We obtain reasonably tight upper and lower bounds on the sum n x ϕ (⌊x/n⌋), involving the Euler functions ϕ and the integer parts ⌊x/n⌋ of the reciprocals of integers.
We obtain a more precise version of an asymptotic formula of A. Dubickas for the number of monic Eisenstein polynomials of fixed degree d and of height at most H, as H → ∞. In particular, we give an explicit bound for the error term. We also obtain an asymptotic formula for arbitrary Eisenstein polynomials of height at most H.
We obtain asymptotic formulas with remainder terms for the hyperbolic summations $$\sum _{mn\le x} f((m,n))$$
∑
m
n
≤
x
f
(
(
m
,
n
)
)
and $$\sum _{mn\le x} f([m,n])$$
∑
m
n
≤
x
f
(
[
m
,
n
]
)
, where f belongs to certain classes of arithmetic functions, (m, n) and [m, n] denoting the gcd and lcm of the integers m, n. In particular, we investigate the functions $$f(n)=\tau (n), \log n, \omega (n)$$
f
(
n
)
=
τ
(
n
)
,
log
n
,
ω
(
n
)
and $$\Omega (n)$$
Ω
(
n
)
. We also define a common generalization of the latter three functions, and prove a corresponding result.
We study polynomials with integer coefficients which become Eisenstein polynomials after the additive shift of a variable. We call such polynomials shifted Eisenstein polynomials. We determine an upper bound on the maximum shift that is needed given a shifted Eisenstein polynomial and also provide a lower bound on the density of shifted Eisenstein polynomials, which is strictly greater than the density of classical Eisenstein polynomials. We also show that the number of irreducible degree n polynomials that are not shifted Eisenstein polynomials is infinite. We conclude with some numerical results on the densities of shifted Eisenstein polynomials.
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