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Let $f$ be an arithmetic function such that $f(n)\ll n^{\alpha}$ for some Real number $\alpha$ in $[0,1)$. Can someone give me examples of such functions other than the sum of divisors function or is …

Let $n$ be a positive integer $\geq 2$ et denote by $ P^{+}(n)$ the greatest prime factor of $n$ my question is as follows: If $a$ and $b$ are two numbers, is there any method to express or to bound $ …

Let $x$ be a sufficiently large number. Is there an explicit or asymptotic formula for the following sum $$\sum_{\substack{n\leq x\\ n=a^2+b^2+c^2}} 1.$$ Any reference would be helpful.

In their paper "Moyenne de certains fonctions arithmétiques sur les entiers friables", Tenenbaum and Wu proved that for the case of the function $\beta$ which is the indicator function of integers tha …

Modular forms are used to solve Fermat's last theorem, Mock modular forms are used in black holes theory. I read also in Quanta magazine that Eisenstein series are used to compute what we call Monster …

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 result t …

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 oscila …

De La Breteche and Tenenbaum established in their paper 'Propriétés statistiques des entiers friables' the asymptotic formula for the above sum. They find it as a corollary; Uniformly for $2 \leq y \l …

Let $x$ and $y$ be two positive real numbers. What is the mean value of the function $\log$ on $y$ friables integers less than $x$ i.e the value of the following sum $$\sum_{\substack{n \leq x \\ P(n …

Let $X$ be a positive real number. Can someone help me by providing an asymptotic formula for this sum. $$\sum_{n \leq X, \; n\, \equiv\, a \mod{b}} \log{n},$$ where $a$ and $b$ are two coprime integ …

Let $\chi_{s,k}$ be the characteristic function of integers $n$ which are expressed as sum of $s$ positive $k$-th powers i.e $\chi_{s,k}(n)=1$ if and only if $n=a_1^k+\cdots+a_s^k.$ Examples of this t …

Let $r_{s,k}(n)$ be the number of representation of a naturel number $l$ as a sum of $s$ positive $k$-th powers. Joung Min Song introduced some conditions to study asymtotic behavior of some positive …

Here is a good paper that can answer your question! https://arxiv.org/pdf/1705.07498.pdf

In their paper Sign changes of Hecke eigenvalues, Matomaki and Radziwill showed that (Theorem 1.2 of the paper) for a large enough $x$ , the number of sign changes of sign changes of in the non-vanish …

When I was doing some task in number theory which involves bounds for the Riemann zeta function, I was stuck on the following question: Let $\zeta$ be the Riemann zeta function and let $\zeta^{(k)}$ …