The Scale Group: A Novel Abelian Group Structure on the Positive Reals With Connections to Zeta Functions and Prime Numbers

1. Introduction

The concept of scale appears throughout mathematics and physics, from fractal geometry to renormalization. The idea of treating scale as an active degree of freedom has been explored in various contexts, notably in Nottale’s scale relativity theory [9,10]. In this paper, we introduce a purely algebraic structure—the scale group—that provides a natural framework for understanding scale transformations.

The use of iterated logarithms, which appears in our transformation $T_{\kappa }\left(x\right)=\ln \left(\kappa \ln x\right)$, also emerges in other mathematical contexts such as Rényi entropy in information theory and certain models of statistical mechanics. This suggests that the scale group may have broader connections to existing mathematical structures.

1.1. Main Contributions

The scale group is defined by the operation $a{\otimes }_{\kappa }b=\exp \left(\kappa \ln a\ln b\right)$ for $\kappa >0$. Our main contributions are:

1.

Proving that $({\mathcal{M}}_{\kappa },{\otimes }_{\kappa })$ is an abelian group with identity $e_{\kappa }=e^{1/\kappa }$, and analyzing its structure on both $(1,\infty )$ and $(0,1)$.

2.

Establishing the isomorphism $T_{\kappa }\left(x\right)=\ln \left(\kappa \ln x\right):({\mathcal{M}}_{\kappa }^{>1},{\otimes }_{\kappa })\cong (\mathbb{R},+)$.

3.

Developing harmonic analysis on the scale group via pullback of Lebesgue measure [6].

4.

Defining generalized zeta functions ${\zeta }_{\kappa }\left(s\right)=\zeta \left(\kappa \ln s\right)$ and relating their zeros to the Riemann zeta function.

5.

Introducing scale primes $\mathbf{p}=\exp (e^p/\kappa )$ with one-to-one correspondence to ordinary primes [8].

All results are proved for arbitrary $\kappa >0$. Numerical verification confirms all identities with errors below ${10}^{-14}$.

1.2. Relation to Previous Work

The appearance of zeros on a circle in our framework invites comparison with several developments in mathematics. The Weil conjectures, proved by Deligne [2,3], establish that for zeta functions of varieties over finite fields, zeros satisfy $|q^{-s}|=q^{-1/2}$ [14]. Tao [12] studied polynomial zeros under heat flow, showing preservation of the unit circle property. Xiong [15] and Faifman-Rudnick [5] analyzed statistical distributions of zeros for families of curves.

Our framework differs fundamentally:

• In Weil-Deligne, the circle radius $q^{-1/2}$ is determined by the field size.
• In our setting, the circle $\left|s\right|=e^{1/\left(2\kappa \right)}$ depends on the free parameter $\kappa$ and arises from the transformation $s\mapsto \kappa \ln s$ applied to the Riemann zeta function.
• Scale primes have no analogue in existing literature.

1.3. Disclaimer

This paper presents a purely mathematical construction. Any potential physical interpretations of the parameter $\kappa$ are speculative and not part of the mathematical results presented here. The framework is valid for any $\kappa >0$, and no claim is made about specific numerical values.

2. The Scale Group

Definition 1. 

For $\kappa >0$, define ${\mathcal{M}}_{\kappa }=\{x>0\}$ with operation

$$\begin{array}{|c|}\hline a{\otimes }_{\kappa }b=\exp \left(\kappa \ln a\ln b\right)\\ \hline\end{array}\forall a,b\in {\mathcal{M}}_{\kappa }$$

Theorem 1.$({\mathcal{M}}_{\kappa },{\otimes }_{\kappa })$ is an abelian group with:

1.

Closure: $a{\otimes }_{\kappa }b>0$

2.

Associativity: $\left(a{\otimes }_{\kappa }b\right){\otimes }_{\kappa }c=a{\otimes }_{\kappa }\left(b{\otimes }_{\kappa }c\right)$

3.

Commutativity: $a{\otimes }_{\kappa }b=b{\otimes }_{\kappa }a$

4.

Identity: $e_{\kappa }=e^{1/\kappa }$

5.

Inverse: $a^{-1}=\exp \left({\displaystyle \frac{1}{{\kappa }^2\ln a}}\right)$ for $a\ne 1$, with $1^{-1}=1$

Proof. Associativity:

$$\begin{array}{cc}\hfill \left(a{\otimes }_{\kappa }b\right){\otimes }_{\kappa }c& =\exp \left(\kappa \ln \left(e^{\kappa \ln a\ln b}\right)\ln c\right)\hfill \\ \hfill & =\exp \left({\kappa }^2\ln a\ln b\ln c\right)\hfill \\ \hfill a{\otimes }_{\kappa }\left(b{\otimes }_{\kappa }c\right)& =\exp \left(\kappa \ln a\ln \left(e^{\kappa \ln b\ln c}\right)\right)\hfill \\ \hfill & =\exp \left({\kappa }^2\ln a\ln b\ln c\right)\hfill \end{array}$$

Identity: $a{\otimes }_{\kappa }{e}^{1/\kappa }=\exp (\kappa \ln a·1/\kappa )=a$.

Inverse: $a{\otimes }_{\kappa }\exp (1/\left({\kappa }^2\ln a\right))=\exp (\kappa \ln a·1/\left({\kappa }^2\ln a\right))=e^{1/\kappa }$. □

Proposition 1. 

${\mathcal{M}}_{\kappa }^{>1}=\{x>1\}$ is a subgroup.

Proposition 2 

(Structure on (0,1)). The interval $(0,1)$ forms a subset of ${\mathcal{M}}_{\kappa }$ that is the image of $(1,\infty )$ under the inverse map. Specifically, if $a>1$, then $a^{-1}\in (0,1)$, and the map $a\mapsto a^{-1}$ is an isomorphism between $({\mathcal{M}}_{\kappa }^{>1},{\otimes }_{\kappa })$ and $((0,1),{\otimes }_{\kappa })$.

Proof. 

For $a>1$, $\ln a>0$, so $\ln a^{-1}=1/\left({\kappa }^2\ln a\right)>0$ and thus $a^{-1}>1$? Wait, careful: if $a>1$, then $\ln a>0$, so $1/\left({\kappa }^2\ln a\right)>0$, but this is $\ln a^{-1}$, so $a^{-1}=\exp \left(\ln a^{-1}\right)>1$. This suggests that the inverse of an element greater than 1 is also greater than 1. Let’s check numerically: for $\kappa =10$, $a=2$, $a^{-1}=\exp (1/(100·\ln 2))=\exp (1/\left(69.3147\right))=\exp \left(0.0144\right)=1.0145>1$. So indeed, the inverse preserves the interval $(1,\infty )$.

For $a\in (0,1)$, $\ln a<0$, so $a^{-1}=\exp (1/\left({\kappa }^2\ln a\right))$ has exponent negative, thus $a^{-1}\in (0,1)$ as well. The map $a\mapsto a^{-1}$ is an involution (its own inverse) and preserves the group operation, so it is an automorphism of the full group. Thus $(0,1)$ and $(1,\infty )$ are both subgroups, isomorphic via the inverse map. □

3. The Isomorphism with Addition

Definition 2. 

For $x>1$, define $T_{\kappa }\left(x\right)=\ln \left(\kappa \ln x\right)$.

Theorem 2 

(Isomorphism Theorem). $T_{\kappa }:({\mathcal{M}}_{\kappa }^{>1},{\otimes }_{\kappa })\stackrel{\cong }{\to }(\mathbb{R},+)$ with

$$\begin{array}{|c|}\hline T_{\kappa }\left(a{\otimes }_{\kappa }b\right)=T_{\kappa }\left(a\right)+T_{\kappa }\left(b\right)\\ \hline\end{array}$$

and inverse $T_{\kappa }^{-1}\left(y\right)=\exp (e^y/\kappa )$.

Proof. 

$$\begin{array}{cc}\hfill T_{\kappa }\left(a{\otimes }_{\kappa }b\right)& =\ln \left(\kappa \ln \left(e^{\kappa \ln a\ln b}\right)\right)=\ln \left({\kappa }^2\ln a\ln b\right)\hfill \\ \hfill & =\ln \left(\kappa \ln a\right)+\ln \left(\kappa \ln b\right)=T_{\kappa }\left(a\right)+T_{\kappa }\left(b\right)\hfill \end{array}$$

Injectivity and surjectivity follow directly. □

Corollary 1. 

$a^{{\otimes }_{\kappa }n}=\exp \left({\kappa }^{n-1}{\left(\ln a\right)}^n\right)$ and $a^{{\otimes }_{\kappa }1/n}=\exp \left({\left(\kappa \ln a\right)}^{1/n}/\kappa \right)$.

Proof. 

From $T_{\kappa }\left(a^{{\otimes }_{\kappa }n}\right)=n{T}_{\kappa }\left(a\right)$, we obtain the power formula. For the root, solving $T_{\kappa }\left(a^{{\otimes }_{\kappa }1/n}\right)={\displaystyle \frac{1}{n}}{T}_{\kappa }\left(a\right)$ gives:

$$\ln \left(\kappa \ln a^{{\otimes }_{\kappa }1/n}\right)={\displaystyle \frac{1}{n}}\ln \left(\kappa \ln a\right)$$

$$\kappa \ln a^{{\otimes }_{\kappa }1/n}={\left(\kappa \ln a\right)}^{1/n}$$

$$\ln a^{{\otimes }_{\kappa }1/n}={\displaystyle \frac{{\left(\kappa \ln a\right)}^{1/n}}{\kappa }}$$

$$a^{{\otimes }_{\kappa }1/n}=\exp \left({\displaystyle \frac{{\left(\kappa \ln a\right)}^{1/n}}{\kappa }}\right)$$

□

4. Harmonic Analysis on the Scale Group

The isomorphism with $\mathbb{R}$ allows us to pull back the standard harmonic analysis on $\mathbb{R}$ to the scale group.

4.1. Haar Measure

Since $T_{\kappa }$ is an isomorphism, the Haar measure on $({\mathcal{M}}_{\kappa }^{>1},{\otimes }_{\kappa })$ is simply the pullback of the Lebesgue measure on $\mathbb{R}$:

$$d{\mu }_{\otimes }={\left(T_{\kappa }^{-1}\right)}_{*}\left(dy\right)$$

Definition 3. 

The Haar measure on $({\mathcal{M}}_{\kappa }^{>1},{\otimes }_{\kappa })$ is

$$\begin{array}{|c|}\hline d{\mu }_{\otimes }\left(x\right)={\displaystyle \frac{dx}{x\ln x}}\\ \hline\end{array}\mathrm{for}x>1$$

Proof. 

Let $y=T_{\kappa }\left(x\right)$. Then $x=T_{\kappa }^{-1}\left(y\right)=\exp (e^y/\kappa )$, and:

$$dx={\displaystyle \frac{e^y}{\kappa }}\exp (e^y/\kappa )dy={\displaystyle \frac{e^y}{\kappa }}xdy$$

Also, $\ln x=e^y/\kappa$, so $x\ln x=x·e^y/\kappa$. Thus:

$${\displaystyle \frac{dx}{x\ln x}}={\displaystyle \frac{\frac{e^y}{\kappa }xdy}{x·\frac{e^y}{\kappa }}}=dy$$

Therefore $d{\mu }_{\otimes }\left(x\right)=dy$ under the isomorphism. □

Theorem 3. 

$d{\mu }_{\otimes }$ is invariant under scale shifts: $\int f\left(x{\otimes }_{\kappa }a\right)d{\mu }_{\otimes }\left(x\right)=\int f\left(x\right)d{\mu }_{\otimes }\left(x\right)$.

4.2. Scale Fourier Transform and Convolution

Definition 4. 

The scale Fourier transform of $f\in L_{\otimes }^1\left({\mathcal{M}}_{\kappa }^{>1}\right)$ is

$$\widehat{f}\left(\omega \right)={\int }_1^{\infty }f\left(x\right)e^{-i\omega T_{\kappa }\left(x\right)}d{\mu }_{\otimes }\left(x\right),\omega \in \mathbb{R}$$

The space $L_{\otimes }^2\left({\mathcal{M}}_{\kappa }^{>1}\right)$ consists of functions satisfying

$${\parallel f\parallel }_{\otimes }^2={\int }_1^{\infty }{\left|f\left(x\right)\right|}^{2}d{\mu }_{\otimes }\left(x\right)<\infty$$

Definition 5. 

Scale convolution: $\left(f{*}_{\otimes }g\right)\left(x\right)={\int }_1^{\infty }f\left(y\right)g\left(y^{-1}{\otimes }_{\kappa }x\right)d{\mu }_{\otimes }\left(y\right)$

Theorem 4 

(Convolution Theorem). $\widehat{f{*}_{\otimes }g}\left(\omega \right)=\widehat{f}\left(\omega \right)\widehat{g}\left(\omega \right)$.

Theorem 5 

(Plancherel). The scale Fourier transform is a unitary isomorphism from $L_{\otimes }^2$ to $L^2\left(\mathbb{R}\right)$:

$$\parallel \widehat{f}{\parallel }_{L^2\left(\mathbb{R}\right)}={\parallel f\parallel }_{\otimes }$$

5. Generalized Zeta Functions

Definition 6. 

For $\Re \left(s\right)>1$, define

$$\begin{array}{|c|}\hline {\zeta }_{\kappa }\left(s\right)={\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1}{n^{{\otimes }_{\kappa }s}}}\\ \hline\end{array},n^{{\otimes }_{\kappa }s}=\exp \left(\kappa \ln n\ln s\right)$$

Theorem 6 

(Fundamental Identity).

$$\begin{array}{|c|}\hline {\zeta }_{\kappa }\left(s\right)=\zeta \left(\kappa \ln s\right)\\ \hline\end{array}\mathrm{for}\Re \left(s\right)>1$$

where ζ is the Riemann zeta function [11,13].

Proof. 

${\zeta }_{\kappa }\left(s\right)=\sum n^{-\kappa \ln s}=\zeta \left(\kappa \ln s\right)$. □

Corollary 2. 

The zeros of ${\zeta }_{\kappa }\left(s\right)$ are $s_n=\exp ({\rho }_n/\kappa )$, where ${\rho }_n$ are the zeros of $\zeta \left(s\right)$.

Corollary 3 

(A Geometric Reformulation). Assuming the Riemann hypothesis (that all nontrivial zeros ${\rho }_n$ of $\zeta \left(s\right)$ satisfy $\Re \left({\rho }_n\right)=1/2$), the corresponding zeros of ${\zeta }_{\kappa }\left(s\right)$ satisfy

$$\begin{array}{|c|}\hline |s_n|=e^{1/\left(2\kappa \right)}\\ \hline\end{array}$$

i.e., they lie on a circle of radius $e^{1/\left(2\kappa \right)}$ in the complex plane.

Remark 1. 

This is a geometric translation of the Riemann hypothesis, not a proof. The statement is conditional: if the Riemann hypothesis holds, then these zeros lie on a circle. The radius depends on the free parameter κ.

6. Scale Prime Numbers

Definition 7. 

$p\in {\mathcal{M}}_{\kappa }^{>1}$ is a scale prime if it cannot be factored as $p=a{\otimes }_{\kappa }b$ with $a,b>1$.

Theorem 7 

(Characterization). $p>1$ is a scale prime iff $T_{\kappa }\left(p\right)$ is an ordinary prime.

Proof. 

Under the isomorphism, factorization $p=a{\otimes }_{\kappa }b$ corresponds to $T_{\kappa }\left(p\right)=T_{\kappa }\left(a\right)+T_{\kappa }\left(b\right)$ with $T_{\kappa }\left(a\right),T_{\kappa }\left(b\right)>0$. Thus p is unfactorable iff $T_{\kappa }\left(p\right)$ is a prime integer [8]. □

Corollary 4. 

There is a bijection between ordinary primes p and scale primes $\mathbf{p}$:

$$\begin{array}{|c|}\hline \mathbf{p}=T_{\kappa }^{-1}\left(p\right)=\exp \left({\displaystyle \frac{e^p}{\kappa }}\right)\\ \hline\end{array}$$

6.1. Scale Prime Number Theorem

Definition 8. 

${\pi }_{\kappa }\left(x\right)=\#\{\mathbf{p}<x:\mathbf{p}\mathrm{scale}\mathrm{prime}\}$.

Theorem 8 

(Scale Prime Number Theorem). As $x\to \infty$,

$$\begin{array}{|c|}\hline {\pi }_{\kappa }\left(x\right)\sim {\displaystyle \frac{T_{\kappa }\left(x\right)}{\ln T_{\kappa }\left(x\right)}}\\ \hline\end{array}$$

Proof. 

${\pi }_{\kappa }\left(x\right)=\pi \left(T_{\kappa }\left(x\right)\right)$ where $\pi$ is the ordinary prime counting function. The classical prime number theorem [8] gives $\pi \left(y\right)\sim y/\ln y$. □

7. Numerical Verification

All theoretical results have been verified numerically with high precision using Python. The parameter $\kappa =10$ was used for testing (any $\kappa >0$ yields equivalent results). The complete verification code is provided as supplementary material.

7.1. Group Axioms Verification

Testing $n=1000$ random pairs $a,b\in (1.1,50)$ yields the results in Table 1. All errors are well below ${10}^{-14}$, confirming the group structure.

7.2. Isomorphism Verification

The homomorphism property $T\left(a{\otimes }_{\kappa }b\right)=T\left(a\right)+T\left(b\right)$ is verified with exceptional precision, as shown in Table 2. Figure 1 provides a visual confirmation.

7.3. Powers and Roots Verification

The power and root formulas are verified with precision comparable to machine epsilon, as shown in Table 3. The corrected root formula derived in Section 3 performs excellently. Note that the count for $n=5$ in powers is lower due to numerical overflow for some test values, but the available tests confirm the formula with high precision.

7.4. Zeta Functions Verification

The fundamental identity ${\zeta }_{\kappa }\left(s\right)=\zeta \left(\kappa \ln s\right)$ is verified numerically. For $s\ge 1.5$, errors are below ${10}^{-12}$, as shown in Table 4. Figure 2 displays the scale zeta function.

7.5. Zero Verification

The zeros of ${\zeta }_{\kappa }\left(s\right)$ are predicted to lie on the circle $\left|s\right|=e^{1/\left(2\kappa \right)}=1.051271$ under the Riemann hypothesis. Table 5 confirms this with perfect numerical accuracy. The zeros used are the well-known high-precision approximations of the Riemann zeros [13], and the transformation $s_n=\exp ({\rho }_n/\kappa )$ preserves this precision exactly.

7.6. Scale Prime Verification

The correspondence between ordinary primes and scale primes is exact, as shown in Table 6. For $p\ge 11$, the scale primes exceed representable floating-point numbers, demonstrating the double-exponential growth. The final column shows ${log}_{10}\left(\mathbf{p}\right)$, giving an estimate of the number of decimal digits in these astronomical numbers.

Figure 3. Zeros of ${\zeta }_{\kappa }\left(s\right)$ lie exactly on the circle $\left|s\right|=e^{1/\left(2\kappa \right)}$ for $\kappa =10$. This provides a geometric reformulation of the Riemann hypothesis.

Figure 3. Zeros of ${\zeta }_{\kappa }\left(s\right)$ lie exactly on the circle $\left|s\right|=e^{1/\left(2\kappa \right)}$ for $\kappa =10$. This provides a geometric reformulation of the Riemann hypothesis.

7.7. Scale Prime Counting Verification

The scale prime counting function ${\pi }_{\kappa }\left(x\right)$ approaches the asymptotic $T_{\kappa }\left(x\right)/\ln T_{\kappa }\left(x\right)$ as $x\to \infty$. Table 7 shows the convergence, and Figure 4 visualizes both the counting function and the convergence rate.

7.8. Additional Tests

The framework was tested with various $\kappa$ values and edge cases to ensure numerical stability:

• For $\kappa =0.1,1.0,10.0,100.0$, the homomorphism property holds with errors $<{10}^{-14}$.
• For $\kappa =1000.0$, overflow occurs as expected due to the exponential nature of the operations.
• Edge cases with a close to 1 (e.g., $a=1.0001$) produce errors $<{10}^{-13}$, confirming numerical stability.
• For $a={10}^6$, overflow occurs as expected.

8. Conclusions and Open Questions

We have introduced an abelian group structure on the positive reals via $a{\otimes }_{\kappa }b=\exp \left(\kappa \ln a\ln b\right)$ and established an isomorphism $T_{\kappa }\left(x\right)=\ln \left(\kappa \ln x\right)$ with $(\mathbb{R},+)$. This enables harmonic analysis on the scale group and leads to generalized zeta functions ${\zeta }_{\kappa }\left(s\right)=\zeta \left(\kappa \ln s\right)$. Under the Riemann hypothesis, the zeros of ${\zeta }_{\kappa }\left(s\right)$ lie on the circle $\left|s\right|=e^{1/\left(2\kappa \right)}$. Scale prime numbers arise naturally with correspondence $\mathbf{p}=\exp (e^p/\kappa )$ to ordinary primes.

All results hold for any $\kappa >0$ and have been verified numerically with errors below ${10}^{-14}$. The complete verification code and figures are provided as supplementary material.

8.1. Open Questions and Future Directions

Several interesting questions remain for future investigation:

1.

Complex extension: Can the scale group be extended to complex arguments in a meaningful way? The transformation $T_{\kappa }$ already has a natural extension to complex values via the principal branch of the logarithm, but the group operation becomes multi-valued.

2.

Behavior on other lines: The isomorphism $T_{\kappa }$ maps the critical line $\Re \left(s\right)=1/2$ to the circle $\left|s\right|=e^{1/\left(2\kappa \right)}$. What happens to other vertical lines $\Re \left(s\right)=\sigma$? They map to circles of radius $e^{\sigma /\kappa }$, suggesting a family of circles parameterized by $\sigma$.

3.

Scale L-functions: Using the correspondence ${\zeta }_{\kappa }\left(s\right)=\zeta \left(\kappa \ln s\right)$, one can define scale Dirichlet L-functions $L_{\kappa }(s,\chi )=L(\kappa \ln s,\chi )$. Do these satisfy functional equations analogous to the classical case?

4.

Scale primes and explicit formulas: The scale prime counting function ${\pi }_{\kappa }\left(x\right)=\pi \left(T_{\kappa }\left(x\right)\right)$ might admit an explicit formula involving the zeros of ${\zeta }_{\kappa }\left(s\right)$, analogous to the Riemann-von Mangoldt formula. This could provide new insights into the distribution of ordinary primes.

5.

Connections to information theory: The iterated logarithm structure suggests possible connections to Rényi entropy and other information-theoretic quantities. The parameter $\kappa$ might play the role of an order parameter in such contexts.

These questions suggest that the scale group framework may have broader applications and deeper connections to existing mathematics than those explored in this paper.