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My question is about some math objects (matrices, polynomials) and operators that satisfy a number of properties which can lead to a theory similar to PNT, RH, Dirichlet functions, abscissa of convergence, Mertens and related functions, and so on. I am wondering if the framework proposed below has been studied before, and I am looking for references. Also I am looking for interesting examples and whether what I discuss makes sense. The closest I could come up with is the Beurling generalized numbers, Beurling zeta function and PNT for Beurling numbers.

What I propose here is built in the exact same way as Beurling numbers, but involves matrices or functions rather than numbers. The construction is as follows.

Let $P_1,P_2, P_3$ and so on be a sequence of (say) matrices or polynomials, over the real or complex numbers, governed by standard operations such as addition, multiplication, and multiplication by a scalar. We require that the product is commutative, associative, and distributive with respect to addition. The neutral element for the product is denoted as $I$. If we are dealing with matrices, we require that we work with commutative matrices. For example, complex numbers represented as matrices, or matrices of the form $P_k=f_k(P_1)$ where $f_k$ is a (possibly infinite degree) polynomial, to guarantee commutativity.

We define the set $G$ as the set of all finite products of the form $P_1^{a_1} P_2^{a_2}\dots$ where $a_1, a_2\dots$ are integers $\geq 0$. Also, $I\in G$. The elements $P_1,P_2,\dots$ are called the primes of $G$. Two important properties are required:

  • The elements of $G$ can be ordered just like the natural numbers. The $k$-th element of $G$ is denoted as $g_k$, with $g_1=I$. I provide an example later in my question.

  • The factorization is unique. This requires the $P_k$'s to be carefully chosen.

Then we have the zeta function and Euler product

$$\zeta_G(s)\equiv \sum_{k=1}^\infty g_k^{-s} = \prod_{k=1}^\infty (1-P_k^{-s})^{-1}$$

where $s=\sigma+it$ is a complex number. It may or may not converge depending on $s$ and the choice of the $P_k$'s. The standard prime numbers, with $G$ being the set of natural numbers, is a particular case. So are Beurling numbers. Many standard arithmetic functions can be defined, for instance the Liouville function

$$L_G(s)=\sum_{k=1}^\infty (-1)^{\lambda(k)}g_k^{-s}=\prod_{k=1}^\infty (1+P_k^{-s})^{-1}=\frac{\zeta_G(2s)}{\zeta_G(s)},$$

where $\lambda(k)=1$ if $g_k$ has an even number of prime factors (counted with multiplicity) and $\lambda(k)=-1$ otherwise. Let us also define $$A(n)=\sum_{k=1}^n \lambda(k).$$ In the classic theory of the Riemann zeta function, the abscissa $\sigma_c$ of convergence for the function $L_G(s)$ is provided by the formula

$$\sigma_c = \lim\sup_{n\rightarrow\infty} \frac{|\log A(n)|}{\log n}.$$

The Riemann hypothesis is then equivalent to the conjecture that $\sigma_c=\frac{1}{2}$. A big question is how can this be extended to my framework. Of course $\sigma_c$ may vary depending on $G$, though it would be nice to find just one non-trivial example of $G$ for which $\sigma_c$ is know, and if possible, equal to $\frac{1}{2}$.

My question

Of course this involves various power series that need to be properly defined, for instance the principal value of the logarithm of (say) a matrix. For instance, if $g\in G$, $$\log (g^s) = s\sum_{k=1}^\infty (-1)^{k+1}\frac{(g-I)^k}{k}$$ if $g$ is close enough to $I$, and there are ways to extend it if $g$ is not close enough to $I$. I am wondering how to do it properly, whether my proposed theory is new (references are welcome), and about finding good examples where factorization is unique. What about the example discussed in the next section? Or an example where abscissa of convergence makes sense, and is known, for $L_G(s)$? What about the existence of a functional equation for $\zeta_G(s)$?

(a question not related to this post, for the future - so ignore it for the moment, but I think it's pretty exciting: is an element of $G$ the sum of four squares $g_1^2+g_2^2+g_3^2+g_4^2$ with $g_1,\dots,g_4 \in G$?)

Example 1

Let $A$ be a square matrix, and $P_k=\exp(\mu_k A)$. Assume $0 <\mu_1<\mu_2<\mu_3\dots$ and for $g\in G$, define $|g|$ as follows:

$$g\equiv P_1^{a_1} P_2^{a_2}\dots = \exp\Big[\Big(\sum_{k=1}^\infty a_k\mu_k\Big)\cdot A\Big] \Rightarrow |g|=\sum_{k=1}^\infty a_k\mu_k.$$ Now we have an order on $G$, with $g<g'$ if and only if $|g|<|g'|$. The order is strict if the $\mu_k$'s are linearly independent over the set of natural numbers. For instance, say $\mu_k = \log p_k$ where $p_k$ is the $k$-th prime number ($p_1=2$). If in addition $A=I=1$ is an $1\times 1$ matrix, then we are dealing with standard prime numbers, and $G$ is the set of natural numbers.

Another interesting thing, if $2I\in G$, is the following. In that case, $g\in G\Rightarrow 2g\in G$ and we can define the Dirichlet eta function as

$$\eta_G(s)\equiv\sum_{k=1}^\infty \delta_k\cdot g_k^{-s}=(1-2^{1-s})\zeta_G(s).$$

This is an algebraic continuation of $\zeta_G$ playing a role similar to that of an analytic continuation: extending the domain of convergence, hopefully in a unique way. Here $\delta_k=-1$ if $g_k$ has $2I$ as one of its prime factors, otherwise $\delta_k=1$. The algebraic continuation of $\zeta_G$ can be stated explicitly as

$$\zeta_G(s)=(1-2^{1-s})^{-1}\eta_G(s).$$

Example 2

Define $P_k$ as $P_k(x) \equiv x+\mu_k$, with $1<\mu_1<\mu_2<\dots$. In this case we are dealing with polynomials. Factorization in $G$ is de facto unique (unless I am missing something) and the product is commutative. One way to define an order on $G$ is as follows. If $g\in G$, then $g$ is a polynomial and $|g| \equiv g(1)$. Then $g<g'$ if and only if $|g|<|g'|$. Some conditions on the $\mu_k$'s are required for the order to be strict.

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    $\begingroup$ "The factorization is unique (note that for Beurling numbers, this is not a requirement)." That's wrong. Beurling's numbers are by definition a free commutative monoid on the (multi)set of primes, which has unique factorization. The associated norms of two elements might coincide, but the elements are still considered distinct. $\endgroup$
    – Wojowu
    Jun 6, 2021 at 20:40
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    $\begingroup$ A starting point is when the number of primes $P_1,P_2,\dots$ is finite. In that case things are much easier, but I consider this to be a trivial case. $\endgroup$ Jun 7, 2021 at 1:50
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    $\begingroup$ Selberg zeta function is taking closed geodesics on a Riemannian manifold instead of primes as input. My favourite reference is J. Elstrodt, F. J. Grunewald, J. Mennicke, “Groups Acting on Hyperbolic Space" $\endgroup$
    – Denis T.
    Jun 7, 2021 at 13:55
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    $\begingroup$ @Vincent Granville, from your posts in the past month or so, it seems you're very interested in the RH. The following link might be of interest to you: facebook.com/groups/229783678691987/permalink/315436670126687/… $\endgroup$
    – user257465
    Jun 9, 2021 at 9:38
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    $\begingroup$ @user257465 It's hard to say if you are serious or just being rude. $\endgroup$
    – efs
    Jun 15, 2021 at 2:27

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