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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.18933 |
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| _version_ | 1866915555983753216 |
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| author | Fuchs, Sebastian |
| author_facet | Fuchs, Sebastian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18933 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fejér-Kernel Prime Indicators Fuchs, Sebastian Number Theory 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. |
| title | Fejér-Kernel Prime Indicators |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.18933 |