total number of real zeros and the total number of zeros of odd order of the function lying in the interval . These two conjectures opened up new directions in the investigation of the Riemann zeta function. Zero-free region The location of the Riemann zeta function's zeros is of great importance in number theory. The prime number theorem is equivalent to the fact that there are no zeros of the zeta function on the line. A better result that follows from an effective form of Vinogradov's mean-value theorem is that whenever and . In 2015, Mossinghoff and Trudgian proved that zeta has no zeros in the region for . This is the largest known zero-free region in the critical strip for . The strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound consequences in the theory of numbers. Other results It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence () contains the imaginary parts of all zeros in the upper half-plane in ascending order, then The critical line theorem asserts that a positive proportion of the nontrivial zeros lies on the critical line. (The Riemann hypothesis would imply that this proportion is 1.) In the critical strip, the zero with smallest non-negative imaginary part is (). The fact that for all complex implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line . It is also known that no zeros lie on a line with real part 1. : where is Lambert W function. Specific values For any positive even integer , where is the -th Bernoulli number. For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraic -theory of the integers; see Special values of -functions. For nonpositive integers, one has for (using the convention that ). In particular, vanishes at the negative even integers because for all odd other than 1. These are the so-called "trivial zeros" of the zeta function. Via analytic continuation, one can show that This gives a pretext for assigning a finite value to the divergent series 1 + 2 + 3 + 4 + ⋯,