相关论文: On the $L^1$ norm of an exponential sum involving …

We obtain an asymptotic formula for the L^1 norm of the exponential sum $M(\alpha) = \sum_{n\le X}\tau(n)e(n\alpha)$ where $\tau(n) = \sum_{d|n} 1$ is the divisor function. In particular, we show that it is $\sim C\sqrt{X}\log X$ with $C =…

We give a relatively simple proof that \[ \int _0^1\left |\sum _{n\leq x}d(n)e(n\alpha )\right |d\alpha \asymp \sqrt x.\]

We show that the $L^1$ norm of an exponential sum of length $X$ and with coefficients equal to the Liouville or M\"{o}bius function is at least $\gg_{\varepsilon} X^{1/4 - \varepsilon}$ for any given $\varepsilon$. For the Liouville…

Let $\tau(n)$ denote the classical divisor function. In this paper, we consider the hyperbolic fractional sum of the divisor function defined by $$ T(x) = \sum_{n_1 n_2 \leqslant x} \tau\left( \left[ \frac{x}{n_1 n_2} \right] \right) =…

We use the circle method to obtain tight bounds on the $L^p$ norm of an exponential sum involving the divisor function for $p > 2$.

We study an asymptotic behavior of the sum $\sum\limits_{n\le x}\frac{\D \tau(n)}{\D \tau(n+a)}$. Here $\tau(n)$ denotes the number of divisors of $n$ and $a\ge 1$ is a fixed integer.

Let $\tau_k(n)$ be the $k$-th divisor function. In this paper, we derive an asymptotic formula for the sum $$ \sum_{1\leq n_1,n_2, \dots, n_{\ell}\leq X^{\frac{1}{r}} \atop 1\leq n_{\ell+1}\le X^{\frac{1}{s}}}\tau_k(n_1^r+n_2^r+\dots…

Let $\tau_3(n)$ be the triple divisor function which is the number of solutions of the equation $d_1d_2d_3=n$ in natural numbers. It is shown that $$ \sum_{1\leq n_1,n_2,n_3\leq \sqrt{x}}\tau_3(n_1^2+n_2^2+n_3^2)=c_1x^{\frac{3}{2}}(\log…

Let $\tau(n)$ stand for the number of divisors of the positive integer $n$. We obtain upper bounds for $\tau(n)$ in terms of $\log n$ and the number of distinct prime factors of $n$.

Let $\tau(n)$ be the number of divisors of $n$. We give an elementary proof of the fact that $$ \sum_{n\le x} \tau(n)^r =xC_{r} (\log x)^{2^r-1}+O(x(\log x)^{2^r-2}), $$ for any integer $r\ge 2$. Here, $$ C_{r}=\frac{1}{(2^r-1)!}…

Let $\mu(n)$ be the M\"{o}bius function and $e(\alpha)=e^{2\pi i\alpha}$. In this paper, we study upper bounds of the classical sum $$S(x,\alpha):=\sum_{1\leq n\leq x}\mu(n)e(\alpha n).$$ We can improve some classical results of Baker and…

Let $\tau$ denote the divisor function, and $f$ be any multiplicative function that satisfies some mild hypotheses. We establish the asymptotic formula or non-trivial upper bound for the shifted convolution sum $\sum_{n \leq…

We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc…

We consider a problem posed by Shparlinski, of giving nontrivial bounds for rational exponential sums over the arithmetic function $\tau(n)$, counting the number of divisors of $n$. This is done using some ideas of Sathe concerning the…

We prove that $$ \sum_{p \leq x} \frac{1}{\tau(p-1)} \asymp \frac{x}{(\log x)^{3/2}}, \quad \quad \sum_{n \leq x} \frac{1}{\tau(n^2+1)} \asymp \frac{x}{(\log x)^{1/2}}, $$ where $\tau(n)=\sum_{d|n}1$ is the number of divisors of $n$, and…

We obtain several asymptotic estimates for the sums of the restricted divisor function $$ \tau_{M,N}(k) = #\{1 \le m \le M, \ 1\le n \le N: mn = k\} $$ over short arithmetic progressions, which improve some results of J. Truelsen. Such…

Let $f$ be a real polynomial with irrational leading co-efficient. In this article, we derive distribution of $f(n)$ modulo one for all $n$ with at least three divisors and also we study distribution of $f(n)$ for all square-free $n$ with…

We study sums of the shape $\sum_{n \leqslant x} f \left( \lfloor x/n \rfloor \right)$ where $f$ is either the von Mangoldt function or the Dirichlet-Piltz divisor functions. We improve previous estimates when $f = \Lambda$ and $f = \tau$,…

We examine exponential sums of the form $\sum_{n \le X} w(n) e^{2\pi i\alpha n^k}$, for $k=1,2$, where $\alpha$ satisfies a generalized Diophantine approximation and where $w$ are different arithmetic functions that might be multiplicative,…

Let us denote by $\tau(n)$ and $\si(n)$ the number and the sum of the divisors of $n$ and by $\vfi$ Euler's function. We give effective upper bounds for $\frac{n}{\vfi(n)}$ in terms of $\vfi(n)$, and for $\frac{\si(n)}{n}$ in terms of…