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The hypothesis that the Riemann zeta function ζ( s ) = ∑^∞ _n=1 n ^{− s} is equal to zero only when s (which is a complex number) is equal to -2k for some natural number k, or when the real part of s is 1/2.
This has not been proved, although it has been shown that there are infinitely many values of s with real part equal to 1/2 for which the Riemann zeta function is equal to zero.