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1996) On the arithmetic structure of sets characterized by sum of digits properties, J. Number Theory 61, 25–38. Montgomery, H. and Soundararajan, K. (2004) Primes in short intervals, Comm. Math. Phys. 252, 589–617. Murty, M. R. and Saidak, F. (2004) Non-abelian generalizations of the Erd˝os–Kac theorem, Canad. J. Math 56, 356–372. Murty, V. K. and Murty, M. R. (1984 a ) An analogue of the Erd˝os–Kac theorem for Fourier coeﬃcients of modular forms, Indian J. Pure Appl. Math. 15, 1090–1101. Murty, V.

Amer. Math. Soc 302, 269–280. Erd˝os, P. and Pomerance, C. (1985) On the normal number of prime factors of ϕ(n), Rocky Mountain J. Math 15, 343–352. Erd˝os, P. and Wintner, A. (1939) Additive arithmetical functions and statistical independence, Amer. J. Math. 61, 713–721. Halberstam, H. (1955) On the distribution of additive number theoretic functions. I, J. London Math. Soc. 30, 43–53. Halberstam, H. (1956) On the distribution of additive number theoretic functions. III, J. London Math. Soc. 31, 15–27.

3 is motivated by the above construction. We use a lower bound sieve and the Bombieri–Vinogradov theorem to construct many primes p such that (p − 1)/2 has no small prime factors and then consider pairs p, of such primes. Using elementary arguments and some applications of the upper bound sieve we show that most of the products p we have constructed have suﬃciently large Carmichael function. Although we do not match the constant 18 that follows from the conjecture, we actually obtain a large number of integers of the required type using this unconditional argument.