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Let $𝒫_{r}$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a, q) = 1$ . Denote by $𝒫_{2} (a, q)$ the least almost-prime $𝒫_{2}$ which satisfies $𝒫_{2} \equiv a (mod q)$ . It is proved that for sufficiently large $q$ , there holds $𝒫_{2} (a, q) ≪ q^{1.8345} .$ This result constitutes an improvement upon that of Iwaniec (1982), who obtained the same conclusion, but for the range $1.845$ in place of $1.8345$ .

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On a problem of Hardy and Littlewood

On a question of P. Turán

On a ternary quadratic form over primes

On an almost pure sieve

On Bombieri's asymptotic sieve

On indefinite quadratic forms in four variables

On power-free numbers and polynomials. I.

On power.free numbers and polynomials. II.

On primitive divisors of Mersenne numbers

On Representing Integers as Products of the p + 1.

On Sieve method and Goldbach problem

On sums over primes

On the Brun-Titchmarsh theorem.

On the composition of the arithmetic functions σ and φ

On the constant β(k) in Rosser's sieve

On the coprimality of values of Euler's function at consecutive integers.

On the difference between consecutive prime numbers

On the Difference Between Consecutive Primes.

On the error term in the linear sieve

On the least almost-prime in arithmetic progression

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