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Let $T(N)$ count pairs $n,n+2$ both prime with $n+2\le N$. The Selberg sieve replaces primality by the necessary condition that the polynomial $F(n)=n(n+2)$ have no small prime divisor. Its strength is a nonnegative quadratic majorant with optimizable divisor weights; its limitation is that a sifted integer need not be prime.

For a prime $p$, let $\nu (p)$ count the forbidden residues of $n$ modulo $p$. We have $\nu (2)=1$, since the two roots coincide modulo two, and $\nu (p)=2$ for every odd prime, where the roots are zero and $-2$. The Chinese remainder theorem makes $\nu (d)$ multiplicative on squarefree integers. With $g(d)=\nu (d)/d$,These local densities constitute the sieve distribution for the twin-prime problem. Since ${\sum }_{p\le z}g(p)=2\log \log z+O(1)$, it is a dimension-two sieve, which is why the expected upper bound has two logarithms in the denominator.

Let $P(R)={\prod }_{p\le R}p$ and choose real Selberg sieve weights ${\lambda }_d$ supported on squarefree $d\le R$, with ${\lambda }_1=1$. If $\gcd (F(n),P(R))=1$, the only supported divisor of $F(n)$ is one. HenceExpanding and using the distribution of the polynomial roots givesHere $[d,e]$ denotes the least common multiple. The divisor-square support reaches $R^2$, although each individual weight is supported only to $R$; keeping these two levels distinct is essential for estimating the remainder.

The Selberg sieve diagonalization solves the main-term minimization exactly. On squarefree integers putMultiplicativity givesAlso, Möbius inversion gives ${\sum }_{r\le R}\mu (r)Y_r={\lambda }_1=1$. Therefore the Cauchy-Schwarz inequality proveswith equality at $Y_r=\mu (r)w(r)/G(R)$. Inverting the divisor sums gives the optimizing weightsIn particular ${\lambda }_1=1$ and $Q(\lambda )=1/G(R)$.

These weights have $|{\lambda }_d|\le 1$, which supplies a simple rigorous remainder estimate. Indeed, multiply the inner sum by ${\prod }_{p\mid d}(1-g(p){)}^{-1}={\sum }_{e\mid d}w(e)$. The terms are $w(em)$ for distinct squarefree integers $em\le R$ with $e\mid d$ and $(m,d)=1$, hence form a subset of the sum defining $G(R)$. Since $\nu ([d,e])\le \nu (d)\nu (e)$ and $\nu (d)\le \tau (d)$,Thus

For this sieve $w(2)=1$ and $w(p)=2/(p-2)$ for odd primes. Here is a direct proof that $G(R)\asymp {\log }^{2}R$, sufficient for the upper bound. PutThe Mertens theorem for reciprocal primes implies $W(z)\asymp {\log }^{2}z$, by taking logarithms and using the convergence of ${\sum }_{p}g(p{)}^2$. Give each divisor $d\mid P(z)$ probability $w(d)/W(z)$. Its expected logarithm isby the Mertens first theorem. With $z=R^{1/8}$ this is at most $\frac{1}{3}\log R$ for large $R$. The Markov inequality shows that at least two thirds of the total product weight has $d\le R$. These terms occur in $G(R)$, so $G(R)\gg W(R^{1/8})\gg {\log }^{2}R$. Conversely $G(R)\le W(R)\ll {\log }^{2}R$, proving the claim without treating an untruncated Euler product as a truncated divisor sum.

Choose, for example, $R=N^{1/4}$. Any twin-prime pair with smaller member greater than $R$ survives this sieve; pairs with smaller member at most $R$ contribute at most $O(R)$. ConsequentlyThe chosen level is deliberately enough to prove the correct logarithmic order; sharper level and remainder analyses improve the numerical upper-bound constant.

The local correction factor also explains the conjectured main-term constant. Relative to two independent primality conditions, the singular series isThe twin-prime constant $C_2$ is positive and convergent because its logarithm has an absolutely convergent tail. The factor two at $p=2$ records that only one parity is allowed, while each odd-prime factor records the two distinct forbidden residues. The Hardy–Littlewood twin-prime asymptotic conjecture predictsThis is a conjecture, not a consequence of the upper-bound calculation.

The parity problem in sieve theory is the obstruction to reversing the argument. The nonnegative square majorizes all survivors, including composites with sufficiently large prime factors; small-divisor information does not isolate the desired prime-factor parity or force both factors to be prime. Therefore the Selberg sieve establishes the correct upper-bound scale and the arithmetic local factors, but by itself does not prove a positive lower bound for twin primes, their infinitude, or the conjectured asymptotic.

The conjecture asserts that infinitely many prime numbers $p$ have $p+2$ prime. The Hardy–Littlewood twin-prime asymptotic conjecture predicts their quantitative frequency and is stronger. The Selberg sieve supplies an upper bound of the predicted order but does not prove infinitude.

This positive convergent product corrects for the dependence of primality of $n$ and $n+2$ at each odd prime. The associated singular series is $2C_2$: the factor at two is two, and at an odd prime the local correction is $(1-2/p)/(1-1/p{)}^2$. It appears in the Hardy–Littlewood twin-prime asymptotic conjecture and in upper bounds from the Selberg sieve.

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