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Keywords:
almost-prime; arithmetic progression; linear sieve; Selberg's $\Lambda ^2$-sieve
almost-prime; arithmetic progression; linear sieve; Selberg's $\Lambda ^2$-sieve
Summary:
Let $\mathcal P_{2}$ denote a positive integer with at most $2$ prime factors, counted according to multiplicity. For integers $a$, $q$ such that $(a,q)=1$, let $\mathcal P_{2}(q,a)$ denote the least $\mathcal P_{2}$ in the arithmetic progression $\{nq+a\}_{n=1}^{\infty }$. It is proved that for sufficiently large $q$, we have $$ \mathcal P_{2}(q,a)\ll q^{1.825}. $$ This result constitutes an improvement upon that of J. Li, M. Zhang and Y. Cai (2023), who obtained $\mathcal P_{2}(q,a)\ll q^{1.8345}.$
References:
[1] Halberstam, H., Richert, H.-E.: Sieve Methods. London Mathematical Society Monographs 4. Academic Press, London (1974). MR 0424730 | Zbl 0298.10026
[2] Heath-Brown, D. R.: Almost-primes in arithmetic progressions and short intervals. Math. Proc. Camb. Philos. Soc. 83 (1978), 357-375. DOI 10.1017/S0305004100054657 | MR 0491558 | Zbl 0375.10027
[3] Heath-Brown, D. R.: Zero-free regions for Dirichlet $L$-functions and the least prime in an arithmetic progression. Proc. Lond. Math. Soc., III. Ser. 64 (1992), 265-338. DOI 10.1112/plms/s3-64.2.265 | MR 1143227 | Zbl 0739.11033
[4] Hooley, C.: On the Brun-Titchmarsh theorem. J. Reine Angew. Math. 255 (1972), 60-79. DOI 10.1515/crll.1972.255.60 | MR 0304328 | Zbl 0252.10045
[5] Iwaniec, H.: A new form of the error term in the linear sieve. Acta Arith. 37 (1980), 307-320. DOI 10.4064/aa-37-1-307-320 | MR 0598883 | Zbl 0444.10038
[6] Iwaniec, H.: On the Brun-Titchmarsh theorem. J. Math. Soc. Japan 34 (1982), 95-123. DOI 10.2969/jmsj/03410095 | MR 0639808 | Zbl 0486.10033
[7] Iwaniec, H., Laborde, M.: $P_2$ in short intervals. Ann. Inst. Fourier 31 (1981), 37-56. DOI 10.5802/aif.848 | MR 0644342 | Zbl 0472.10048
[8] Jurkat, W. B., Richert, H.-E.: An improvement of Selberg's sieve method. I. Acta Arith. 11 (1965), 217-240. DOI 10.4064/aa-11-2-217-240 | MR 0202680 | Zbl 0128.26902
[9] Laborde, M.: Buchstab's sifting weights. Mathematika 26 (1979), 250-257. DOI 10.1112/S0025579300009803 | MR 0575644 | Zbl 0429.10028
[10] Levin, B. V.: On the least almost prime number in an arithmetic progression and the sequence $k^2x^2+1$. Usp. Mat. Nauk 20 (1965), 158-162 Russian. MR 0188173 | Zbl 0154.30002
[11] Li, J., Zhang, M., Cai, Y.: On the least almost-prime in arithmetic progression. Czech. Math. J. 73 (2023), 177-188. DOI 10.21136/CMJ.2022.0478-21 | MR 4541095 | Zbl 07655761
[12] Linnik, Y. V.: On the least prime number in an arithmetic progression. I. The basic theorem. Mat. Sb., Nov. Ser. 15 (1944), 139-178 Russian. MR 0012111 | Zbl 0063.03584
[13] Linnik, Y. V.: On the least prime number in an arithmetic progression. II. The Deuring-Heilbronn phenomenon. Mat. Sb., Nov. Ser. 15 (1944), 347-368 Russian. MR 0012112 | Zbl 0063.03585
[14] Mertens, F.: A contribution to analytic number theory: On the distribution of primes. J. Reine Angew. Math. 78 (1874), 46-62 German \99999JFM99999 06.0116.01. DOI 10.1515/crll.1874.78.46 | MR 1579612
[15] Motohashi, Y.: On almost-primes in arithmetic progressions. III. Proc. Japan Acad. 52 (1976), 116-118. DOI 10.3792/pja/1195518371 | MR 0412128 | Zbl 0361.10039
[16] Pan, C. D., Pan, C. B.: Goldbach Conjecture. Science Press, Beijing (1992). MR 1287852 | Zbl 0849.11080
[17] Xylouris, T.: On the least prime in an arithmetic progression and estimates for the zeros of Dirichlet $L$-functions. Acta Arith. 150 (2011), 65-91. DOI 10.4064/aa150-1-4 | MR 2825574 | Zbl 1248.11067
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