Related papers: Additive divisor problem for multiplicative functi…

Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $\lambda_{A}(n) =(-1)^{\Omega_A(n)}$ where $\Omega_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville…

This article provides new asymptotic results for the summatory Mobius function $\sum_{p \leq x} \mu(p+a) =O \left (x(\log x)^{-c} \right )$ and the summatory Liouville function $\sum_{p \leq x} \lambda(p+a) =O \left (x(\log x)^{-c} \right…

Let $F({\bf x})={\bf x}^tQ_{\bf x}+\mathbf{b}^t{\bf x}+c\in\mathbb{Z}[{\bf x}]$ be a quadratic polynomial in $\ell (\ge 3 )$ variables ${\bf x} =(x_{1},...,x_{\ell})$, where $F({\bf x})$ is positive when ${\bf x}\in\mathbb{R}_{\ge…

Let $\lambda$ denote the Liouville function. We show that as $X \rightarrow \infty$, $$ \int_{X}^{2X} \sup_{\alpha} \left | \sum_{x < n \leq x + H} \lambda(n) e(-\alpha n) \right | dx = o ( X H) $$ for all $H \geq X^{\theta}$ with $\theta >…

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 $A_f(1,n)$ be the normalized Fourier coefficients of a $GL(3)$ Maass cusp form $f$ and let $a_g(n)$ be the normalized Fourier coefficients of a $GL(2)$ cusp form $g$. Let $\lambda(n)$ be either $A_f(1,n)$ or the triple divisor function…

Given unitary automorphic cuspidal representations $\pi$ and $\pi'$ defined on $GL_n(\mathbb{A}_E)$ and $GL_m(\mathbb{A}_F)$, respectively, with $E$ and $F$ solvable algebraic number fields we deduce a prime number theorem for the…

Consider the divisor sum $\sum_{n\leq N}\tau(n^2+2bn+c)$ for integers $b$ and $c$. We extract an asymptotic formula for the average divisor sum in a convenient form, and provide an explicit upper bound for this sum with the correct main…

We establish new Bombieri-Vinogradov type estimates for a wide class of multiplicative arithmetic functions and derive several applications, including: a new proof of a recent estimate by Drappeau and Topacogullari for arithmetical…

Let $\lambda$ denote the Liouville function. We show that for all sufficiently large integers $N$, the (non-trivial) convolution sum bound $$ \left|\sum_{1 \leq n < N} \lambda(n) \lambda(N-n)\right| < N-1 $$ holds. This (essentially)…

We show that the expected asymptotic for the sums $\sum_{X < n \leq 2X} \Lambda(n) \Lambda(n+h)$, $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$, and $\sum_{X < n \leq 2X} \Lambda(n) d_k(n+h)$ hold for almost all $h \in [-H,H]$, provided that…

A representation of divisor function $\tau(n)\equiv \sigma_{0}(n)$ by means of logarithmic residue of a function of complex variable is suggested. This representation may be useful theoretical instrument for further investigations of…

We prove an asymptotic formula for the shifted convolution of the divisor functions $d_k(n)$ and $d(n)$ with $k \geq 4$, which is uniform in the shift parameter and which has a power-saving error term, improving results obtained previously…

We give an asymptotic formula for the divisor sum $\sum_{c<n\leq N}\tau\left((n-b)(n-c)\right)$ for integers $b<c$ of the same parity. Interestingly, the coefficient of the main term does not depend on the discriminant as long as it is a…

Let \sigma(n) = \sum_{d \mid n}d be the usual sum-of-divisors function. In 1933, Davenport showed that that n/\sigma(n) possesses a continuous distribution function. In other words, the limit D(u):= \lim_{x\to\infty} \frac{1}{x}\sum_{n \leq…

Let $\pi$ be a cuspidal automorphic representation of a general linear group over the rational numbers. We establish a subconvex bound for the standard $L$-function of $\pi$ in the $t$-aspect. More generally, we address the spectral aspect…

We establish a result of Bombieri-Vinogradov type for the Dirichlet coefficients at prime ideals of the standard $L$-function associated to a self-dual cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n$ over a number field $F$…

In this article, we present relations for the Euler totient function $\varphi(n)$ and the number of divisors $\tau(n)$ in terms of finite sums of integer parts of rational numbers or greatest common divisors of pairs of integers. Some of…

Given a multiplicative function $f$, we let $S(x,f)=\sum_{n\leq x}f(n)$ be the associated partial sum. In this note, we show that lower bounds on partial sums of divisor-bounded functions result in lower bounds on the partial sums…

We deduce asymptotic formulas for the sums $\sum_{n_1,\ldots,n_r\le x} f(n_1\cdots n_r)$ and $\sum_{n_1,\ldots,n_r\le x} f([n_1\cdots n_r])$, where $r\ge 2$ is a fixed integer, $[n_1,\ldots,n_r]$ stands for the least common multiple of the…