Related papers: Sums of divisors on arithmetic progressions

We study primitive divisors of terms of the sequence P_n=n^2+b, for a fixed integer b which is not a negative square. It seems likely that the number of terms with a primitive divisor has a natural density. This seems to be a difficult…

Given a rational $a=p/q$ and $N$ nonnegative $d$-dimensional real vectors $u_1$, ..., $u_N$, we show that it is always possible to choose $(d-1)+\lceil (pN-d+1)/q\rceil$ of them such that their sum is (componentwise) at least…

Let $s(n)$ denote the sum of the proper divisors of the natural number $n$. We show that the number of $n \leq x$ such that $s(n)$ is a sum of two squares has order of magnitude $x/\sqrt{\log x}$, which agrees with the count of $n \leq x$…

We prove that there is a constant $c > 0$ depending only on $M \geq 1$ and $\mu \geq 0$ such that $$\int_y^{y+a}{|g(t)| \, dt} \geq \exp (-c/(a\delta))\,, a \in (0,\pi]\,,$$ for every $g$ of the form $$g(t) = \sum_{j=0}^n{a_j…

The inequalities concern the sum of s powers of primes with non-integer exponent c>1. Here s =2,3,4,or 5. The equations are similar, taking integer part before summing; here s = 3 or 5. New ranges of c are found in all cases for which many…

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite…

A well known result of Newman says that upto a limit, multiples of $3$ with even number of 1's in binary representation always exceed multiples of $3$ with odd number of 1's. The phenomenon of preponderance of even number of 1's is now…

We prove that the the discrepancy of arithmetic progressions in the $d$-dimensional grid $\{1, \dots, N\}^d$ is within a constant factor depending only on $d$ of $N^{\frac{d}{2d+2}}$. This extends the case $d=1$, which is a celebrated…

We give explicit bounds on sums of $d(n)^2$ and $d_4(n)$, where $d(n)$ is the number of divisors of $n$ and $d_4(n)$ is the number of ways of writing $n$ as a product of four numbers. In doing so we make a slight improvement on the upper…

We show that there exists a positive constant C such that the following holds: Given an infinite arithmetic progression A of real numbers and a sufficiently large integer n (depending on A), there needs at least Cn geometric progressions to…

We consider the positive divisors of a natural number that do not exceed its square root, to which we refer as the {\it small divisors\/} of the natural number. We determine the asymptotic behavior of the arithmetic function that adds the…

For a fixed unit vector $a=(a_1,a_2,\ldots,a_n)\in S^{n-1}$, we consider the $2^n$ sign vectors $\varepsilon=(\varepsilon^1,\varepsilon^2,\ldots,\varepsilon^n)\in \{+1,-1\}^n$ and the corresponding scalar products $\varepsilon\cdot…

We study symmetric nonnegative forms and their relationship with symmetric sums of squares. For a fixed number of variables $n$ and degree $2d$, symmetric nonnegative forms and symmetric sums of squares form closed, convex cones in the…

Let $d(n; r_1, q_1, r_2, q_2)$ be the number of factorization $n=n_1n_2$ satisfying $n_i\equiv r_i\pmod{q_i}$ ($i=1,2$) and $\Delta(x; r_1, q_1, r_2, q_2)$ be the error term of the summatory function of $d(n; r_1, q_1, r_2, q_2)$ with…

It is not difficult to think of applications that can be modelled as graph problems in which placing some facility or commodity at a vertex has some positive or negative effect on the values of all the vertices out to some distance, and we…

The \textit{sepr-sequence} of an $n\times n$ real matrix $A$ is $(s_1,\ldots,s_n)$, where $s_k$ is the subset of those signs of $+,-,0$ that appear in the values of the $k\times k$ principal minors of $A$. The $12\times 12$ matrix…

Let $s_1, s_2, \ldots$ be the sequence of positive integers, arranged in increasing order, that are representable by any binary quadratic form of fixed discriminant $D$. We show that \[ \limsup_{n \rightarrow \infty} \frac{s_{n+1}-s_n}{\log…

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…

For an integer b>=2, let s_b(n) be the sum of the digits of the integer n when written in base b, and let S_b(N) be the sum of s_b(n) over n=0,...,N-1, so that S_b(N) is the sum of all b-ary digits needed to write the numbers 0,1,...,N-1.…

Let $D$ be a subset of a finite commutative ring $R$ with identity. Let $f(x)\in R[x]$ be a polynomial of positive degree $d$. For integer $0\leq k \leq |D|$, we study the number $N_f(D,k,b)$ of $k$-subsets $S\subseteq D$ such that…