Fejér-dirichlet Lift Constructs Entire Functions with Explicit Factorizations and Zero-Free Intervals for Odd Primes

The distribution of prime numbers, a fundamental question in mathematics, receives fresh insight through a newly developed technique called the Fejér-Dirichlet lift. Sebastian Fuchs advances the field by constructing entire functions that exhibit a remarkable connection between prime and composite numbers, effectively turning divisor information into explicit mathematical forms. This innovative approach not only creates functions vanishing at primes and positive at composites, but also establishes a clear threshold guaranteeing the absence of real zeros within specific intervals, and identifies boundary zeros with precise multiplicity. Furthermore, the research provides a constructive method for understanding the appearance of zeta functions through the lift’s spectrum, linking geometric weights to polylogarithm-zeta factorizations and offering a powerful new tool for exploring number theory.

Smooth Prime Indicators and Analytic Properties

This paper presents a comprehensive investigation into the creation of functions that effectively identify prime numbers, focusing on constructing ‘smooth’ indicators with well-defined analytical properties crucial for both theoretical understanding and potential applications. The team leverages harmonic analysis to examine the frequency components and smoothness of these indicators, employing polylogarithms and the Riemann zeta function as fundamental building blocks. A key innovation is the introduction of a tangent-matched indicator, F♯, designed with specific analytical properties at prime numbers, enhancing both its performance and theoretical understanding. The paper rigorously demonstrates these properties, exploring the behaviour of the indicators with varying parameters and providing a detailed analysis of their zero structure on the real axis.

The paper is meticulously organized, beginning with an introduction outlining the problem and contributions, followed by a section establishing the necessary mathematical background. Subsequent sections detail the construction of the indicators, analyze their analytical properties, and explore their global zero structure. Appendices provide detailed proofs and technical information, ensuring a complete and transparent presentation of the research. The research demonstrates a strong connection to existing work, building upon previous results and highlighting its unique contributions. The paper is written in a clear and precise style, making it accessible to researchers in the field. The indicators developed have potential applications in improving prime number algorithms, developing new cryptographic schemes, and furthering research in number theory and signal processing. Overall, this work represents a significant advancement in our understanding of prime number indicators, making this paper a valuable resource for experts in analytic number theory and related areas.

Prime Interpolation via Fejér Dirichlet Lift

This research introduces the Fejér-Dirichlet lift, a novel method that connects discrete information about prime numbers with continuous analytical functions. The lift transforms divisor information at integers into entire functions, effectively interpolating the divisor sum Ta(n) = (a ∗1)(n) while possessing a factorized Dirichlet series, bridging the gap between number theory and analysis. The team developed two specific indicators, F(x, q) and F♯(x, q), that interpolate the prime/composite pattern at integer arguments. The original indicator, F(x, q), vanishes at primes and is positive at composite integers greater than or equal to four.

In contrast, the refined, tangent-matched variant, F♯(x, q), enforces a vanishing slope at every integer, structurally controlling the function’s real-zero geometry near primes. Measurements reveal that F♯(x, q) exhibits a unique behaviour, proving that for any fixed q greater than 1, the open interval (p −1, p) is free of real zeros for every odd prime p greater than or equal to a specific threshold, P0(q). Furthermore, the prime p itself is demonstrated to be a boundary zero of multiplicity exactly two. Detailed analysis of the original indicator, F(x, q), reveals the presence of a “companion” zero, xp(q), within the interval (p −1, p).

The team quantified the displacement of this companion zero, establishing that it decays exponentially with a rate of q−p, specifically showing that p −xp(q) = log q K(q, p) q−p1 + O(q−p), where K(q, p) = 1 2 S′′ q (p) −(log q)2 q−p. The research also establishes a Polylog, Zeta factorization identity, linking the framework to established mathematical constants and functions. These results demonstrate the power of the Fejér-Dirichlet Lift as a versatile framework for constructing prime indicators with precisely specifiable analytical properties.

Primes, Composites, and the Fejér-Dirichlet Lift

This research presents a novel method, termed the Fejér-Dirichlet lift, which establishes a connection between discrete number theory and continuous analysis. The lift transforms information about divisors of integers into entire functions, exhibiting explicit Dirichlet series factorizations, and allows for the construction of functions with specific properties related to prime and composite numbers. Specifically, the team constructed an entire function that vanishes at prime numbers but takes positive values at composite integers, demonstrating a clear distinction between these two types of numbers within the continuous domain. Furthermore, the researchers developed a refined version of this lift, yielding an entire function that provides insight into the appearance of certain mathematical series through its spectral properties.

They established a link between this function and the polylogarithm-zeta function, deepening the understanding of relationships between different areas of mathematics. Analysis of the function reveals that, for sufficiently large primes, intervals between prime numbers are free of real zeros, and at each prime number, the function exhibits a specific contact point with multiplicity two. The authors acknowledge that determining a precise Weierstrass product for the constructed function remains a challenge, requiring more detailed control over zero locations and growth characteristics. The team provides code and data to reproduce their figures and verify their results, ensuring transparency and facilitating further research. While the current work focuses on specific parameter regimes, future investigations could explore the behaviour of the lift with alternating or negative parameters.