on bounded gaps between primes

Overview

The twin prime conjecture predicts infinitely many prime pairs $p,p+2$. Zhang and Maynard did not prove this conjecture, but they proved the first unconditional bounded-gap theorems for primes by two different mechanisms. Zhang kept the Goldston-Pintz-Yildirim framework and supplied a restricted Bombieri-Vinogradov theorem beyond the classical $1/2$ barrier for smooth moduli (Zhang, 2014; Goldston, Pintz & Yildirim, 2009). Maynard changed the sieve weights themselves; his multidimensional Selberg sieve obtains bounded gaps using only the classical Bombieri-Vinogradov theorem (Maynard, 2015).

Prime Tuples and Distribution Level

The twin prime conjecture asserts that

$$\underset{n\to \infty }{\mathrm{liminf}}(p_{n+1}-p_n)=2.$$

The bounded-gap problem asks for the weaker statement

$$\underset{n\to \infty }{\mathrm{liminf}}(p_{n+1}-p_n)<\infty .$$

This weaker statement does not specify the gap. It says only that some finite even difference occurs infinitely often among prime differences.

The flexible language is that of admissible tuples. A finite set

$$\mathcal{H}=\{h_1,\dots ,h_k\}\subset \mathbb{Z}$$

is admissible if, for every prime $p$, the residues $h_1,\dots ,h_k(\mathrm{mod}p)$ do not cover all residue classes modulo $p$. Equivalently, there is no congruence obstruction forcing one of the integers

$$n+h_1,\dots ,n+h_k$$

to be composite for every $n$. The prime $k$-tuples conjecture predicts that every admissible tuple has infinitely many translates consisting entirely of primes. For bounded gaps it is enough to prove that some fixed admissible tuple has infinitely many translates containing at least two primes.

The analytic input is distribution of primes in arithmetic progressions. For an arithmetic function $\gamma$, write

$$\Delta (\gamma ;q,a)=\sum _{\begin{array}{c}n\le x\\ n\equiv a(q)\end{array}}\gamma (n)-\frac{1}{\phi (q)}\sum _{\begin{array}{c}n\le x\\ (n,q)=1\end{array}}\gamma (n).$$

For primes, one takes $\gamma =\vartheta$ or $\Lambda$, where $\vartheta (n)=\log n$ if $n$ is prime and $0$ otherwise. A Bombieri-Vinogradov type estimate says that these errors are small on average over moduli:

$$\sum _{q\le x^{\theta }}\underset{(a,q)=1}{\max }|\Delta (\vartheta ;q,a)|{\ll }_A\frac{x}{(\log x{)}^A}.$$

The classical Bombieri-Vinogradov theorem gives this for every $\theta <1/2$. The Elliott-Halberstam conjecture predicts it for every $\theta <1$.

The number $1/2$ is the central threshold. In sieve arguments the weights are supported on divisors up to a parameter $R$, but after squaring the weight one must control moduli as large as roughly $R^2$. Thus the classical Bombieri-Vinogradov range $q\le x^{1/2-o(1)}$ naturally corresponds to a sieve support

$$R\le x^{1/4-o(1)}.$$

Zhang’s proof gets past this by proving a distribution theorem beyond $x^{1/2}$ for restricted, smooth moduli. Maynard’s proof avoids needing $R>x^{1/4}$ by changing the sieve optimization.

The GPY Positivity Criterion

For an admissible tuple $\mathcal{H}$, define

$$P_{\mathcal{H}}(n)=\prod _{i=1}^k(n+h_i).$$

The Goldston-Pintz-Yildirim method assigns a nonnegative weight $\lambda (n{)}^2$ to integers $n$ for which $P_{\mathcal{H}}(n)$ has few small prime factors. The basic sums are

$$S_1=\sum _{n\sim x}\lambda (n{)}^2$$

and

$$S_2=\sum _{n\sim x}\left(\sum _{i=1}^k\vartheta (n+h_i)\right)\lambda (n{)}^2.$$

If all weighted $n$ had at most one prime among $n+h_1,\dots ,n+h_k$, then the inner prime sum would be at most about $\log x$. Hence the inequality

$$S_2>(\log 3x)S_1$$

forces the existence of some $n\sim x$ for which at least two of the shifts are prime. Repeating this for arbitrarily large $x$ gives bounded gaps, with a bound at most

$$\underset{i}{\max }{h}_i-\underset{i}{\min }{h}_i.$$

A standard GPY weight is

$$\lambda (n)=\frac{1}{(k+\ell )!}\sum _{\begin{array}{c}d\mid P_{\mathcal{H}}(n)\\ d\le R\end{array}}\mu (d){\left(\log \frac{R}{d}\right)}^{k+\ell }.$$

This is a truncated divisor approximation to the condition that many of the shifts $n+h_i$ are prime. The parameter $\ell$ is not a minor technicality: it gives the weight enough flexibility to favor tuples with several prime components rather than only tuples in which all components behave like primes.

Opening the square gives

$$S_1=\sum _{d,e\le R}{\lambda }_d{\lambda }_e\#\{n\sim x:[d,e]\mid P_{\mathcal{H}}(n)\}.$$

For each pair $d,e$, the congruence $[d,e]\mid P_{\mathcal{H}}(n)$ restricts $n$ to a set of residue classes modulo $[d,e]$. Since $\mathcal{H}$ is admissible, this count has a main term governed by the singular series of $\mathcal{H}$. Thus $S_1$ is a divisor-sum calculation.

The second sum is different. Opening the square in $S_2$ gives terms of the form

$$\sum _{\begin{array}{c}n\sim x\\ n\equiv c([d,e])\end{array}}\vartheta (n+h_i).$$

After shifting the residue class, this is a prime-counting problem in arithmetic progressions modulo $[d,e]$. Therefore the error term in $S_2$ is controlled by Bombieri-Vinogradov type estimates for moduli up to approximately

$$[d,e]\le R^2.$$

This is precisely where distribution of primes enters the sieve.

The GPY calculation has two competing requirements. The main term becomes strong enough to prove bounded gaps if the divisor support is slightly larger than $x^{1/4}$. But classical Bombieri-Vinogradov controls the corresponding error terms only when $R^2\le x^{1/2-o(1)}$, hence $R\le x^{1/4-o(1)}$. This is the GPY barrier. Conditional on any level of distribution $\theta >1/2$, GPY proves bounded gaps; unconditionally it proves the weaker statement

$$\underset{n\to \infty }{\mathrm{liminf}}\frac{p_{n+1}-p_n}{\log p_n}=0.$$

Zhang’s Modification of GPY

Zhang’s proof retains this one-dimensional GPY weight but changes its support. In the notation of the paper, one takes

$$D=x^{1/4+\varpi },\varpi =\frac{1}{1168},$$

and restricts divisors to be composed only of small primes. More concretely, let

$$P=\prod _{p<D_1}p,D_1=x^{\varpi },$$

and use a weight of the form

$$\lambda (n)=\sum _{d\mid (P_{\mathcal{H}}(n),P)}\mu (d)g(d),$$

where $g(d)$ is a logarithmic cutoff supported on $d<D$. Thus every divisor $d$ appearing in the weight is smooth: all prime factors of $d$ are smaller than $D_1$.

This restriction has two effects. First, it does not destroy the main term. The GPY main terms are still large enough when $k$ is very large, because the contribution of divisors with a large prime factor is negligible for the chosen parameters. Second, it changes the error term in exactly the needed way. After squaring the weight, the moduli have size up to

$$D^2=x^{1/2+2\varpi },$$

which is beyond the classical Bombieri-Vinogradov range, but they are smooth moduli. Zhang proves that this restricted family satisfies a Bombieri-Vinogradov type estimate strong enough for the GPY error term.

The crucial theorem may be summarized as follows. For the residue classes $C_i(d)$ naturally produced by the condition $d\mid P_{\mathcal{H}}(n)$,

$$\sum _{\begin{array}{c}d<D^2\\ d\mid P\end{array}}\sum _{c\in C_i(d)}|\Delta (\vartheta ;d,c)|{\ll }_A\frac{x}{(\log x{)}^A}.$$

This is not a full level $1/2+2\varpi$ theorem for all moduli. It is an averaged distribution theorem for precisely the smooth moduli arising from Zhang’s modified GPY weight. That restriction is why the theorem is both provable and sufficient.

The Smooth-Moduli Mechanism

The reason smooth moduli are useful is factorization. If $d\mid P$ and $d$ is in the difficult range just above $x^{1/2}$, then $d$ can be written as

$$d=rq$$

with $r$ lying in a convenient size range. Because all prime factors of $d$ are small, the factor $r$ can be adjusted with much finer resolution than for a general modulus. This makes the characteristic function of the allowed moduli behave like a well-factorable weight.

This factorization is the entry point for the dispersion method. The distribution error is first converted, by Cauchy-Schwarz and orthogonality of additive characters, into sums measuring cancellation in congruence conditions. The flexible factorization $d=rq$ separates variables so that the modulus can be split between different parts of the convolution produced by the von Mangoldt function.

Zhang uses a Heath-Brown identity to decompose $\vartheta$ into Dirichlet convolutions. The resulting estimates fall into three regimes.

Analytic Regimes in Zhang's Proof

Type I estimates handle convolutions in which one variable is long and the other variables are short. The long variable can be averaged effectively after the modulus has been factored.

Type II estimates handle bilinear sums in which two variables have comparable size. The dispersion method reduces these to Kloosterman-type sums, where square-root cancellation gives the saving.

Type III estimates handle trilinear sums in the range where three variables are all substantial. This is the hardest case. Zhang reduces the problem to complete exponential sums over finite fields and invokes estimates of Birch-Bombieri type, ultimately resting on Deligne’s work, to obtain cancellation.

The Type III range is the new analytic core. Without it, the dispersion method covers the easier convolution ranges but fails in the balanced trilinear regime created by the Heath-Brown decomposition. The complete exponential sums that appear are not arbitrary; their phases come from rational functions generated by the congruence manipulations. The proof must verify that these phases are nondegenerate enough for deep algebraic-geometric bounds to apply.

After these estimates are assembled, the GPY positivity calculation can be run with

$$k_0=3.5\times 10^6.$$

Zhang proves that every admissible $k_0$-tuple has infinitely many translates containing at least two primes. Choosing an admissible tuple of that size inside an interval of length $7\times 10^7$ gives

$$\underset{n\to \infty }{\mathrm{liminf}}(p_{n+1}-p_n)<7\times 10^7.$$

The proof mechanism is therefore precise. Zhang does not solve the parity problem, and he does not prove Elliott-Halberstam. He shows that the exact moduli needed by a modified GPY weight have enough factorization to allow distribution beyond $x^{1/2}$.

Maynard’s Change of Sieve Weights

Maynard’s proof begins from the same positivity principle but changes the weight space. Instead of one divisor $d$ of the product $P_{\mathcal{H}}(n)$, the weight keeps separate divisor variables for the separate shifts:

$$w_n={\left(\sum _{\begin{array}{c}{d}_i\mid n+h_i\\ 1\le i\le k\end{array}}{\lambda }_{d_1,\dots ,d_k}\right)}^2.$$

This is the central structural change. The GPY weight asks whether the product $P_{\mathcal{H}}(n)$ has small prime factors. Maynard’s weight asks, separately for each coordinate, which of the individual numbers $n+h_i$ has small prime factors. The extra degrees of freedom allow the sieve to focus its mass on tuples with several prime components.

There is also a preliminary $W$-trick. One fixes

$$W=\prod _{p\le w}p$$

and restricts $n$ to a residue class modulo $W$ for which all $n+h_i$ are coprime to $W$. This removes small-prime congruence obstructions uniformly and lets the remaining sieve sums behave cleanly. After this restriction, the tuple’s admissibility is encoded in a common singular factor that appears in both numerator and denominator.

Opening the square in

$$\sum _{N\le n<2N}{w}_n$$

produces congruence conditions

$$[d_i,e_i]\mid n+h_i(1\le i\le k).$$

Because the shifts are distinct and the small primes have been handled by $W$, the Chinese remainder theorem gives a main term as long as the moduli are pairwise compatible. The total modulus is roughly

$$\prod _{i=1}^k[d_i,e_i].$$

Thus the support of ${\lambda }_{d_1,\dots ,d_k}$ is chosen so that this product is at most about $R^2$, with

$$R=N^{\theta /2-o(1)}.$$

When $\theta =1/2$ from Bombieri-Vinogradov, this is still only $R=N^{1/4-o(1)}$. Maynard’s point is that this is now enough.

Evaluation of Maynard’s Sums

Maynard studies the weighted expression

$$S(N,\rho )=\sum _{N\le n<2N}\left(\sum _{i=1}^k{1}_{\mathbb{P}}(n+h_i)-\rho \right)w_n.$$

If $S(N,\rho )>0$, then at least one $n$ has more than $\rho$ prime entries among $n+h_1,\dots ,n+h_k$. Thus the goal is to make the ratio

$$\frac{{\sum }_i{\sum }_{N\le n<2N}{1}_{\mathbb{P}}(n+h_i)w_n}{{\sum }_{N\le n<2N}{w}_n}$$

large.

The coefficients are parameterized by a smooth function $F$ on the simplex

$${\mathcal{R}}_k=\left\{(t_1,\dots ,t_k)\in [0,1{]}^k:\sum _{i=1}^k{t}_i\le 1\right\}.$$

Schematically,

$${\lambda }_{d_1,\dots ,d_k}\approx \left(\prod _{i=1}^k\mu (d_i)\right)F\left(\frac{\log d_1}{\log R},\dots ,\frac{\log d_k}{\log R}\right),$$

with the product support condition encoded by the simplex.

The denominator is governed by

$$I_k(F)={\int }_{{\mathcal{R}}_k}F(t_1,\dots ,t_k{)}^{2}d{t}_1\cdots d{t}_k.$$

The numerator for the $m$-th shift is governed by

$$J_k^{(m)}(F)={\int }_{[0,1{]}^{k-1}}{\left({\int }_0^{1}F(t_1,\dots ,t_k)d{t}_m\right)}^2\prod _{i\ne m}d{t}_i.$$

The form of $J_k^{(m)}$ has an important meaning. When $n+h_m$ is prime, divisibility conditions in the $m$-th coordinate essentially disappear, since a prime $n+h_m$ has no small divisor from the sieve range. Thus the $m$-th coordinate is integrated out linearly before squaring. This is exactly what the inner integral in $J_k^{(m)}$ records.

After the divisor sums are evaluated, the common singular factors cancel in the ratio, and the expected number of primes detected by the weight is approximately

$$\frac{\log R}{\log N}\frac{{\sum }_{m=1}^k{J}_k^{(m)}(F)}{I_k(F)}.$$

Since $R=N^{\theta /2-o(1)}$, this becomes

$$\frac{\theta }{2}\frac{{\sum }_{m=1}^k{J}_k^{(m)}(F)}{I_k(F)}.$$

Define

$$M_k=\underset{F}{\sup }\frac{{\sum }_{m=1}^k{J}_k^{(m)}(F)}{I_k(F)}.$$

Maynard’s theorem can be stated in the following operational form: if $r\ge 1$ and

$$M_k>\frac{2r}{\theta },$$

then infinitely many translates of an admissible $k$-tuple contain at least $r+1$ primes.

This is the point at which the proof diverges decisively from Zhang’s. For two primes, take $r=1$. Under Bombieri-Vinogradov one has $\theta =1/2$, so it is enough to have

$$M_k>4.$$

Maynard proves that $M_{105}>4$, and an explicit admissible $105$-tuple of diameter $600$ gives

$$\underset{n\to \infty }{\mathrm{liminf}}(p_{n+1}-p_n)\le 600.$$

The Variational Mechanism

The quantity $M_k$ is not a formal decoration; it is the central optimization problem. Maynard proves that

$$M_k\to \infty (k\to \infty ),$$

with an explicit lower bound of logarithmic order. Conceptually, the test functions $F$ are chosen so that their mass is spread across many coordinates, while the marginal integrals appearing in the $J_k^{(m)}$ remain large. This is a high-dimensional effect. The one-dimensional GPY weight does not have enough independent coordinates to exploit it.

This unboundedness has a strong consequence. For every fixed $m$, one can choose $k$ such that

$$M_k>4m$$

under the classical Bombieri-Vinogradov level $\theta =1/2$. Hence some translate of an admissible $k$-tuple contains at least $m+1$ primes infinitely often. Therefore

$$\underset{n\to \infty }{\mathrm{liminf}}(p_{n+m}-p_n)<\infty$$

for every fixed $m\ge 1$.

The key proof is therefore not an improvement in the distribution of primes. It is a proof that the Selberg sieve has a better multidimensional choice of weights than the GPY product-divisor ansatz. Once those weights are chosen, the distribution estimates needed to evaluate the sums remain in the classical Bombieri-Vinogradov range.

Comparison of the Two Mechanisms

Component	Zhang	Maynard
Shared baseline	GPY positivity criterion	GPY positivity criterion
Main innovation	Restricted distribution beyond $1/2$	Multidimensional weight optimization
Sieve support	$D=x^{1/4+\varpi }$	$R=N^{1/4-o(1)}$ under Bombieri-Vinogradov
Moduli controlled	Smooth moduli up to $x^{1/2+2\varpi }$	General moduli within the classical average range
Deep technical core	Dispersion method, Type I/II/III estimates, algebraic exponential sums	Evaluation of multidimensional Selberg weights and optimization of $M_k$
First bound	$7\times 10^7$	$600$
Qualitative strength	At least two primes in very large admissible tuples	Arbitrarily many primes in bounded intervals

The shortest accurate summary is this. Zhang crossed the GPY barrier by proving that the particular moduli generated by a smooth-divisor GPY weight have better distribution than arbitrary moduli. Maynard crossed the same barrier by showing that the GPY weight was not the right optimization problem; a multidimensional Selberg weight extracts enough information from the classical distribution theorem.

Why the Twin Prime Conjecture Remains Open

Both methods prove bounded gaps, not gap $2$. They show that some fixed bounded interval contains two primes infinitely often. This implies that at least one even difference below the bound occurs infinitely often as a prime difference, but the methods do not identify the difference and do not force it to be $2$.

The obstruction is the parity problem. Sieve weights can efficiently show that integers avoid small prime factors, and they can prove that a weighted tuple contains several prime-like entries on average. They have much less power to distinguish primes from products of two large primes with the precision needed to force a specific twin-prime pattern.

The significance of the two papers is therefore structural. Zhang showed that a restricted but deep improvement of Bombieri-Vinogradov is enough to activate GPY. Maynard showed that a different sieve geometry makes the classical Bombieri-Vinogradov theorem enough. These are two distinct mechanisms for producing bounded gaps, and neither removes the final parity obstruction separating bounded gaps from twin primes.

See Also

• on sum-product in finite fields via entropy: Distribution in finite fields provides a useful comparison point for how arithmetic structure can force cancellation in exponential and combinatorial sums.
• on the sum-product conjecture’s falsity: Another number-theoretic topic where the precise form of arithmetic structure matters more than a heuristic random model.
• on the lower bound of Ramsey number: Shares the theme of using weighted averaging to convert a global existence problem into a positivity calculation.

References

🐻  Goldston, D.A., Pintz, J. & Yildirim, C.Y. 2009. Primes in tuples. I. Annals of Mathematics 170(2), 819–862.

🐻  Maynard, J. 2015. Small gaps between primes. Annals of Mathematics 181(1), 383–413.

🐻  Zhang, Y. 2014. Bounded gaps between primes. Annals of Mathematics 179(3), 1121–1174.