A natural sum U of sieve weights is bounded by the product of (1-1/p) via an identity reducing it to a controlled sum of squared theta coefficients, sufficient for short-interval applications.
Friedlander, Sifting short intervals II, Math
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A weaker but simpler sieve inequality
A natural sum U of sieve weights is bounded by the product of (1-1/p) via an identity reducing it to a controlled sum of squared theta coefficients, sufficient for short-interval applications.