Zeros of Partial Summs of the Riemann Zeta Function

Abstract

The semiperiodic behavior of the zeta function $\zeta(s)$ and its partial sums $\zeta_N(s)$ as a function of the imaginary coordinate has been long established. In fact, the zeros of a $\zeta_N(s)$, when reduced into imaginary periods derived from primes less than or equal to $N$, establish regular patterns. We show that these zeros can be embedded as a dense set in the period of a surface in $\mathbb{R}^{k+1}$, where $k$ is the number of primes in the expansion. This enables us, for example, to establish the lower bound for the real parts of zeros of $\zeta_N(s)$ for prime $N$ and justifies the use of methods of calculus to find expressions for the bounding curves for sets of reduced zeros in $\mathbb{C}$.