Prime-Sinc Relation With Universal Combinatorial Product

초록

In this paper, we investigate a fundamental connection between prime numbers and the sinc function, which we term the prime-sinc relation (PSR). At the core of our analysis lie two classical facts: (i) every natural number greater than one has a unique prime factorization, and (ii) the normalized sinc function vanishes precisely at all nonzero integers, thereby allowing representation as an infinite product of structured polynomials. Exploiting these facts, we introduce a novel set-theoretic operation termed the universal combinatorial product (UCP), from which we construct two classes of entire functions as infinite products of distinct sinc functions. This construction leads directly to our main result for the PSR. By leveraging this foundational relationship, we introduce a novel expression for well-known functions (such as the Riemann zeta function), highlighting its mathematical significance. From a practical perspective, the proposed PSR theorem provides a unifying representation framework for signal processing. In particular, we introduce a new class of atom sets based on PSR, which enables flexible spectral filter design with arbitrary shapes and offers an efficient representation for implicit neural networks that learn functional mappings. Furthermore, the PSR framework suggests broader potential across signal processing tasks, including waveform shaping, signal denoising, and sequence design, with relevance to interference mitigation, quality enhancement, and signal detection. © 1991-2012 IEEE.

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