📝 SummarXiv

Abstract

In this paper, we consider the representation of the Riemann zeta function $\zeta$ defined by Abel's summation formula. Using the differential equations, we show that $|(1-s)\zeta(s) -\bar{s}\zeta(1-\bar{s})| \neq 0$ for any point $s$ in the critical strip except the critical line. This result suggests an asymmetry of $(1-s)\zeta(s) $ across the critical strip. This does not contradict the Riemann functional equation but prove that non-trivial zeros cannot lie off the critical line. These results are consistent with the Riemann Hypothesis and suggest that non-trivial zeros lie on the critical line.