Ternary Egyptian fractions with prime denominator

Abstract

AbstractFor a prime number p, let $$A_3(p)= | { m \in \mathbb {N}: \exists m_1,m_2,m_3 \in \mathbb {N}, \frac{m}{p}=\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3} } |$$ A 3

( p )

=

|

{ m ∈ N : ∃

m 1

,

m 2

,

m 3

∈ N ,

m p

=

1

m 1

+

1

m 2

+

1

m 3

}

|

. In 2019 Luca and Pappalardi proved that $$x (\log x)^3 \ll \sum _{p \le x} A_{3}(p) \ll x (\log x)^5$$

x

( log x )

3

≪

∑

p ≤ x

A 3

( p )

≪ x

( log x )

5

. We improve the upper bound, showing $$\sum _{p \le x} A_{3}(p) \ll x (\log x)^3 (\log \log x)^2$$

∑

p ≤ x

A 3

( p )

≪ x

( log x )

3

( log log x )

2

.