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Fejér-Kernel Prime Indicators

Sebastian Fuchs

TL;DR

This work constructs a Fejér-kernel-based prime indicator $\mathcal P$ by regularizing the sine-quotient divisor-encoder with Fejér cosine polynomials, yielding a function that vanishes exactly at odd primes and remains positive off the integers. It proves precise smoothness properties, including $C^{1}$ regularity and explicit second-derivative jumps at squares, and establishes a divisor-filter identity at integers that motivates smooth analogues. The paper then develops two smooth divisor-analogue functionals, $\mathcal P_{\tau}$ and $\mathcal P_{\sigma}$, via weighted Fejér sums with a smooth cutoff, showing that $\mathcal P_{\tau}(n;\kappa)\to\tau(n)-2$ and $\mathcal P_{\sigma}(n;\kappa)\to\sigma(n)-n-1$ as $\kappa\to\infty$, and offering conjectures about companion zeros near odd primes in the smooth setting. The work further provides a coherent analytic framework and numerical guidance for evaluating these sums with stable, $O(\sqrt{x})$ complexity and discusses the Fejér–Dirichlet lift as a generalization tying these constructions to Dirichlet series and L-functions. Although not aimed at prime counting or PNT implications, the results illuminate a structural bridge between harmonic-analytic kernels and arithmetic functions, with potential for future smooth-arithmetic generalizations and spectral interpretations.

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.

Fejér-Kernel Prime Indicators

TL;DR

This work constructs a Fejér-kernel-based prime indicator by regularizing the sine-quotient divisor-encoder with Fejér cosine polynomials, yielding a function that vanishes exactly at odd primes and remains positive off the integers. It proves precise smoothness properties, including regularity and explicit second-derivative jumps at squares, and establishes a divisor-filter identity at integers that motivates smooth analogues. The paper then develops two smooth divisor-analogue functionals, and , via weighted Fejér sums with a smooth cutoff, showing that and as , and offering conjectures about companion zeros near odd primes in the smooth setting. The work further provides a coherent analytic framework and numerical guidance for evaluating these sums with stable, complexity and discusses the Fejér–Dirichlet lift as a generalization tying these constructions to Dirichlet series and L-functions. Although not aimed at prime counting or PNT implications, the results illuminate a structural bridge between harmonic-analytic kernels and arithmetic functions, with potential for future smooth-arithmetic generalizations and spectral interpretations.

Abstract

A prime indicator is constructed by applying the Fejér identity to the sine-quotient encoder of trial division. For integers , holds exactly for odd primes; . For all non-integers one has . The function is piecewise and its second derivative has jumps precisely at the squares , with explicit sizes. Replacing the sharp cut-off by a smooth transition yields analogues and with integer limits and as , obtained from locally uniform convergence of derivative series. For large , numerical evidence indicates companion zeros near odd primes for and an asymmetric pair for . No assertion is made beyond integer input, and no statements are claimed about the prime number theorem or zero distributions of -functions. The appendix includes two illustrative prime-counting sums.

Paper Structure

This paper contains 36 sections, 30 theorems, 155 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Analytic construction of $\mathcal{P}$
  3. Motivation
  4. Regularisation
  5. Definition
  6. Smoothness properties of $\mathcal{P}$
  7. Global $C^{1}$-smoothness
  8. Second-derivative jumps
  9. The Prime–Zero Property of $\mathcal{P}(x)$
  10. Analytic Structure and Computational Complexity
  11. Lower and upper bounds for composite integers
  12. Computational Complexity
  13. Smooth analogues of arithmetic functions: setup
  14. Motivation
  15. Structure of the construction
  16. ...and 21 more sections

Key Result

Proposition 2.4

For an integer $i \ge 2$, the function $F(\cdot,i)$ defined in eq:Fxi is an entire function with the following properties:

Figures (9)

  • Figure 1: Central finite-difference profile of $\mathcal{P}"(x)$ on $[3.5,16.5]$. Discontinuities at $x=4,9,16$ (integer squares) match Proposition \ref{['prop:second']}; the observed jump heights agree with $\Delta_{m^2}\mathcal{P}"=2\pi^2\,[m^2\sin^2(\pi/(m+1))]^{-1}$.
  • Figure 2: Numerical profile of $\mathcal{P}(x)$ for $2\le x\le 50$. Dotted vertical lines mark odd primes; $\mathcal{P}(x)$ vanishes at these positions in accordance with Theorem \ref{['thm:primezero']}.
  • Figure 3: Local behavior of $\mathcal{P}(x)$ near $x=13$ (an odd prime). A zero occurs at $x=13$ without a sign change, consistent with nonnegativity and the odd prime–zero property.
  • Figure 4: The smooth cutoff function $\phi_{\kappa}(u)$ for different values of the steepness parameter $\kappa$. A smooth transition from 1 to 0 occurs around $u=1$. For larger $\kappa$, the transition becomes steeper while $C^\infty$-smoothness is preserved.
  • Figure 5: Companion zeros of the smooth divisor–counting analogue $\mathcal{P}_{\tau}(x;\kappa)$ on $[2,8]$ for $\kappa\in\{2,5,10,100\}$. Vertical dashed lines mark the odd primes $3,5,7$. Consistent with Conjecture \ref{['conj:companion-zeros']}, for larger $\kappa$ a pair of real zeros appears on either side of each odd prime and moves toward $p$; the value at $x=p$ is strictly negative for finite $\kappa$ and decays like $\mathrm e^{-2\kappa/(p+1)}$.
  • ...and 4 more figures

Theorems & Definitions (82)

  • Definition 2.1
  • Remark 2.2: On the choice of summation limit
  • Remark 2.3: Integer evaluation
  • Proposition 2.4: Properties of the Fejér term
  • proof
  • Proposition 2.5: Resonant partial-fraction identity and convergence
  • proof
  • Definition 2.6: Nearest-lattice notation
  • Lemma 2.7: Nearest-pole dominance bounds
  • proof
  • ...and 72 more