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Given four positive integers $n,$ $m,$ $l$ and $k \geq 2.$ I want to find a lower bound for this expression $$|\sqrt[k]{n}+\sqrt[k]{m}-\sqrt[k]{l}|$$ in terms of these integers. Many thanks

Your product is the following $$\prod_{y<p\leq x} \left(1+\frac{1}{p}\right).$$ You can use the fact that $$\prod_{y<p\leq x} \left(1+\frac{1}{p}\right)\leq \prod_{y<p\leq x} \left(1-\frac{1}{p}\right …

When I was doing some task of analytic number theory I was stuck on computing this sum $$S:=\frac{1}{L} \sum_{q \in \mathcal{Q}} \phi(q) \overline{a}^{\frac{1}{2}},$$ where $\overline{a}$ is the inv …

Let $q\geq 2$ be an integer and $\alpha \in R$ such that $(q-1)\alpha \in R \setminus Z.$ For every positve integer $n$ there exists a unique sequence $(a_j(n)_{j\geq 0},$ $a_j(n) \in \{0,1,...,q-1\}$ …

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 …

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 …

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 $X$ 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 $\mu$ be the Mobius function. In his paper "Explicit estimates on several summatory functions involving the Moebius function", Olivier Ramaré proves the following effective bound: $$\left|\sum_{n\ …

Let $f$ be a primitive form of an even weight $k$ and level $N\geq 1.$ By the theory of Hecke operators, $$\lambda_f(n)=\frac{\hat{f}(n)}{n^{\frac{k-1}{2}}}$$ is a real number. Studying the distributi …

In their paper 'Sign changes of Hecke eigenvalues', Matomaki and Radziwill established in Lemma 6.2 the following result: There exists absolute positive constants $c$ and $\eta$ such that uniformly in …

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 $f$ be a primitive form of an even weight $k\geq 2$ for the full modular group $SL_2(Z)$ and $\lambda_f(n)$ be the $n$-th normalized Fourier coefficient. Define a multiplicative function $g$ by $ …

Let $f$ be a primitive form of an even weight $k\geq 2$ for the full modular group $SL_2(Z)$ and let $\lambda_f(n)$ be the $n$-th normalized Fourier coefficient of $f.$ Can someone provide me with an …

Let $f$ be a primitive form of an even weight $k\geq 2$ for the full modular group $SL_2(Z)$. Following from , Proposition 2.3 from [Z. Rudnick and P. Sarnak, Zeros of principal $L$-functions and ra …