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Michael Hardy
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  1. For $x>1$ $$ N(x)=\sum_{0<n<x \\n \equiv 1 (mod \ 4)\\n\ squarefree} 1 $$$$ N(x)=\sum_{0<n<x \\n \equiv 1 \pmod 4\\ n\text{ squarefree}} 1 $$ 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 $x>1$ $$ M(x)=\sum_{0<n<x \\n \equiv 1 (mod \ 6)\\n\ squarefree} 2^{\alpha (n)} $$$$ M(x)=\sum_{0<n<x \\n \equiv 1 \pmod 6\\ n\text{ squarefree}} 2^{\alpha (n)} $$ 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?

  1. For $x>1$ $$ N(x)=\sum_{0<n<x \\n \equiv 1 (mod \ 4)\\n\ squarefree} 1 $$ 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 $x>1$ $$ M(x)=\sum_{0<n<x \\n \equiv 1 (mod \ 6)\\n\ squarefree} 2^{\alpha (n)} $$ 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?

  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?

How to calculateestimate sums over arithmetic functions like $\sum \mu (n)$progressions?

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five
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How to calculate arithmetic functions like $\sum \mu (n)$?

  1. For $x>1$ $$ N(x)=\sum_{0<n<x \\n \equiv 1 (mod \ 4)\\n\ squarefree} 1 $$ 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 $x>1$ $$ M(x)=\sum_{0<n<x \\n \equiv 1 (mod \ 6)\\n\ squarefree} 2^{\alpha (n)} $$ 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?