On the zeros of sums of the Riemann zeta function Original Research Article

In this article, we study the zeros of ζ(σ0+s)±ζ(σ0−s) for a fixed image. We give a complete description where the zeros of the function are, except for image. It turns out that the behavior of zeros of the function with image is very different from that of the function with image. Roughly speaking, zeros of the function for image tend to be located on the imaginary axis or the real axis. On the other hand, almost all zeros of the functions for image are arbitrarily close to image and there are fewer zeros in any strip which does not contain these axes. We have the analogues for the function ζ(σ0+s)+aζ(σ0−s) (image and a=1; image and a≠0,1).