For a function $f : ℕ \to ℕ$ , define $N_{f}^{\times} (x) = # {n \leq x : n = k f (k) for some k}$ . Let $τ (n) = \sum_{d | n} 1$ be the divisor function, $ω (n) = \sum_{p | n} 1$ be the prime divisor function, and $φ (n) = # {1 \leq k \leq n : (k, n) = 1}$ be Euler’s totient function. We prove that

\begin{array}{l} (1) & N_{τ}^{\times} (x) ≍ \frac{x}{{(log x)}^{1 / 2}}; \\ (2) & N_{ω}^{\times} (x) = (1 + o (1)) \frac{x}{log log x}; \\ (3) & N_{φ}^{\times} (x) = (c_{0} + o (1)) x^{1 / 2}, \end{array}

where

c_{0} = 1.365 \dots

Keywords:

AMSC: 11N37, 11N64, 11B83

References

1. R. Balasubramanian and K. Ramachandra , On the number of integers $n$ such that $n d (n) \leq x$ , Acta Arith. 49 (1988) 313–322. Crossref, Web of Science, Google Scholar
2. P. Bateman , The distribution of values of the Euler function, Acta Arith. 21 (1972) 329–345. Crossref, Google Scholar
3. M. Balazard and G. Tenenbaum , Sur la répartition des valeurs de la fonction d’Euler, Compos. Math. 110(2) (1998) 239–250. Crossref, Web of Science, Google Scholar
4. P. Erdős and L. Mirsky , The distribution of values of the divisor function $d (n)$ , Proc. London Math. Soc. s3-2 (1952) 257–271. Crossref, Google Scholar
5. K. Ford , The distribution of totients, Ramanujan J. 2 (1998) 67–151. Crossref, Web of Science, Google Scholar
6. G. H. Hardy and S. Ramanujan , The normal number of prime factors of a number $n$ , Q. J. Math. 48 (1917) 76–92. Google Scholar
7. R. Hall and G. Tenenbaum , Divisors, Cambridge Tracts in Mathematics, Vol. 90 (Cambridge University Press, 1988). Crossref, Google Scholar
8. N. M. Korobov , Estimates of trigonometric sums and their applications, Uspekhi Mat. Nauk 13 (1958) 185–192 (Russian). Google Scholar
9. A. A. Karatsuba and S. M. Voronin , The Riemann Zeta-Function (Walter de Gruyter, 1992). Crossref, Google Scholar
10. F. Luca and A. Sankaranarayanan , The distribution of integers $n$ divisible by $l^{ω (n)}$ , Publ. Inst. Math. (Beograd) (N.S.) 76(90) (2004) 89–99. Crossref, Google Scholar
11. H. Maier and C. Pomerance , On the number of distinct values of Euler’s $φ$ -function, Acta Arith. 49 (1988) 263–275. Crossref, Web of Science, Google Scholar
12. M. R. Murty , Problems in Analytic Number Theory, 2nd edn. (Springer, 2008). Google Scholar
13. G. Tenenbaum , Introduction to Analytic and Probabilistic Number Theory, Graduate Studies in Mathematics, Vol. 163, 3rd edn. (American Mathematical Society, 2015). Crossref, Google Scholar
14. I. M. Vinogradov , A new estimate for $ζ (1 + i t)$ , Izv. Akad. Nauk SSSR, Ser. Mat. 22 (1958) 161–164 (Russian). Google Scholar