Numbers with small prime factors, and the least $$k$$-th power non-residue.(English)Zbl 0211.37801

Mem. Am. Math. Soc. 106, 106 p. (1971).
Summary: For a positive integer $$n$$ and real $$x,y$$ with $$x\ge 1$$, $$y\ge 0$$, define $$\Psi_n(x,y)$$ to be the number of integers $$m$$ such that $$1\le m\le x$$, $$(m,n) = 1$$, and $$m$$ has no prime factor greater than $$y$$. Define also $$B_l(n,x,y)$$ to be the number of integers $$m$$ such that $$1\le m\le x$$, $$m\equiv l\pmod n$$, and $$m$$ has no prime factor greater than $$y$$.
In §3, the author attempts a comprehensive review of previous research on the functions $$\Psi_n$$ and $$B_l$$. (One paper that was missed is [J. van de Lune and E. Wattel, Math. Centrum, Amsterdam Afd. Zuivere Wisk. ZW 1968-007, 35 p. (1968; Zbl 0215.06502)]. Errors are pointed out in several papers.
In §4, there is a more or less expository derivation (based on work of V. Ramaswami) of an asymptotic formula for $$B_l(n, x, x^{1/a})$$, valid for $$1\le a\le (\log x)^{1/2}$$ and $$(l,n) =1$$.
In §5, asymptotic formulas for $$\Psi_n(x,y)$$) are derived. For example,
$\Psi_n(x,x^{1/a}) = n^{-1} \varphi(n)\rho(a)x+ \varphi(n)b(n)\rho(a-1)(x/\log x) +$
$+ O\left( \{\rho(a - 3) \xi(n) + an/\varphi(n)\} (x/\log^2x) + 2^{\omega(n)}x^{(1 - 1/a)} \right)$
whenever $$3 <a\le (\log x)^{1/2}$$. Here $$\varphi(n)$$ is Euler’s function, $$\omega(n)$$ is the number of distinct prime factors of $$n$$,
$b(n) = n^{-1} \left\{1 - \gamma - \sum_{p\mid n} (p-1)^{-1} \log p\right\},$
$$\gamma$$ is Euler’s constant, $$\xi(n) = O(\{\log \log (3n)\}^3)$$ and $$\rho(a)$$ is a positive continuous non-increasing function defined recursively by $$\rho(a) = 1$$ $$0\le a \le 1)$$,
$\rho(a) = \rho(N) - \int_n^a v^{-1} \rho(v-1)\,dv\qquad (N < a \le N+1,\ N=1,2,\ldots).$
In Theorem 5.55, an asymptotic formula is given for the number of integers $$m$$ such that $$1\le m\le x$$, $$(m,n) = 1$$, and $$C(n)$$ has no prime factor greater than $$m^{1/a}$$ $$(a$$ fixed, $$a >1)$$. Let $$C(n)$$ be the multiplicative group of residue classes mod $$n$$ which are relatively prime to $$n$$, let $$C_k(n)$$ denote the subgroup of $$k$$-th powers, and write $$\nu_k(n) = [C(n) : C_k(n)]$$.
In §6, results of the type (1) are used to get $$O$$-estimates for $$g_1(n,k)$$ the smallest positive integer in $$C(n)$$ but not in $$C_k(n)$$. For example, let $$w)$$ be any integer $$\ge 2$$, and let $$a_w$$ be the unique root of the equation $$\rho(a) = w^{-1}$$. If $$n,k$$ are any integers such that $$\nu_k(n)\ge w$$, then for each $$\delta>0$$,
$g_1(n,k) = O_{w,\delta} \left(n^{3/(8a_w) + \delta}\right),$
the implied constant depending at most on $$w,\delta$$. If a certain arithmetical condition on $$n$$ and $$k$$ is satisfied, then the exponent $$3/(8a_w)$$ can be replaced by $$1/(4a_w)$$. (With more effort, this arithmetical condition can be dispensed with, and we get, the result
$g_1(n,k) = O_{w,\delta} \left(n^{1/(4a_w) + \delta}\right),$
whenever $$\nu_k(n)\ge w$$. See a forthcoming paper by the author.)
In §7, several specific estimates are given for $$g_1(p,k)$$ $$(p$$ prime), an example being
$g_1(p,k) \le 4.7p^{1/4}\log p\quad\text{for }(k,p -1) >1.$
For large $$p$$, this improves an inequality of it{A. Brauer} [Math. Z. 33, 161–176 (1931; Zbl 0001.05703)] for $$g_1(p,2)$$ (Brauer’s result holds for $$p\equiv 7\pmod 8$$; on p. 77 of the present paper, it is erroneously stated that he proved a similar result for $$p \equiv 1\pmod 8$$.)
The paper concludes with a specific upper estimate for the number of distinct prime factors of $$n$$.
Reviewer: Karl K. Norton

MSC:

 11N25 Distribution of integers with specified multiplicative constraints 11N37 Asymptotic results on arithmetic functions 11N69 Distribution of integers in special residue classes

Citations:

Zbl 0215.06502; Zbl 0001.05703
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