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7h
comment Using the Brun Sieve to show very weak approximation to twin prime conjecture
Halberstam and Richert in Sieve Methods (Dover, 2011) prove using Brun's sieve that there are infinitely many p such that p+2 has at most 8 prime factors. Including some introductory material the exposition takes 67 pages. The key is their definition of the characteristic function on p. 58. The basic idea is simple enough but doesn't look like it lends itself to anything one could describe as a "straightforward exercise."
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revised Does anyone recognize this sequence?
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answered At what rate are composites removed in a set after each prime multiple is cancelled out?
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comment Proof of inequality involving multiplicative function?
Terms in the binomial expressions on the left with m factors are all covered by terms in mth powers of the expression on the right. Once we see the LHS can be written as a product of binomials we can compare the two sides. Your hint prompted me to look at the LHS again. The key (which I forgot or didn't know) is that $n=p_k\#,~\binom{k}{m}$ is the number of squarefree divisors of n having $\nu(d)=m.$
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comment Proof of inequality involving multiplicative function?
@user1952009: edited to reflect that.