# Questions tagged [arithmetic-functions]

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### Sign changes of a sequence

Let $f$ be an arithmetical function. Suppose that $f(n)>0$ if $n$ is in an integer set $A$ and that $f(n)<0$ for another integer set $B.$ Is there a result from number theory or an elementary ...
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### Where can I find the problem by Lagarias?

Jeffrey Lagarias proved, unconditionally, that: $$\sigma(n)<H_n+2\exp(H_n)\log(H_n)\qquad n>1$$ This was posed as a problem in: J. C. Lagarias, Problem 10949: A generous bound for divisor ...
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### $\frac{\sigma(n)}{n} < e \ln \ln (n)$ is true?

In Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187–213 (pdf) we find the following result: If the Riemann hypothesis is true ...
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### Arithmetical function comparable to sine function [closed]

I was wondering if there exists or can we construct (using known arithmetic functions) an arithmetical function that has the same behaviour of the function sine or comparable to it (I mean that ...
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### Is $T(n)=\sum_{k=1}^{n}\frac{\lambda(k)\Lambda(k)}{k} \geq 0$ and what is the upper bound of $T(x)=\sum_{n\leq x} \lambda(n)\Lambda(n)$?

Let $\Lambda(n)$ denote the von Mangoldt function: $\Lambda(n)=\log p$ when $n=p^e$ is a prime power ($e\ge 1$) and $\Lambda(n)=0$ otherwise. and $\lambda(n)$ be Liouville Function, , I'm interested ...
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### Arithmetic expansion of harmonic sum

Note: I have modified the initial question as follows: Let $w_1, w_2, \ldots, w_d$ be positive weights, and $x_1, x_2, \ldots, x_d$ be positive variables. Now, let us consider the following harmonic ...
### Is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound?
Pillai showed in 1929 that the function $A(n)$ giving the number solutions of the equation $\phi(x)=n$ is unbounded in (S. Pillai, On some functions connected with $\varphi(n)$, Bull. Amer. Math. Soc. ...