Selberg/GPY Sieve Majorant in Analytic Number Theory

• The Selberg/GPY sieve majorant is a family of positive-definite weights that dominates the indicator function for multiple primes in admissible k-tuples, driving advances in prime gap research.
• Its construction utilizes a quadratic form in divisor sums with smooth cutoff functions and a variational principle to optimize coefficients and control error terms.
• By leveraging numerical optimization and distribution hypotheses like GEH, the majorant provides explicit asymptotic bounds essential for establishing record low gaps between consecutive primes.

The Selberg/GPY sieve majorant is a family of positive-definite weights central to modern analytic number theory, constructed to dominate the indicator of having at least $m+1$ primes among a set of $k$ shifted integers $n+h_1,\dots,n+h_k$ for admissible $k$-tuples. Its construction, optimization, and error-analysis underpin current record bounds for $H_m=\liminf_{n\to\infty}(p_{n+m}-p_n)$, and it forms the technical heart of breakthroughs on bounded gaps between primes. Two principal sources for this theory are “Variants of the Selberg sieve, and bounded intervals containing many primes” (Polymath, 2014) and “On a weighted sum over multiplicative functions and its applications to the GPY sieve” (Liu, 2022).

1. Definition and Formal Structure of the Majorant

Given an admissible $k$-tuple $\mathcal{H}=(h_1,\dots,h_k)$, the Selberg/GPY majorant is a quadratic form in divisor sums associated to the polynomial $P(n) = \prod_{i=1}^k (n + h_i)$. For a compactly supported smooth function $F:[0,\infty)\to\mathbb R$, define the Selberg divisor-sum

$$\lambda_F(n) = \sum_{d \mid n} \mu(d) F\Big(\frac{\log d}{\log x}\Big),$$

supported on $d$ whose prime divisors all exceed $x^{-S(F)}$ with $S(F) = \sup\{t : F(t)\neq0\}$. The full sieve weight is a sum of squares of such combinations:

$\nu(n) = \Big(\sum_{d\mid P(n)} \lambda_d \Big)^2,$

where each $\lambda_d$ is as above. This quadratic form ensures positivity and enforces that $\nu(n)$ majorizes the indicator of “$n + h_1,\dots,n + h_k$ contain at least $m+1$ primes,” up to error terms.

The normalization parameter

$$B = \frac{\phi(W)}{W} \log x,\qquad W = \prod_{p \leq w} p,\quad w = \log\log\log x,$$

ensures main terms in sum asymptotics simplify, appearing with factors $B^{-k}\frac{x}{W}$ or $B^{1-k}\frac{x}{W}$ as appropriate (Polymath, 2014).

Smoothened versions of the majorant, crucial for explicit calculations, use weights

$$\lambda_d = \mu(d)P\Big( \frac{\log(R/d)}{\log R} \Big ) = \mu(d)\left( \frac{\log(R/d)}{\log R} \right)^m$$

with $P(x) = x^m$, $m>k$, $d\leq R$, and all primes dividing $d$ less than a smoothness parameter $z$. These are aggregated over divisor sets $$D_n = \{ d \leq R : d\mid Q(n),\ p\mid d \implies p<z \}$$ (Liu, 2022).

2. Variational Principle and Optimization

The selection of coefficients in the majorant is governed by a variational principle. For function $F$ as above, set

$I(F) = \int_{[0,\infty)^k} F(t)^2\,dt,$

$$J_i(F) = \int_{[0,\infty)^{k-1}} \left( \int_0^{\infty} F(t_1,\dots,t_k)\,dt_i \right)^2\, d\widehat t_i.$$

The supremum

$$M_k = \sup_{F \not\equiv 0,\ \text{supp} F\subset\{t_1+\dots+t_k\leq1\}} \frac{\sum_{i=1}^k J_i(F)}{I(F)}$$

measures the efficacy of the sieve weight. Explicit bounds for $H_m$ follow once $M_k > \frac{2m}{\vartheta}$, where $\vartheta$ is the distribution exponent of primes in arithmetic progressions available under the Bombieri–Vinogradov or Elliott–Halberstam type hypotheses (Polymath, 2014).

Variants involving truncated or enlarged support (e.g., allowing $F$ on an $\epsilon$-enlarged simplex or with vanishing marginal constraints) offer flexibility and enable stronger results, particularly under the generalized Elliott–Halberstam conjecture.

Numerical optimization for $M_k$ is executed by reducing the infinite-dimensional variational problem to a generalized eigenvalue problem via a finite basis of symmetric polynomials or Krylov-subspace methods. Explicit computational results include $M_{50} > 4.00238$ and derived records $H_1\leq246$ unconditionally, $H_1\leq6$ under GEH (Polymath, 2014).

3. Level-of-Distribution Hypotheses and Error Control

Three hypotheses are used to control the error terms in sum asymptotics:

• Bombieri–Vinogradov $EH[\vartheta]$: For all $\vartheta<1/2$, uniform error in sums of the von Mangoldt function over moduli $q\leq x^\vartheta$.
• Motohashi–Pintz–Zhang $MPZ[\varpi,\delta]$: Like $EH[\vartheta]$, but allows some savings for $q\leq x^{1/2+2\varpi}$, $q$ $x^{\delta}$-smooth.
• Generalized Elliott–Halberstam $GEH[\vartheta]$: Extends $EH$-type error bounds to general convolutions of smooth sequences over the relevant range.

Error terms in the prime-sum and non-prime-sum asymptotics vanish provided the total support width $\sum_i (S(F_i) + S(G_i))$ is less than the relevant exponent ($\vartheta$ or $1/2+2\varpi$), ensuring the validity of main-term asymptotics in the majorant analysis (Polymath, 2014).

4. Key Bilinear Forms, Asymptotic Expansions, and Lemmas

Central to the analysis are quadratic and bilinear forms:

$$Q_g = \sum_{\substack{d_1, d_2 \leq R \ p\mid [d_1,d_2]\implies p<z}} \lambda_{d_1} \lambda_{d_2} g([d_1,d_2]),$$

with $g$ multiplicative, often encoding local prime counts. Their main-term asymptotics take the form

$$Q_g \sim \frac{I_k(1,u)}{(\log R)^k} \prod_{p}(1-g(p))(1-1/p)^{-k},$$

where $I_s(t,v)$ is computable recursively. Weighted partial summation, Mellin inversion, and Buchstab-type recursions underpin the error estimation and explicit coefficient formulas (e.g., for smoothing factors $f(u;k,m)$). All stated error terms are rigorously controlled under the imposed distribution hypotheses (Liu, 2022).

Relevant lemmas include:

• Discrepancy bounds (Eq. (2.1), (Polymath, 2014)).
• Analytic properties for Dirichlet series with multiplicative coefficients.
• Bounds on sums of divisor functions localized by smooth kernels (Liu, 2022).

5. Innovations and Numerical Optimization over GPY

Several key innovations distinguish the Selberg/GPY majorant from earlier sieve-theoretic approaches:

• Extended support sets for cutoff functions $F$ allow flexibility and strengthen bounds, especially under $GEH$.
• Explicit “parity barrier” identification: It is shown that purely sieve-theoretic (i.e., parity-insensitive) majorants cannot deliver $H_1 < 6$, formalizing longstanding heuristics about limitations of the method (Polymath, 2014).
• Numerical large-scale variational optimization, especially for large $k$, delivers explicit constants and bounds (e.g., $M_{50} > 4.00238$). Computational methods employed include reduction to finite-dimensional generalized eigenvalue problems based on symmetric polynomials and iterates of the integral operator $\mathcal{L}$.
• Sharper asymptotic formulas for weighted sums, including all main and lower-order coefficients with explicit smoothing factors (e.g., $f(u;k,m)$), improving tractability for computations and supporting further numerical refinement (Liu, 2022).

Zhang’s original smoothened GPY sieve only achieved a lower bound on the key weighted sum, without full asymptotic expansions or tractable coefficient formulas. Later developments yield explicit, numerically accessible expansions:

$$S \sim \frac{N}{(\log R)^k} \prod_p \left(1-\frac{\nu_p}{p} \right) \left(1 - \frac{1}{p} \right)^{-k} \left\{\frac{k\theta}{2}I_{k-1}(1,u) - I_k(1,u)\right\}$$

for explicit, computable $I_s(1,u)$ and $u = \theta/(2\delta)$ (Liu, 2022).

6. Consequences for Small Gaps and Prime k-tuple Results

The optimized majorants feed into the main asymptotic relations for prime tuples. For any $k$ such that $M_k>2m/\vartheta$ (with $\vartheta$ reflecting the available level of distribution), one achieves the tuple result $DHL[k,m+1]$ and thereby

$H_m \leq H(k) \ll m e^{(4-\frac{24}{181})m}$

unconditionally (or $H_m \ll m e^{2m}$ under $EH$ or $GEH$). For $m=1$, $k=50$ suffices for $M_{50}>4$, yielding $H_1 \leq 246$ unconditionally. With $GEH$, the construction collapses to $k=3$ and $H_1 \leq 6$, matching the “parity barrier” limit (Polymath, 2014).

Further, the methods yield explicit upper bounds for $H_m$ for specific small values of $m$, and show that advances hinge on either stronger distributional hypotheses or circumventing the intrinsic limitations of parity-blind sieve constructions.

7. Comparative and Structural Overview

The evolution of the Selberg/GPY majorant reflects an iterative strengthening along several axes:

• Adoption of multidimensional sieve-theoretic weights amenable to explicit calculation and optimization.
• Extension of permissible smooth cutoff supports via analytic and combinatorial innovations.
• Transition from mere lower bounds to full asymptotic expansions, facilitating sharper numerical work.
• Integration of advanced distributional hypotheses and explicit error control, thereby improving both conditional and unconditional bounds.

The structural properties—positivity, “majorant” behavior, upper bounds, and tractable error terms—remain central, guaranteeing each step aligns rigorously with major sieve-theoretic frameworks and analytic number theory doctrines (Polymath, 2014, Liu, 2022).