Sieve methods and Brun–Titchmarsh

Let $P(z)$ be the product of primes below a finite threshold, and consider integers in a finite set or with nonnegative finitely supported weights. An integer survives if it is divisible by none of the selected primes. Expanding the product of one minus each divisibility indicator gives an alternating sum over squarefree divisors of the prime product. The coefficients are exactly Möbius values. Summing first over integers gives the Legendre sieve identity, an arithmetic form of inclusion–exclusion with no approximation yet.

For initial integers through a real endpoint, the count divisible by a divisor is its floor quotient. Substituting the endpoint divided by the divisor plus a bounded remainder suggests a density product. However, the number of divisors of the prime product doubles with every selected prime. Bounding every individual remainder and then summing absolute values can overwhelm the intended main term. A local estimate that looks harmless in isolation is not harmless after an exponentially large inclusion–exclusion expansion.

Practical sieve methods replace the full alternating expansion by carefully chosen weights with restricted support. For an upper bound, the resulting divisor-weight expression must majorize the survival indicator pointwise; multiplying by nonnegative input weights and summing preserves the inequality. Selberg’s method uses a square of a short divisor sum to guarantee nonnegativity and then optimizes its coefficients. The permitted divisor range balances the size of the main term against accumulated errors. This is a structural explanation, not a proof of the full optimized inequality.

Survival only means absence of the selected prime factors. A composite with both factors above the sieve threshold still survives, and the integer one always survives. Conversely, a selected small prime is removed even though it is prime. Recovering a prime count therefore requires accounting for these exceptions and choosing the threshold appropriately. Basic divisor-density information also has a parity limitation: it need not distinguish integers with an even number of prime factors from those with an odd number sufficiently well to give sharp prime lower bounds. An upper bound should never be reported as a matching asymptotic.

For real $x>q\geq2$ and an integer $a$ coprime to $q$, the classical Brun–Titchmarsh inequality gives the displayed upper bound. This formulation has no unspecified asymptotic remainder and is uniform in the eligible residue class and modulus within its stated range. As the modulus approaches the endpoint, the logarithm of their ratio shrinks, making the bound weak. At or below the modulus the displayed denominator is zero or negative, so this version cannot be applied.

For a fixed modulus and large endpoint, the bound is roughly twice the prime-number-theorem main term. That does not mean actual counts are twice their expected scale, nor does the inequality provide a positive lower bound. When the modulus grows as a fixed power of the endpoint, the ratio-logarithm matters substantially and cannot be replaced by the endpoint logarithm without changing the constant. The distinction is useful when summing over many moduli: an individually uniform upper bound can control a difficult range even when a uniform asymptotic has not been established there.