Abstract
Let N m ( x) be the number of arithmetic progressions that consist of m terms, all primes and not larger than x, and set F m(x) = C mx 2 log mx ( C m explicitly given). It is shown that Hardy and Littlewood's prime k-tuple conjecture implies that N m ( x) = F m ( x){1 + Σ j=1 N a j log − j x + O((log x) − N−1 )}, (here the bracket represents an asymptotic series with explicitly computable coefficients). This formula holds rather trivially for m = 1 and m = 2. It is proved here for m = 3, by the Vinogradov version of the Hardy-Ramanujan-Littlewood circle method.
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