What question did this study set out to answer?

The aim is to investigate the sifting function and its role in establishing the infinitude of twin primes using the Krafft transformation.

February 6, 2026Open Access

On the Non-Triviality of the Analytic Sieve of Dimension κ=2 for Small Sifting Parameters

Key Points

The aim is to investigate the sifting function and its role in establishing the infinitude of twin primes using the Krafft transformation.
Analyzed the sifting function for arithmetic set $A_n$.
Established a reflection principle for the sifting parameters.
Demonstrated the behavior of the Krafft transformation as a Type II bilinear operator.
Utilized Montgomery-Vaughan $L^2$ estimates and Weylean equidistribution for analysis.
Established that twin primes' existence is linked to the positivity of the sifting function $S(A_n)$ for $A_n$ at dimension $A=2$.
Confirmed that the sequence of twin primes is infinite with $ ext{liminf}_{n o ext{infinity}} (p_{n+1} - p_n) = 2$.
Showed that the error term is strictly subordinate to the main term in the analytic framework.

Abstract

We investigate the sifting function S (Aₙ, P, z) for an arithmetic set Aₙ derived from the Krafft transformation (1798). By establishing a reflection principle for the sifting parameters, we show that the existence of twin primes is equivalent to the positivity of S (Aₙ) for a sifting dimension =2. We address the parity problem by demonstrating that the Krafft transformation acts as a non-degenerate Type II bilinear operator, K (a, b), which lifts the multiplicative group structure of the reduced residues modulo 6 into the space of sifting indices. Combined with Montgomery-Vaughan L² estimates and Weylean equidistribution, this structural localization ensures that the error term remains strictly subordinate to the main term. This provides an analytic proof that the sequence of twin primes is infinite, confirming ₍ (p₍+₁ - pₙ) = 2.

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