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Number Theory

By Benacerraf, P

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Bull. Acad. Polon. Sci. Cl. III 4 (1956), 207–209 and Ann. Pol. Math. 4 (1958), 201–213. b) Let N = card Then p < x, p prime : ␸( p + ␯ + 1) − a␯ < ␧, 1 ≤ ␯ ≤ k . ␸( p + ␯) 18 Chapter I N > C(a, ␧) · x/((log x)k+2 · log log x) Y. Wang. A note on some properties of the arithmetical functions ␸(n), ␴(n) and d(n). Acta Math. Sinica 8 (1958), 1–11. x log x where N is defined in b). c) N > c1 (a, ␧) P. Erd˝os and A. Schinzel. Distributions of the value of some arithmetical functions. Acta Arith. 6 (1961), 473–485.

The unitary analogue of Pillai’s arithmetical function. Collect. Math. 40 (1989), 19–30. E. Ingham. Some asymptotic formulae in the theory of numbers. J. London Math. Soc. 2 (1927), 202–208. ␸(n)␸(n + k) = b) n≤x · 1+ p|k c) 1 3 x 3 1 − 2) p( p 2 (1 − 2/ p 2 ) · p + O(x 2 log2 x) 1+ ␸(n)/␸(n + k) = x p|/k n≤x d) 1+ ␸(n + k)/␸(n) = x n≤x p|/k 1 p 2 ( p − 1) + O(log2 x) 1 − 1) + O(log2 x) p2 ( p L. Mirsky. Summation formula involving arithmetic functions. Duke Math. J. 16 (1949), 261–272. 31 Asymptotic formulae for generalized Euler functions 1) Let F = { f 1 (x), .

Quart. 22 (1984), 371. 6) If ␻(n) ≥ 2, then ␴k (n) · Jk (n) ≤ n 2k − pk + 1 p|n J. S´andor. Note on the function ␴ and ␸. Bull. Number Theory Rel. Topics 12 (1988), 78–80. § I. 5 Unitary analogues of Jk , ␴k , d Let Jk∗ denote the unitary analogue of the Jordan totient function. Then: 1) Jk∗ (n) + d ∗ (n) ≤ ␴k∗ (n) 2) Jk∗ (n) + ␴k∗ (n) ≤ n k · d ∗ (n) 3) ␴ ∗ (n)J ∗ (n) 1 < k 2kk <1 ␨ (2k) n Euler’s ϕ-function 13 4) d ∗ (n) · n k ≤ Jk∗ (n) (d ∗ (n))2 ≤ n 2k where d ∗ and ␴k∗ are the unitary analogues of d and ␴k J.