Zeros of Partial Sums of the Riemann Zeta Function

Abstract

The semiperiodic behavior of the zeta function ζ(s) and its partial sums ζN(s) as a function of the imaginary coordinate has been long established. In fact, the zeros of a ζN(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 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 ζN(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 ℂ.

Keywords:

• Primary 11M26
• Secondary 11M41, 11Y35, 40A25
• Zeta function
• Dirichlet series
• zeros of exponential sums
• series convergence