In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0.

Is it weird? The zeros of the eta function include all the zeros of the zeta function: the negative even integers (real equidistant simple zeros); the zeros along the critical line, none of which are known to be multiple and over 40% of which have been proven to be simple, and the hypothetical zeros in the critical strip but not on the critical line, which if they do exist must occur at the vertices of rectangles symmetrical around the x-axis and the critical line and whose multiplicity is unknown.

In the equation η(s) = (1−21−s) ζ(s), "the pole of ζ(s) at s=1 is cancelled by the zero of the other factor" and as a result η(1) is neither infinite nor zero. So what about that? The problem of proving this without defining the zeta function first was signaled and left open by E. Landau in his 1909 treatise on number theory: "Whether the eta series is different from zero or not at the points s=/=1, whether these are poles of zeta or not, is not readily apparent here." Finally, Sondow gave direct proof.