# Tagged Questions

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### Is there a way to show that $d(n)$, which counts the number of divisors of $n$ is non-increasing? [on hold]

Is there a way to show that $d(n)$, which counts the number of divisors of $n$ is non-increasing? I'm trying to use the Cauchy condensation test to show that $\sum_{n\ge{2}}\frac{d(n)}{n\log^2n}$ is ...
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### Properties of Arithmetic Functions

I was recently working on arithmetic functions and using Perron's formula to obtain asymptotic estimates. One observation I made was that the Dirichlet series often can be written in terms of the ...
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### Order of a function related to divisors

Let $f(n)=\max(\{d(ab):\ a,b\le n\})$ where $d(m)$ is the number of divisors of $m.$ What is the order of $f$? In particular I'm looking for an asymptotic upper bound.
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### Chebyshev's first $\vartheta(x)$ function question

This was an exercise using the first Chebyshev function, $\vartheta(x)= \sum_{p \leq x} \log p.$ The question is simply how to prove (2) below, the rest is my two thoughts on how to proceed. [Edit: ...
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### Comparing average values of an arithmetic function

Suppose $f(n)$ is a positive real-valued arithmetic function such that $$\frac1n\sum_{k=1}^nf(k)\sim g(n)$$ for $g(x)$ a monotonic increasing function. What can be said about the asymptotic behavior ...
### Asymptotics for almost all $x$
Theorem 2.2 in Shparlinski 2006 says: For all positive integers $n\le x$ except possibly $o(x)$ of them, the bound $$M(x)\ll\frac{x}{\log x}\exp\left((C+o(1))(\log\log\log x)^2\right)$$ holds. ...
At the last line of the proof: $\lambda^{-1}(n)=\mu(n)\lambda(n)=\mu^2(n)=|\mu(n)|$. Why $\mu(n)\lambda(n)=\mu^2(n)$? How to prove this?