Find the asymptotic expansion of a sum involving fractional parts for a real power α as x approaches infinity | Step-by-Step Solution

What We're Solving:

We're looking at sums of the form $\sum_{n=1}^{x} \{n^{\alpha}\}$ where $\{y\}$ denotes the fractional part of $y$ (i.e., $\{y\} = y - \lfloor y \rfloor$), and we want to find how this sum behaves as $x \to \infty$.

The Approach:

The key insight is to use the Euler-Maclaurin formula combined with properties of fractional parts:

• Fractional parts are periodic-like functions that oscillate between 0 and 1
• The Euler-Maclaurin formula helps us convert discrete sums into integrals plus correction terms
• For irrational α, we get different behavior than for rational α due to distribution properties

Step-by-Step Solution:

Step 1: Rewrite using fractional part identity Start with $\{y\} = y - \lfloor y \rfloor$, so: $$\sum_{n=1}^{x} \{n^{\alpha}\} = \sum_{n=1}^{x} n^{\alpha} - \sum_{n=1}^{x} \lfloor n^{\alpha} \rfloor$$

Step 2: Handle the first sum The sum $\sum_{n=1}^{x} n^{\alpha}$ has a known asymptotic expansion:

• If α > -1: $\sum_{n=1}^{x} n^{\alpha} = \frac{x^{\alpha+1}}{\alpha+1} + \frac{x^{\alpha}}{2} + O(x^{\alpha-1})$

Step 3: Analyze the floor function sum This is the tricky part! The behavior depends heavily on whether α is rational or irrational:

• For irrational α: Use Weyl's equidistribution theorem
• For rational α = p/q: The fractional parts have period q, leading to different asymptotics

Step 4: Apply distribution results For irrational α, the sequence $\{n^{\alpha}\}$ is equidistributed mod 1, which means: $$\sum_{n=1}^{x} \{n^{\alpha}\} \sim \frac{x}{2} + \text{smaller order terms}$$

The Framework:

Here's the general result structure:

For irrational α > 0: $$\sum_{n=1}^{x} \{n^{\alpha}\} = \frac{x}{2} + O(x^{\theta})$$ where θ < 1 depends on α.

For rational α = p/q with gcd(p,q) = 1: The expansion involves periodic terms and the structure is more complex.

Memory Tip:

Think of fractional parts as "cutting off the integer part" - they measure how far each $n^{\alpha}$ is above its floor. For irrational powers, these fractional parts spread out evenly (equidistribution), giving us the clean asymptotic behavior. The key is always: integer part + fractional part = whole number, so study each piece separately!