On the Riemann Hypothesis

It is a common method of mathematics to try to get insight from the simpler cases of a problem. In case of the Riemann Hypothesis our only such hope is the Basel problem. 

Euler's solution can be generalized, but those I have tried become useless rather fast. 

Another solution of the Basel problem seems general enough to be possible to be applied to higher powers: the solution using Parseval's identity (theorem). Using Parseval's identity one can find the values of \(\zeta\) for all even integers.

Why just don't we apply the same to \(s=3\), and find odd powers as well?. The reason boils down to the fact that \(L^p\) is Hilbert iff \(p=2\). The proof relies on the Parallelogram law in turn, which is founded on the notion of inner product space. 

An inner product is defined for two vectors only [this is related to the two values of binary logic but here is not the place to pursue that]. 

Why? Sentimental reasons can be provided, such as `logic is binary', etc. but there is no logical necessity. One should in principle be able to define inner product for any number of vectors. In particular, for  \(s=3\) one can define $$\langle\cdot,\cdot,\cdot\rangle: V\times V\times V\to \mathbb{R}$$ by

$$\langle u,v,w\rangle=\sum_i u_i v_i w_i.$$

One can create differential geometry accordingly:

$$ds^3=g_{abc}dx^a dx^b dx^c.$$

Next step, we should create an analogue Fourier analysis. Important problems arise here, such as `why don't we have triply-periodic functions' [not the usual answer], etc. 

Fourier analysis is based on the Hilbert inner product of two functions

$$\langle f,g\rangle=\int_a^b f(x)\bar{g}(x)\ dx.$$

This requires two tools

1. Generalization/analogue of complex conjugation, which in turn faces us with the old problem of `triplets',
2. Inner product of three functions.

The second we already have, but for the first we must define a product 

$$\star: V\times V\times V\to V,$$

such that our natural expectations are satisfied. We expect

1. $$(x_1,0,0)(x_2,0,0)(x_3,0,0)=(x_1 x_2 x_3,0,0)$$
2. $$(0,1,0)^3=(-1,0,0)$$
3. $$j^3=0, \quad j\neq 0 \Rightarrow (0,0,1)^3=(0,0,0)$$

We define basic trigonometric functions by the following system of ODEs

• $$\frac{d}{dz}\text{st} z=\text{ct} z\ \text{tt}z$$
• $$\frac{d}{dz}\text{ct} z=-\text{st} z\ \text{tt}z$$
• $$\frac{d}{dz}\text{tt} z=\text{st} z\ \text{ct}z$$