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 frac(1, 2) ≤ σ0 ≤ frac(3, 4). It turns out that the behavior of zeros of the function with σ0 < frac(1, 2) is very different from that of the function with σ0 > frac(3, 4). Roughly speaking, zeros of the function for σ0 < frac(1, 2) tend to be located on the imaginary axis or the real axis. On the other hand, almost all zeros of the functions for σ0 > frac(3, 4) are arbitrarily close to Re (s) = ± (σ0 - frac(1, 2)) and there are fewer zeros in any strip which does not contain these axes. We have the analogues for the function ζ (σ0 + s) + a ζ (σ0 - s) (σ0 > frac(3, 4) and | a | = 1; σ0 > frac(1, 2) and | a | ≠ 0, 1).
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory