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  1. For $x>1$ $$ N(x)=\sum_{0<n<x \\n \equiv 1 \pmod 4\\ n\text{ squarefree}} 1 $$ How to estimate $N(x)$'s order? (Like $N(x) \sim Ax$)
  2. Furthermore, for $n=p_1p_2\cdots p_v$, define $\alpha (n)=v$ ,then for $x>1$ $$ M(x)=\sum_{0<n<x \\n \equiv 1 \pmod 6\\ n\text{ squarefree}} 2^{\alpha (n)} $$ How to estimate $M(x)$'s order?

In fact, I want to learn some methods to estimate these sums. Are there some books, articles or lessons about it?

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  • $\begingroup$ For 1, we have a recurrence relation $$\sum_{\substack{a\leq\sqrt{x}\\ a=\text{odd}}}N(\lfloor{\frac{x}{a^2}}\rfloor)≈\frac{x}{4}$$ Hence, as a rough estimation if $N \sim Ax$, $A$ should be $\frac{2}{\pi^2}$. $\endgroup$
    – Alapan Das
    Commented Dec 15, 2021 at 4:16
  • $\begingroup$ Two 'black box' methods include the Selberg-Delange method and Wirsing's theorems. See e.g. here. If you don't have a congruence condition, these problems are just summation of multiplicative functions. To deal with the congruence condition you have to show cancellation in your sums twisted by a Dirichlet character. $\endgroup$
    – Ofir Gorodetsky
    Commented Dec 15, 2021 at 11:32

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