The notes that readers can find below attempt to follow an empirical path to gather a collection of clues about Riemann’s hypothesis. The main direction of this route is to try to achieve heuristic ¹ proof of the existence of a bijection between Riemann non-trivial zeros and prime numbers.

For this purpose, one can write the following facts:

The Riemann hypothesis might be described such that all the non-trivial zeros of the Riemann zeta function are to be found in the critical line L, defined by:

$\forall a\in {ℝ}^{+}:z=\frac{1}{2}+ia\in \text{L}$;(2)

that is, according to this hypothesis, one can write the structure of the set ${\text{Z}}_R$ of non-trivial zeros of the Riemann zeta function using:

$\exists z_I\in {\text{Z}}_R\subset ℂ:\zeta \left(z_I\right)=0\Rightarrow z_I=\frac{1}{2}\pm i{a}_I\wedge a_I\in \text{A}\subset {ℝ}^{+}$. (3)

1) If the set ${\text{Z}}_R$, identifies the set of complex non-trivial zeros of the Riemann function, then if the Riemann Hypothesis holds, one can also write: ${\text{Z}}_R\subset \text{L}$.

2) The positive definite real set A, defined in the Equation (3), could be called the set of the Riemann function's auxiliary (non-trivial) zeros.

3) The non-trivial zeros as written in the Equation (3) are observed in pairs, as the complex conjugate of a non-trivial zero is also a zero. That is, the following expression holds:

$\forall z_I\in {\text{Z}}_R:\zeta \left(z_I\right)=0\Rightarrow \exists z_I^{\ast }\in {\text{Z}}_R\to \zeta \left(z_I^{\ast }\right)=0$, (4)

4) Here, one could consider the role of the set of auxiliary non-trivial zeros A, which, as a real set, is invariant upon conjugation of the ${\text{Z}}_R$ elements. To contemplate this, and taking into account the nature of the complex zeros, the set M of positive modules of the zeros will be used instead:

$\forall z_I\in {\text{Z}}_R:m_I=\left|z_I\right|=+{\left|\frac{1}{4}+a_I^2\right|}^{\frac{1}{2}}\to m_I\in \text{M}\subset {ℝ}^{+}$.(5)

Heuristic computational evidence [11] - [13] that many non-trivial zeros exist, all within the critical line L.

Thus, one can assume there is already a piece of computational heuristic evidence in the literature pointing towards an infinite number of Riemann function non-trivial zeros in L.

It has been proven in various ways, as seen in [14] , starting with Euclid [15] [16] , that the number of prime natural numbers is infinite. Such affirmation constitutes the so-called prime number theorem. Several proofs of the theorem can be invoked and are available from various sources, as explained in the well-structured book by Derbyshire [4] .

Let us name P the set of primes. The set P can be assumed as an ordered set, that is:

$\text{P}=\left\{p_1;p_2;p_3;\cdots ;p_N;\cdots \right\}\subset \mathbb{N}\to p_1<p_2<p_3<\cdots <p_N<\cdots$,(6)

being $\mathbb{N}$ the set of natural numbers.

If evidence exists, even empirical ² , showing that a bijective map between the set of prime numbers P and the Riemann zeta function non-trivial zeros is present, then one can write:

$\begin{array}{l}\forall z_I:\zeta \left(z_I\right)=0\to z_I\in {\text{Z}}_R\wedge \forall p_I\in \text{P}\\ \Rightarrow {\phi }_Z:{\text{Z}}_R\to \text{P}\wedge {\phi }_Z^{-1}:\text{P}\to {\text{Z}}_R\end{array}$ (7)

However, one can also consider that the positive real set M forms the critical line L. Then alternatively, one can also write, if the previous bijective map in the Equation (7) exists, that there also might exist a bijective map between the sets M and P, like:

$\phi :\text{M}\to \text{P}\wedge {\phi }^{-1}:\text{P}\to \text{M}$ (8)

One must note that one can consider the set of Riemann function auxiliary zeros A as an ordered set, an order transmitted to the set of the modules of the zeros M. That is:

$\text{A}=\left\{a_1;a_2;a_3;\cdots ;a_N;\cdots \right\}\subset {ℝ}^{+}\to a_1<a_2<a_3<\cdots <a_N<\cdots$

and therefore

$\text{M}=\left\{m_1;m_2;m_3;\cdots ;m_N;\cdots \right\}\subset {ℝ}^{+}\to m_1<m_2<m_3<\cdots <m_N<\cdots$ (9)

1) One can also hypothesize that a trivial correspondence between the ordered sets M and P might exist.

Then, expressing such a correspondence using the diagonal Cartesian product, which is defined now as:

$\begin{array}{l}\text{M}=\left\{m_I|I=1,2,3,\cdots \right\}\wedge \text{P}=\left\{p_I|I=1,2,3,\cdots \right\}\Rightarrow \\ {\text{F}}_2=\text{M}\ast \text{P}=\left\{f_I=\left(a_I;p_I\right)|I=1,2,3,\cdots \right\}\end{array}$ (10)

2) The described mathematical construction might be applied to any pair of ordered sets. According to the definition (10), the diagonal Cartesian product differs from the usual definition of the Cartesian product. Diagonal product tuples are accepted as elements if they hold a pair of elements of the implied sets containing the same ordering numeral. It is also not difficult to see that the diagonal Cartesian product contains the underlying structure of the so-called inward product of two vectors, for example, [17] , equivalent to the so-called Hadamard or diagonal matrix product.

3) As defined in the Equation (10), the resultant diagonal Cartesian product set ${\text{F}}_{\text{2}}=\text{M}\ast \text{P}$, contains ordered pairs of elements, which can supposedly constitute an initial set of two-dimensional points. These points could later be associated with an implicit form of the map $\phi$ in the equation . Meanwhile, the reversed diagonal Cartesian product set ${\text{G}}_{\text{2}}=\text{P}\ast \text{M}$, can be associated with the inverse map ${\phi }^{-1}$.

The empirical evidence of the existence of some diagonal Cartesian product connected with a bijective map $\phi$ between the sets M and P can be easily observed when plotting the points of the diagonal Cartesian product between the sets M and P. Plotting the points of ${\text{F}}_{\text{2}}=\text{M}\ast \text{P}$ presents $\phi :\text{M}\to \text{P}$ and doing the same with ${\text{G}}_{\text{2}}=\text{P}\ast \text{M}$ a representation of ${\phi }^{-1}:\text{P}\to \text{M}$ is obtained.

1) Owing to the ordered nature of both M and P sets, one can draw the previous Cartesian products in a three-dimensional fashion using, as shown in Figure 1 , the modules of the zeros in the Z-axis, the order of natural numbers in the Y-axis, and the ordered prime numbers in the X-axis. Also, Figure 2 shows a similar arrangement but interchanges the X-axis’s role with the Z-axis. Both Figures are constructed using the graphical tool of reference [18] . The increasing size of the points in both figures indicates a greater closeness of the points to the observer in the plot.

2) Such three-dimensional visualizations can be associated with an extension of the two-dimensional diagonal Cartesian products as defined in the equation (10), but now constructed, using the ordered set of natural numbers $\mathbb{N}$, as a new coordinate:

${\text{F}}_{\text{3}}=\text{M}\ast \mathbb{N}\ast \text{P}\wedge {\text{G}}_{\text{3}}=\text{P}\ast \mathbb{N}\ast \text{M}$(11)

Thus, as shown in Figure 1 and Figure 2 below, an empirical form representing the diagonal Cartesian product exists, which can be associated and visualized like a bijective map between the sets M and P.

This indicates that one can safely consider the number of non-trivial Riemann zeta function zeros on the line L, which might be connected in a one-to-one correspondence with the prime numbers P.

Figure 1 shows a shape resembling the positive branch of the hyperbolic sine, while Figure 2 acquires the form of the positive branch of the hyperbolic tangent.

Consequently, considering the bijection between M and P, the number of non-trivial zeros of the Riemann function could be considered heuristically infinite.

Therefore, if the Riemann function’s non-trivial zeros set ${\text{Z}}_R$ is shown to be heuristically infinite, there is no need to consider that some could be placed outside the critical line L.

It is sufficient to contemplate the existence of the diagonal Cartesian product associated with the bijective map, involving every module of the Riemann function non-trivial zero in ${\text{Z}}_R$ with every prime number in P.

However, one can obtain a non-linear correlation between the elements of the sets ${\text{Z}}_R$ and P in the way recently described [19] and already used [20] to find a relation between the cardinality of prime numbers and Mersenne numbers.

The specific functional form of the plot, which has to involve statistical gear if calculated, will be just briefly resumed here when no order variable is included, leaving a deeper study for further insight, if necessary. The function chosen is of the form:

$y^{\nu }=a{x}^{\mu }+b$,

and numerical results are obtained with the first $N=128$ primes and auxiliary zeros of the Riemann function. It is interesting to note that a better correlation in the statistical sense:

$\left(r=0.9998|a=1.0135|\mu =1.336|b=-0.01301|\nu =0.8668\right)$ (12)

has been found when the roles of prime numbers and auxiliary zeros of the Riemann function are $\left\{y,x\right\}$; while a lower correlation index is found when the roles are reversed as $\left\{x,y\right\}$:

$\left(r=0.9915|a=0.8297|\mu =0.5871|b=0.2051|v=0.5871\right)$. (13)

Such a result seems coherent with intuition because representing natural numbers with real numbers yields a better fit than representing real numbers with a set of natural numbers.

However, in both cases, the results shown in Equations (12) and (13) are good heuristic evidence of a non-linear relation between prime numbers and Riemann function zeros, at least for these 128 dimensions.

However, it is unclear whether augmenting the dimensions of the fittings and plots beyond higher dimensions will result in a random succession different from the hyperbolic-like branches shown in Figure 1 and Figure 2 or the close non-linear relationships in Equations (12) and (13).