On the zeros of sums of the Riemann zeta function

In this article, we study the zeros of ζ(σ0+s)±ζ(σ0−s) for a fixed σ0∈R. We give a complete description where the zeros of the function are, except for . It turns out that the behavior of zeros of the function with is very different from that of the function with . Roughly speaking, zeros of the function for tend to be located on the imaginary axis or the real axis. On the other hand, almost all zeros of the functions for are arbitrarily close to 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) ( and |a|=1; and |a|≠0,1).