Continuous crop circles drawn by Riemann's zeta function.

Let η (s) = ∑ n = 1 ∞ (− 1) n + 1 n − s be the alternating zeta function. For a real number τ we define certain complex numbers b M , m (τ) and consider finite Dirichlet series υ M (τ , s) = ∑ m = 1 M b M , m (τ) m − s and η N (τ , s) = ∑ M = 1 N υ M (τ , s). Computations demonstrate some remarkable properties of these finite Dirichlet series, but nothing was supported by a proof so far. First, numerical data show that η N (τ , s) approximates η (s) with high accuracy for s in the vicinity of 1 / 2 + i τ ; this allows one to surmise that (*) η (s) = ∑ M = 1 ∞ υ M (τ , s). Moreover, it looks that lim M → ∞ ⁡ m υ M (τ , 1 − σ + i t) ‾ m υ M (τ , σ + i t) = η (σ + i t) m η (1 − σ + i t) ‾ m ; in other words, the individual summands in expected expansion (⁎) satisfy with an increasing accuracy a counterpart of the classical functional equation. Let ϒ M (τ , σ + i t) = υ M (τ , σ + i t) / η (σ + i t). When M , τ , and either σ or t are fixed, and the fourth parameter varies, the plot of ϒ M (τ , σ + i t) on the complex plane contains numerous almost ideally circular arcs with geometrical parameters closely related to the non-trivial zeros of the zeta function. [ABSTRACT FROM AUTHOR]