A Geometric Perspective on the Riemann Zeta Function’s Partial Sums

The Riemann Zeta Function, ζ(s), is an important complex function whose behavior has implications for the distribution of the prime numbers among the natural numbers. Most notably, the still unsolved Riemann Hypothesis, which states that all non-trivial zeros of the zeta function have real part one-half, would imply the most regular distribution of primes possible in the context of current theory. The Riemann Zeta Function is the simplest of the Dirichlet series and is represented in its Dirichlet series form as ζ(s) = ∞ n+1 n −s = 1−s+2−s+3−s+ . . . . This series only converges when the real part of s, (s), is greater than 1, outside the area of the complex plane relevant to the distribution of the primes. This area is called the critical strip: {s ∈ C : 0 < (s) < 1}. The result of our investigation of the geometric distribution will be to draw connections between the partial sums of the Dirichlet series and the value of ζ(s) with s in the critical strip despite the series’ divergence. This article will illustrate connections between existing theory of the Riemann Zeta Function and geometric analysis of the partial sums through visual representations. From the connections between the visually accessible geometry and this theory, we illuminate and explore potentially provable improvements of the theory based on symmetry among the partial sums. 1. The Importance of the Riemann Zeta