Let $b_n$ denote the function such that $$b_n=\begin{cases}1 & n \text{ is the sum of two squares}\\0&\text{otherwise}\end{cases}$$ How do I show the sum $$\sum_{n \leq x} \frac{b_n}{n} = O(\sqrt{\log{x}})$$? I need a fairly elementary proof which uses the fact that about half the primes are congruent to $1$ mod $4$ in the sum.
2026-03-29 22:44:18.1774824258
How to show this series grows like $\sqrt{\log{x}}$?
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