- #1
Boorglar
- 210
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I'm doing the exercises from Introduction to Analytic Number Theory by A.J. Hildebrand (online pdf lecture notes) from Chapter 2: Arithmetic Functions II - Asymptotic Estimates, and some of them leave me stumped...
1. Homework Statement
Problem 2.14:
Obtain an asymptotic estimate with error term [itex]O(x^{\frac{1}{3}})[/itex] for the number of squarefull integers [itex] ≤ x [/itex], i.e., for the quantity [itex]S(x) = \left\{n ≤ x : p | n => p^{2} | n\right\} [/itex].
2. Homework Equations
The text describes a method known as the "convolution method" to evaluate sums of arithmetic functions asymptotically. In our case, the arithmetic function would be the characteristic function of the squarefull integers, a(n) = 1 if n is squarefull and 0 otherwise.
If [itex]a = f * g[/itex] (Dirichlet convolution), then [tex]\sum_{n ≤ x} a(n) = \sum_{d ≤ x} g(d) F\left(\frac{x}{d}\right)[/tex] where [itex]F(x)[/itex] is the summatory function of [itex]f[/itex].
I am not even sure how to start. I tried expressing [itex]a(n) = 1 * (\mu * a)[/itex] thus using [itex]f = 1[/itex] and [itex]g = \mu * a[/itex]. But I can't find a way to estimate the sum [tex]\sum_{d ≤ x} g(d) \left\lfloor{\frac{x}{d}}\right\rfloor[/tex] Here [itex]\mu[/itex] means the Mobius function.
1. Homework Statement
Problem 2.14:
Obtain an asymptotic estimate with error term [itex]O(x^{\frac{1}{3}})[/itex] for the number of squarefull integers [itex] ≤ x [/itex], i.e., for the quantity [itex]S(x) = \left\{n ≤ x : p | n => p^{2} | n\right\} [/itex].
2. Homework Equations
The text describes a method known as the "convolution method" to evaluate sums of arithmetic functions asymptotically. In our case, the arithmetic function would be the characteristic function of the squarefull integers, a(n) = 1 if n is squarefull and 0 otherwise.
If [itex]a = f * g[/itex] (Dirichlet convolution), then [tex]\sum_{n ≤ x} a(n) = \sum_{d ≤ x} g(d) F\left(\frac{x}{d}\right)[/tex] where [itex]F(x)[/itex] is the summatory function of [itex]f[/itex].
The Attempt at a Solution
I am not even sure how to start. I tried expressing [itex]a(n) = 1 * (\mu * a)[/itex] thus using [itex]f = 1[/itex] and [itex]g = \mu * a[/itex]. But I can't find a way to estimate the sum [tex]\sum_{d ≤ x} g(d) \left\lfloor{\frac{x}{d}}\right\rfloor[/tex] Here [itex]\mu[/itex] means the Mobius function.