Related papers: On a restricted linear congruence

We consider the problem of finding the set of permutations $r_j$ of $\{1,\cdots , n\}$ such that $\sum_{i=1}^n \prod_{j=1}^k r_j(i)$ is maximized or minimized. While the set of permutations maximizing this value are easily determined,…

We calculate the triple correlations for the truncated divisor sum $\lambda_{R}(n)$. The $\lambda_{R}(n)$'s behave over certain averages just as the prime counting von Mangoldt function $\Lambda(n)$ does or is conjectured to do. We also…

Let $\sigma_i$, $i=1,\ldots,n$, denote positive Borel measures on $\mathbb{R}^d$, let $\mathcal{D}$ denote the usual collection of dyadic cubes in $\mathbb{R}^d$ and let $K:\,\mathcal{D}\to[0,\infty)$ be a~map. In this paper we give…

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…

In a recent paper, Lapkova uses a Tauberian theorem to derive the asymptotic formula for the divisor sum $\sum_{n \leq x} d( n (n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove her result by…

We consider sums of the form \[\sum_{j=0}^{n-1}F_1(a_1n+b_1j+c_1)F_2(a_2n+b_2j+c_2)... F_k(a_kn+b_kj+c_k),\] in which each $\{F_i(n)\}$ is a sequence that satisfies a linear recurrence of degree $D(i)<\infty$, with constant coefficients. We…

In this paper, we prove several new infinite families of Ramanujan--like congruences satisfied by the coefficients of the generating function $U_t(a,q)$ which is an extension of MacMahon's generalized sum-of-divisors function. As a…

We define an $f$-restricted partition $p_f(n,k)$ of fixed length $k$ given by the bivariate generating series \begin{align*} Q_f(z,u) \coloneqq 1+\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} p_f(n,k) u^kz^n…

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 $p_{-t}(n)$ denote the number of partitions of $n$ into $t$ colors. In analogy with Ramanujan's work on the partition function, Lin recently proved in \cite{Lin} that $p_{-3}(11n+7)\equiv0\pmod{11}$ for every integer $n$. Such…

A cubic partition consists of partition pairs $(\lambda,\mu)$ such that $\vert\lambda\vert+\vert\mu\vert=n$ where $\mu$ involves only even integers but no restriction is placed on $\lambda$. This paper initiates the notion of generalized…

We call a CNF formula linear if any two clauses have at most one variable in common. Let m(k) be the largest integer m such that any linear k-CNF formula with <= m clauses is satisfiable. We show that 4^k / (4e^2k^3) <= m(k) < ln(2) k^4…

We estimate the number of solutions of certain diagonal congruences involving factorials. We use these results to bound exponential sums with products of two factorials $n!m!$ and also derive asymptotic formulas for the number of solutions…

We obtain several asymptotic formulas for the sum of the divisor function $\tau(n)$ with $n \le x$ in an arithmetic progressions $n \equiv a \pmod q$ on average over $a$ from a set of several consecutive elements from set of reduced…

Inspired by Cohen and te Riele~\cite{Cohen1996}, who computationally verified that for every $n \leq 400$ there exists $k$ such that $\sigma^k(n) \equiv 0 \pmod{n}$ (where $\sigma^k$ denotes the $k$-fold iteration of the sum-of-divisors…

Let $p$ be a large prime number. We prove that any integer $\lambda$ modulo $p$ can be represented in the form $$ m!n! +\sum_{i=1}^{47}n_i!\equiv \lambda \pmod p, $$ with $\max\{m,n,n_1,\ldots,n_{47}\}\ll p^{1300/1301}.$ This improves the…

We focus on three pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong -- each is vitiated by divergent series. We…

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \begin{equation*} \sum\limits_{i+j+k=p^{r}\atop{i,j,k\in \mathcal{P}_{p}}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} (\bmod p^{r}),…

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 revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x^2+5y^2. Making use of Ramanujan's_1\psi_1 summation formula we establish a new Lambert series identity for…