Zeros of the Alternating Zeta Function on the Line R(S) = 1

which is easily established, first for R > ( ) s 1 by combining terms in the convergent Dirichlet series, and then by using analytic continuation to extend the result to the entire s -plane (see Theorem 13.11 in [1]). The factor ( ) 1 2 − −s has a simple zero at s = 1 that cancels the pole of ζ ( ) s , so (1) shows that ζ ( ) s is an entire function of s and that it vanishes at each zero of the factor ( ) 1 2 − −s with the exception of s = 1, at which point ζ ( ) log 1 2 = . The other zeros of the factor ( ) 1 2 − −s lie on the line R = ( ) s 1, and occur at the points s k i = + 1 2 2 π / log , where k is a nonzero integer. This note deduces the vanishing of ζ ( ) s at the zeros of ( ) 1 2 − −s , except for s = 1, without using (1). Instead, we use identity (4) relating the partial sums