We present a new approach to analytic number theory based on the "Prime Framework." This paper introduces the key ideas and proofs underlying the framework, which reconstructs classical results—from unique factorization to the Prime Number Theorem—using an axiomatic system. By embedding natural numbers into fiber algebras over a smooth manifold and enforcing a canonical representation via a coherence inner product, we define intrinsic primes and prove their unique factorization. Further, a specially constructed operator encodes the divisor structure of numbers, and its spectral analysis recovers the Euler product for the Riemann zeta function and leads to analytic results such as explicit prime-counting formulas and an internal analogue of the Riemann Hypothesis.
Traditional number theory assumes the existence of natural numbers as a given and builds its structure from that starting point. The Prime Framework, in contrast, constructs natural numbers internally by embedding them into a rich geometric–algebraic setting. This whitepaper gently introduces the fundamental ideas behind this approach and outlines the proofs that lead to a new derivation of classical results in number theory.
At the core of the Prime Framework are four interrelated axioms:
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Smooth Reference Manifold:
A smooth, connected, and orientable manifold ( M ) serves as the geometric backdrop. The manifold provides the necessary structure to support local algebraic data. -
Fiber Algebras:
For each point ( x ) in ( M ), there is an associated algebra ( C_x ) (typically a Clifford algebra). These algebras capture the local arithmetic structure and allow for the representation of numbers. -
Symmetry Group Action:
A Lie group ( G ) acts by isometries on ( M ) and lifts to ( C_x ). This symmetry ensures that local representations remain consistent under transformations, reinforcing the objectivity of the construction. -
Coherence Inner Product:
Each fiber algebra ( C_x ) is equipped with a G-invariant, positive-definite inner product. This inner product enforces a minimal-norm condition, selecting a unique, canonical representation (or embedding) for each number.
Together, these axioms provide a self-contained foundation from which numbers and their properties are derived.
(For further details, see the discussions in the foundational documents.)
Instead of assuming the existence of natural numbers, the Prime Framework constructs them via a universal number embedding:
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Multi-base Representation:
Every natural number ( N ) is represented in every base ( b \geq 2 ) by its digit sequence. These sequences are encoded into distinct graded components of the fiber algebra ( C_x ). -
Canonical Embedding:
The coherence inner product forces all these representations to agree. The unique minimal-norm element in the set of all representations is taken as the canonical embedding ( \hat{N} ) of ( N ).
This approach not only reconstructs the numbers from first principles but also sets the stage for defining arithmetic operations intrinsically.
Within the framework, an embedded number ( \hat{N} ) is defined to be an intrinsic prime if it cannot be nontrivially factored within ( C_x ). In other words, if
[ \hat{N} = \hat{A} \cdot \hat{B}, ]
then one of ( A ) or ( B ) must equal 1.
A central result is the unique factorization theorem:
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Existence:
Every composite number ( \hat{N} ) can be factored into a product of intrinsic primes. The recursive application of the factorization process, enabled by the finite nature of ( N ), ensures termination in a product of primes. -
Uniqueness:
The coherence inner product ensures that the canonical embedding is unique. Any alternative factorization would lead to conflicting representations and a violation of the minimal-norm condition.
Thus, the Prime Framework mirrors the classical Fundamental Theorem of Arithmetic, proving that every natural number factors uniquely into intrinsic primes.
A key innovation is the introduction of a linear operator ( H ) on the Hilbert space ( \ell^2(\mathbb{N}) ):
[ H(\delta_N) = \sum_{d \mid N} \delta_d, ]
where ({\delta_N}_{N \in \mathbb{N}}) is the standard orthonormal basis. This operator encodes the divisor structure of numbers.
Analyzing the spectral properties of ( H ) involves:
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Formal Determinant:
Forming the determinant ( D(u) = \det(I - uH) ) and expanding it reveals a multiplicative structure linked to the intrinsic primes. -
Intrinsic Zeta Function:
By substituting ( u = p^{-s} ) into the determinant, we recover the intrinsic zeta function:[ \zeta_P(s) = \frac{1}{D(s)} = \prod_{\text{intrinsic } p} \frac{1}{1 - p^{-s}}, ]
which is the classical Euler product representation of the Riemann zeta function for (\Re(s) > 1).
The intrinsic zeta function provides the analytic backbone for deriving classical number theory results:
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Prime Number Theorem:
By applying Mellin inversion, contour integration, and Tauberian arguments within the framework, one obtains the asymptotic formula[ \pi(X) \sim \frac{X}{\ln X}, ]
where (\pi(X)) counts the number of intrinsic primes less than or equal to ( X ).
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Explicit Formulas:
Detailed analysis further leads to explicit formulas for both the prime-counting function ( \pi(X) ) and the nth prime ( p_n ).
A striking feature of the Prime Framework is that the symmetry and spectral properties of the operator ( H ) enforce a functional equation for (\zeta_P(s)). This, in turn, implies that all nontrivial zeros of (\zeta_P(s)) lie on the critical line ( \Re(s) = \frac{1}{2} )—an internal analogue of the Riemann Hypothesis.
The Prime Framework offers a new way to approach number theory:
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Foundationally New:
By deriving natural numbers from first principles within a geometrically and algebraically rich setting, the framework reinterprets classical results from a novel perspective. -
Unification of Algebra and Analysis:
The intrinsic construction leads to a unique factorization of numbers, while the operator-based spectral analysis recovers key analytic number theory results—including an internal version of the Riemann Hypothesis. -
A Promising Road Ahead:
This approach not only re-establishes the core theorems of number theory but also paves the way for deeper exploration of the interplay between geometry, algebra, and analysis.
The detailed constructions and rigorous proofs offer a robust platform for future research in both foundational mathematics and analytic number theory.
Review the proofs here: 1. Axioms, 2. Numbers, 3. Intrinsic Primes, 4. Prime Operator, 5. Analysis