The prime number theorem over integers of power-free polynomial values

Human–AI partner teams are positioned to transform mathematical creativity, shifting discovery from incremental, bot-tom-up reasoning to a broader mode of inquiry that spans the full landscape of mathematics and science. This paper examines that transition by advancing Galois Smartnetwork Field Theory (Galois SNFT) as a framework for co-evolutionary human–machine reasoning—one that integrates mathematics, computation, and physics through the organizing power of higher structures mathematics, especially symmetry. To accelerate the inclusion of mathematical research into the computational infrastructure, Galois SNFT extends Neural Network Field Theory (NNFT) approaches by adding mathematics as a cornerstone to physics and computation. Digging deep into Modern Symmetry Theory’s convergence toward a Grand Unified Symmetry (GUS) framework with frontier mathematics from Clausen, Scholze, Lurie, Bhatt, Pridham, Barwick, and Haine, Galois SNFT deploys three symmetry properties (phase stability, glocal propagation, and symmetry constraint) to analyze the Millennium Prize Problems (MPP). The MPP can be partitioned into Langlands, physics, and orthogonal arms. Within this landscape, the Riemann Hypothesis (regarding the distribution of prime numbers along a critical line) is particularly suited to a symmetry-based analysis via phase stability, making it a compel-ling test case for co-evolutionary human–AI mathematical discovery.

This paper analyzes the Binary Goldbach Conjecture (bGC) through a deterministic structural lens, employing a Failure Mode Analysis (FMA) framework to map prime and composite inventories onto the Left-Right Partition Table (LRPT). The FMA framework identifies the specific structural conditions—categorized into three distinct Tiers—that render the existence of a counterexample (a “Failure State”) structurally inadmissible under standard density constraints. We establish structural identities governing the conservation of partition elements, demonstrating that the count of Prime-Prime (<i>P P</i>) pairs functions as a necessary deterministic residual. The analysis identifies tiered inadmissible failure states where, in each Tier, the exhaustion of composite inventories mathematically forces prime-prime partitions into existence to preserve information conservation. Numerical analysis for N up to 10<sup>6</sup> shows how the tiered FMA framework quantifies the structural mechanisms through which the bGC remains valid mathematically. Furthermore, as explained in Appendix I, by leveraging the midpoint symmetry of Goldbach primes, the FMA approach yields a ”Mirror Search” mechanism for distal primes that demonstrates superior discovery efficiency compared to sequential scanning methods guided by the Prime Number Theorem. The analysis also reveals, as detailed in Appendix II, that the failure state (<i>P P</i> (<i>N</i>) = 0) implies a deterministic dependency between partition components, allowing the primality characteristic function on the interval [3, 2N − 3] to be determined by testing π(N) fewer odd integers.

Abstract In 1970, Apostol introduced the Möbius function of order k for an integer $k\ge 1$ . In 2001, it was generalized by Bege to the Möbius function $\mu _{k,m}$ of two parameters for any integers $m\ge k\ge 1$ . In this article, we will establish three kinds of asymptotic formulas related to $\mu _{k,m}$ for $m\ge k\ge 2$ in a unified elementary method. These formulas are related to the shifted convolution sums of Fourier coefficients of holomorphic cusp forms, Bergelson and Richter’s dynamical generalization of the prime number theorem, and the Titchmarsh divisor problem.

Duality between prime factors and the prime number theorem for arithmetic progressions—II

Abstract Assuming the Generalized Riemann Hypothesis and a pair correlation conjecture for the zeros of Dirichlet L -functions, we establish the truth of a conjecture of Montgomery (in its corrected form stated by Friedlander and Granville) on the magnitude of the error term in the prime number theorem in arithmetic progressions. As a consequence, we obtain that, under the same assumptions, the Elliott–Halberstam conjecture holds true. As another consequence, under the same assumptions, we will show that the number of Dirichlet characters $$\chi (\text {mod}\,q)$$ χ ( mod q ) for which $$L(\frac{1}{2},\chi )=0$$ L ( 1 2 , χ ) = 0 is of order less than $$q^{1/2+\varepsilon }$$ q 1 / 2 + ε .

On a variant of the prime number theorem in arithmetic progressions

A Variant of the Prime Number Theorem, 2

Analytic number theory is a branch of number theory, which examines arithmetic phenomena using the real and complex analysis. The most important in the topic is the analysis of prime numbers, the distribution of which contains profound regularities and irregularities, which have puzzled mathematicians over the centuries. The current paper gives a thorough and strict description of analytic techniques in number theory, along with the Riemann zeta function and its use in the distribution of prime numbers. We start with classical proofs of results on primes and arithmetic functions to derive the analytic theory of the zeta function, its analytic continuation, functional equation, and Euler product representation. We then look at the relationship between the zeta function with the prime distribution which eventually leads to the Prime Number Theorem and its generalizations. Another aspect of number theory that was covered in the paper is zero-free regions, explicit formulas, and the Riemann Hypothesis, and their far-reaching consequences. It is presented as a full self-contained exposition, which can be published in a peer-reviewed journal.

Spectrum RCS(n) = ∑cos(tjlnn), j∈[1, ∞), based on imaginary values of non-trivial zeros of the zeta function ζ(s) = ζ(½±itj) = 0, Im(s) = ±tj, results in 'high peaks' (i.e., resonant values of cosine amplitudes at the prime powers n = pk, k∈ℕ in the negative part of the spectrum). The spectrum of the rth root RCS(n1/r) = ∑cos(tjlnn1/r), r∈ℕ exclusively reaches its resonance values in the values of the degree n = pr. Owing to that fact, prime numbers p1 can be separated from prime powers p1/r, r≥2. The spectrum of the dth degree RCS(nd) = ∑cos(tjlnnd), d∈ℕ exclusively reaches its resonant values exclusively in the values of the root p1/d, d≥2. The spectrum of the dth degree 'compresses' the number axis. For an arbitrary real interval (a, b), all the resonances pd of all prime numbers from the interval (ad, bd) are contained in the interval (a, b). The imaginary sine spectrum-ISS and the composite spectrum of the RCS and ISS is developed.

In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that | ψ ( x ) − x | and | ϑ ( x ) − x | are bounded from above by x log ⁡ x ( log ⁡ x − log ⁡ log ⁡ x ) 8 π for all x ≥ 101 and x ≥ 2 657 respectively, where ψ ( x ) and ϑ ( x ) are the Chebyshev ψ and ϑ functions. Using the extra precision offered by these results, we also prove new explicit descriptions for the error in each of Mertens' theorems which improve earlier bounds by Schoenfeld.

Abstract We present an explicit formula for a weighted sum over the zeros of the Riemann zeta function. This weighted sum is evaluated in terms of a sum over the prime numbers, weighted with the help of the Hermite polynomials. Then we apply this result to deduce, under the Riemann hypothesis, a formula describing the local distribution of zeros of the Riemann zeta function lying along the critical line. For this application we make use of the Prime Number Theorem in conjunction with Kac’s formula for the distribution of values of trigonometric polynomials.

We consider equidistribution of angles for certain hyperbolic lattice points in the upper half-plane. Extending work of Friedlander and Iwaniec, we show that for the full modular group equidistribution persists for matrices with a^{2}+b^{2}+c^{2}+d^{2}=p with p prime; at least if we assume sufficiently good lower bounds in the hyperbolic prime number theorem by Friedlander and Iwaniec. We also investigate related questions for a specific arithmetic co-compact group and its double cosets by hyperbolic subgroups. The general equidistribution problem was studied by Good, and in this case, we show, that equidistribution holds unconditionally when restricting to primes.

The distribution of prime numbers has long been viewed as a balance between order and randomness. In this work, we investigate the relationship between entropy, periodicity, and primality through the computational framework of the binary derivative. We prove that periodic numbers are composite in all bases except for a single trivial case and establish a set of twelve theorems governing the behavior of primes and composites in terms of binary periodicity. Building upon these results, we introduce a novel scale-invariant entropic measure of primality, denoted p(s'), which provides an exact and unconditional entropic probability of primality derived solely from the periodic structure of a binary number and its binary derivatives. We show that p(s') is quadratic, statistically well-defined, and strongly correlated with our earlier BiEntropy measure of binary disorder. Empirical analyses across several numerical ranges demonstrate that the variance in prime density relative to quadratic expectation is small, binormal, and constrained by the central limit theorem. These findings reveal a deep connection between entropy and the randomness of the primes, offering new insights into the entropic structure of number theory, with implications for the Riemann Hypothesis, special classes of primes, and computational applications in cryptography.

RSA algorithm, a widely used public-key cryptosystem, relies on the difficulty of factoring large composite numbers into their prime factors. However, advancements in computational power and factorization techniques have introduced potential threats to its security. As RSA remains a cornerstone of modern cryptography, the need for improved in its security measures is paramount in the face of evolving computational challenges. This study presents a cryptanalytic examination of the RSA encryption scheme and proposes an improvement that leverages prime number distribution patterns to strengthen data security. Several mathematical methods which involve RSA key generation and its encryption/decryption process, ASCII Table, mapping as well as the Sieve of Eratosthenes were applied in the study. The research analyses how RSA public parameters and encoding methods can reveal structural weaknesses when subjected to mathematical scrutiny. To address these vulnerabilities, a modified scheme is introduced, in which plaintext characters are mapped using prime distribution patterns. This substitution increases ciphertexts randomness and minimizes predictable patterns between plaintexts and ciphertexts. Experimental evaluation demonstrates that the proposed improvement enhances resistance to analytical attacks while maintaining RSA operational compatibility. The study contributes to the on-going development of more secure and efficient public-key cryptographic systems.

This study introduces a unified analytical framework, the λ-Overlap Law, which provides a deterministic proof of Goldbach’s Strong Conjecture. The approach derives directly from the Prime Number Theorem and the explicit inequalities of Dusart, establishing that for every even integer E ≥ 4, there exist two primes p and q satisfying p + q = E. The method defines the prime-density kernel λ(x) = 1/(x ln x) and demonstrates that its mirrored forms λ1(E/2 − t) and λ2(E/2 + t) necessarily intersect within a finite interval proportional to (ln E)². This intersection guarantees the existence of at least one symmetric prime pair for every E. The paper distinguishes intuitive heuristic representations (such as the rabbit-motion and circle analogies) from the formal analytical derivation based on covariance, overlap integrals, and continuity arguments. Empirical validation for 106 ≤ E ≤ 10¹8 confirms the analytic predictions, while the geometric λ-circle model illustrates the inherent symmetry of prime distributions. The resulting formulation unifies probabilistic, analytic, and geometric interpretations into a self-consistent proof framework, positioning λ symmetry as a fundamental principle governing additive properties of primes.

This article provides a proof of the famous Prime Number Theorem by establishing an analogous statement of the same in terms of the second Chebyshev Function (x).We shall be extensively using complex analytic techniques in addition to certain meromorphic properties of the Reimann Zeta Function (s) and its Analytic Continuation Property a priori using Riemann's Functional Equation in order to establish our desired result.

In 2022, Bergelson and Richter gave a new dynamical generalization of the prime number theorem by establishing an ergodic theorem along the number of prime factors of integers. They also showed that this generalization holds as well if the integers are restricted to be squarefree. In this paper, we present the concept of invariant averages under multiplications for arithmetic functions. Utilizing the properties of these invariant averages, we derive several ergodic theorems over squarefree numbers and squarefull numbers. These theorems have significant connections with the Erdős–Kac theorem, the Bergelson–Richter theorem, and the Loyd theorem.

The development of prime numbers has always been accompanied by humanitys pursuit of logical rigor and breakthroughs in abstract laws. The evolutionary trajectory of data science has always been closely linked to leaps in computing power and the explosive growth of data volume, providing robust support for in-depth explorations of prime number laws. In this paper, Python tools (e.g., NumPy) are used to acquire data on prime numbers and random matrices. Through normalization processing, comparison of statistical distributions, and visualization research, the statistical correlation between prime numbers (natural numbers greater than 1 that are divisible only by 1 and themselves) and random matrices (matrices whose elements are random numbers) is explored. The paper finds that there is a significant similarity between the distribution law of prime numbers (zeros of the Riemann zeta-function) and the distribution of eigenvalues of random matrices (Gaussian Unitary Ensemble (GUE) model). This statistical insight provides a new perspective for understanding the randomness and orderliness characteristics of prime number distribution in number theory and the application of random matrix theory in interdisciplinary mathematics.

The soft-reset process, or a sequence of charge emissions from a floating storage node through a transistor biased in a subthreshold bias condition, is modeled by a master (Kolmogorov-Bateman) equation. The Coulomb interaction energy after each one-charge emission leads to a stepwise potential increase, giving correlated emission rates represented by Boltzmann factors. The governing probability distribution function is a hypoexponential type, and its cumulants describe characteristics of the single-charge Coulomb interaction at room temperature on a mesoscopic scale. The cumulants are further extended into a complex domain. Starting from three fundamental assumptions, i.e., the generation of non-degenerated states due to single-charge Coulomb energy, the Markovian property of each emission event, and the independence of each state, a moment function is identified as a product of mutually prime elements (algebraically termed as prime ideals) comprising the eigenvalues or the lifetimes of the emission states. Then, the algebraic structure of the moment function is found to be highly analogous to that of an integer uniquely factored into prime numbers. Treating the lifetimes as analogs of the prime numbers, two types of zeta functions are constructed. Standard analyses of the zeta functions analogous to the prime number problem or the Riemann Hypothesis are performed. For the zeta functions, the analyticity and poles are specified, and the functional equations are derived. Also, the zeta functions are found to be equivalent to the analytic extension of the cumulants. Finally, between the number of emitted charges and the lifetime, a logarithmic relation analogous to the prime number theorem is derived.