Year: 2025
Type: preprint
Abstract: A $C^1$ prime indicator $\mathcal{P}\colon\mathbb{R}\to\mathbb{R}$ is constructed by applying the Fejér identity to the sine-quotient encoder of trial division. For integers $n\ge 2$, $\mathcal P(n)=0$ holds exactly for odd primes; $\mathcal P(2)>0$. For all non-integers $x>1$ one has $\mathcal P(x)>0$. The function is piecewise $C^\infty$ and its second derivative has jumps precisely at the squares $m^2$, with explicit sizes. Replacing the sharp cut-off by a smooth transition yields $C^\infty$ analogues $\mathcal{P}_τ$ and $\mathcal{P}_σ$ with integer limits $\mathcal{P}_τ(n;κ)\to τ(n)-2$ and $\mathcal{P}_σ(n;κ)\to σ(n)-n-1$ as $κ\to\infty$, obtained from locally uniform convergence of derivative series. For large $κ$, numerical evidence indicates companion zeros near odd primes for $\mathcal{P}_τ$ and an asymmetric pair for $\mathcal{P}_σ$. No assertion is made beyond integer input, and no statements are claimed about the prime number theorem or zero distributions of $L$-functions. The appendix includes two illustrative prime-counting sums.
Source: arXiv (Cornell University)
Author: Fuchs, Sebastian
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4415164197
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