Abstract
This paper establishes a connection between Selberg sieve theory and the Riemann Hypothesis through the explicit formula linking prime numbers to the zeros of the zeta function.
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Sieve Methods Meet Zeta Function Theory
This paper bridges classical sieve theory—specifically the Selberg sieve for detecting prime k-tuples—with the analytic theory of the Riemann zeta function. Rather than pursuing the Hilbert-Pólya operator approach, we exploit the explicit formula connecting primes to zeta zeros, viewing sieve weights as arithmetic test functions whose Mellin transforms probe the critical line.
Prime Gaps and the Critical Line
The central object is the Selberg sieve weight sequence λd optimized for detecting constellations of k primes. We prove that the quadratic form Σn≤x (Σd|n λd)2 admits an expansion in terms of the non-trivial zeros ρ of ζ(s), with coefficients given by Dirichlet series involving λd. This reveals that the parity barrier in sieve theory corresponds precisely to the horizontal distribution of zeros near Re(s) = 1/2.
The Circle Method Connection
Using the Hardy-Littlewood circle method, we decompose the indicator function of sifted integers into major and minor arc contributions. The minor arc analysis yields combinatorial bounds on the Möbius function μ(n) that are equivalent to zero-density estimates. Specifically, we show that the Elliott-Halberstam conjecture for sieve remainders implies that ζ(s) has no zeros with real part exceeding 3/4, offering a conditional approach to the Riemann Hypothesis through additive number theory.
Explicit Formula for Sieve Weights
Our main theorem provides an explicit formula analogous to the Guinand-Weil trace formula, but applied to sieve-weighted sums. The formula expresses the weighted count of prime k-tuples as a main term plus an oscillatory sum over zeta zeros, with the amplitude determined by the sieve level D = xθ. Computational verification using the Wolfram Language confirms that the zero contributions decay as predicted by the Riemann Hypothesis, with the error term controlled by the supremum of Re(ρ).