectured that these zeros probably occur for those complex numbers the real part of which equals 1/2, but this is still unproven. The investigations of this difficult problem have resulted, as is often the case in mathematics, in an enrichment of the knowledge of the properties of complex numbers.