A geometric approach to the Riemann Hypothesis

Luca · #1 $Old$ November 3rd, 2009, 04:11 AM

Hi,
I am trying to verify with anybody generally intrigued by the Riemann Hypothesis whether they might find of any interest a geometrical approach I have developed. It allows to derive a stricter asymptotic expression for the $n^{th}$ Remainder, $R_n(\sigma+it)$ , of the Alternating Ordinary Dirichlet Series (strictly related to the Riemann Zeta Function), then resulting in a possibly novel Reformulation of The Riemann Hypothesis.

Honestly, I do not currently know whether said equivalent hypothesis could represent an easier challenge than the Riemann Hypothesis, or a more difficult one, or whether it would simply turn the RH into an equally difficult task. But I would be happy to have some feedback by other interested readers.

In fact, by means of standard analysis it is easy to show that
$R_n(\sigma + it) = O(n^{-\sigma}) \;\; as \;\; n\rightarrow\infty.$
However, the described geometric approach allows to derive a stricter asymptotic condition, namely
$R_n(\sigma + it) = \Theta(n^{-\sigma}) \;\; as \;\; n\rightarrow\infty$ (see Big-Theta definition at the end of pg. 6).

The fact that $R_n(\sigma + it) = \Theta(n^{-\sigma}) \;\; as \;\; n\rightarrow\infty$ , imposes then a very specific condition on the possible values of the limit of a certain ratio of partial sums of said Alternating Ordinary Dirichlet Series, finally resulting in the equivalent hypothesis summarised in the following Abstract

For any $s \in \mathbb{C}$ with $\Re(s)>0$ , denote by $S_n(s)$ the $n^{th}$ partial sum of the alternating Dirichlet series $1-2^{-s}+3^{-s}-\cdots$ . We first show that $S_n(s)\neq 0$ for all n greater than some index N(s). Denoting by $D=\left\{s \in \mathbb{C}: \; 0< \Re(s) < \frac{1}{2}\right\}$ the open left half of the critical strip, define for all $s\in D$ and n > N(s) the ratio $P_n(s) = S_n(1-s) / S_n(s)$ . We then prove that the limit $L(s)=\lim_{N(s)<n\to\infty}P_n(s)$ exists at every point s of the domain D. Finally, we show that the function L(s) is continuous on D if and only if the Riemann Hypothesis is true.

An effective way to visualize in one's mind the above result refers to fig.2 in the attached manuscript (downloadable also from arxiv.org/abs/0907.2426).

The limit function L(s) exists regardless of whether or not the the RH is true. If the RH is true, then L(s) is a continuous function and its modulus is the function plotted in Fig. 2. If the RH is not true, then L(s) still coincides with the function whose modulus is plotted in Fig. 2, excepts at the locations of the off-the-critical-line zeros, where it will feature discontinuities L(s) = 0.

Fig. 9 gives then an idea of how said hypothetical off-the-critical-line zeros would affect the convergence pattern of said ratios Pn(s).

Thanks for your kind attention,

Luca