Fej\'er-Kernel Prime Indicators

TL;DR

This paper introduces a novel prime indicator function based on Fejér kernels, which exactly identifies odd primes and exhibits specific smoothness and jump discontinuities, with potential numerical and theoretical implications.

Contribution

The paper constructs a new $C^1$ prime indicator using Fejér identities, providing explicit properties and smooth analogues, and explores their numerical behavior near primes.

Findings

Exact prime detection at odd primes for the indicator

Smooth analogues converge to specific prime-related functions

Numerical evidence of zeros near primes for smooth variants

Abstract

A $C^1$ prime indicator $\mathcal{P}\colon\mathbb{R}\to\mathbb{R}$ is constructed by applying the Fej\'er 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 $\mathcal{P}_\tau$ and $\mathcal{P}_\sigma$ with integer limits $\mathcal{P}_\tau(n;\kappa)\to \tau(n)-2$ and $\mathcal{P}_\sigma(n;\kappa)\to \sigma(n)-n-1$ as $\kappa\to\infty$, obtained from locally uniform convergence of derivative series. For large $\kappa$, numerical evidence indicates companion zeros near odd primes for $\mathcal{P}_\tau$ and an asymmetric pair for $\mathcal{P}_\sigma$. 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.