Abstract
We obtain an asymptotic formula for the cube-full numbers in an arithmetic progression n ≡ l mod q , where q , l = 1 . By extending the construction derived from Dirichlet’s hyperbola method and relying on Kloosterman-type exponential sum method, we improve the very recent error term with x 118 / 4029 < q .
Highlights
Introduction and Main ResultsLet k > 1 be a fixed integer and n be a positive integer
In 2013, Liu and Zhang [19] investigated the distribution of square-full numbers in arithmetic progressions and got an asymptotic formula
1 ck(l, q)x(1/k) + Oq(127/92)+εx(7/46), n≤x,n∈P3 k 3 n≡l(mod q) where the error terms had been corrected by Watt [MR3265055]
Summary
Let k > 1 be a fixed integer and n be a positive integer. We call n a powerful number (or k-full number) if n 1 or for a prime p dividing n, pk divides n. In 2013, Liu and Zhang [19] investigated the distribution of square-full numbers in arithmetic progressions and got an asymptotic formula. 1 ck(l, q)x(1/k) + Oq(127/92)+εx(7/46), n≤x,n∈P3 k 3 n≡l(mod q) where the error terms had been corrected by Watt [MR3265055]. They divided the above sum into four parts as shown in Figure 2 and discussed them separately. An asymptotic formula of cube-full numbers in an arithmetic progression is derived. Note that compared with the result in [20], we improve the error term when x(118/4029) < q.
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