Worked solution: Maynard's multidimensional sieve: bounded gaps between primes (2013)

Step 4 of 8: Turning the sieve into two explicit multiple integrals

In plain words

Once the weights are built from a smooth function $F$ of $k$ variables, the two key sums — the total weight $S_1$ , and the weighted prime-count $S_2$ — become, after some standard but intricate sieve-theoretic bookkeeping, essentially just two multiple integrals of $F$ over the unit cube: one measuring the 'size' of $F$ squared, the other measuring how much $F$ contributes to each single coordinate direction.

This is the payoff of the multidimensional setup: an originally number-theoretic, divisor-counting problem has been converted into ordinary calculus, where $F$ can be chosen and optimized freely.

I_k(F)=\int_0^1\!\cdots\!\int_0^1 F(t_1,\dots,t_k)^2\,dt_1\cdots dt_k,\qquad J_k^{(m)}(F)=\int_0^1\!\cdots\!\int_0^1\Big(\int_0^1 F\,dt_m\Big)^2 dt_1\cdots dt_{m-1}\,dt_{m+1}\cdots dt_k

Detailed analysis

Write $S_1=\sum_n w_n$ and $S_2=\sum_n \big(\sum_i \chi_{\mathbb{P}}(n+h_i)\big)w_n$ (restricted to $n$ in a fixed residue class mod a small modulus $W$ , a routine device removing small-prime obstructions; Maynard 2013, §4, eqs. 4.2–4.3). Maynard's Proposition 4.1 evaluates both sums asymptotically, for $\lambda_{d_1,\dots,d_k}$ built from a smooth function $F$ supported on $\{(t_1,\dots,t_k): \textstyle\sum t_i\le 1\}$ , in terms of the two integrals $I_k(F)=\int_0^1\cdots\int_0^1 F(t_1,\dots,t_k)^2\,dt_1\cdots dt_k$ and $J_k^{(m)}(F)=\int_0^1\cdots\int_0^1\big(\int_0^1 F\,dt_m\big)^2 dt_1\cdots dt_{m-1}dt_{m+1}\cdots dt_k$ (Maynard 2013, §4, Proposition 4.1).

Concretely, $S_1\sim c_1 N(\log R)^k I_k(F)$ and $S_2\sim c_2 N(\log R)^{k+1}\sum_{m=1}^k J_k^{(m)}(F)/\log N$ for explicit constants $c_1,c_2>0$ depending only on $\mathcal{H}$ and $W$ , valid whenever the primes have a 'level of distribution' $\theta>0$ — a quantitative statement about how evenly primes are spread across arithmetic progressions of modulus up to about $N^\theta$ — and $R=N^{\theta/2-\delta}$ for small $\delta>0$ .

This reduces the whole sieve-theoretic problem to pure analysis: everything about $\mathcal{H}$ , $N$ , $R$ has been packaged into these constants, and the only remaining freedom is the choice of the function $F$ , which the next step will optimize.

Terms in this step

Level of distribution: A number $\theta>0$ such that primes up to $N$ are known to be spread evenly across arithmetic progressions of modulus up to $N^\theta$ , on average, with error smaller than any power of $\log N$ . The classical Bombieri–Vinogradov theorem gives $\theta$ arbitrarily close to $1/2$ .