Related papers: Explicit upper bound for an average number of divi…

In this paper we study universal quadratic polynomials which arise as sums of polygonal numbers. Specifically, we determine an asymptotic upper bound (as a function of $m$) on the size of the set $S_m\subset\mathbb{N}$ such that if a sum of…

In this paper, we consider sums of four generalized polygonal numbers whose parameters are restricted to integers with a bounded number of prime divisors. With some restriction on m modulo 30, we show that for n sufficiently large, it can…

We prove a new upper bound for the number of smooth values of a polynomial with integer coefficients. This improves Timofeev's previous result unless the polynomial is a product of linear polynomials with integer coefficients. As an…

We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.

We study the extent to which divisors of a typical integer $n$ are concentrated. In particular, defining the Erd\H{o}s-Hooley $\Delta$-function by $\Delta(n) := \max_t \# \{d | n, \log d \in [t,t+1]\}$, we show that $\Delta(n) \geq (\log…

How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked whether a polynomial bound is possible, we prove a quasi-optimal bound $\text{(input…

In this article, we study extreme values of quadratic character sums with multiplicative coefficients $\sum_{n \le N}f(n)\chi_d(n)$. For a positive number $N$ within a suitable range, we employ the resonance method to establish a…

We say that $d$ is an exponential unitary divisor of $n=p_1^{a_1}... p_r^{a_r}>1$ if $d=p_1^{b_1}... p_r^{b_r}$, where $b_i$ is a unitary divisor of $a_i$, i.e., $b_i\mid a_i$ and $(b_i,a_i/b_i)=1$ for every $i\in \{1,2,...,r\}$. We survey…

The height of a polynomial $f(x)$ is the largest coefficient of $f(x)$ in absolute value. Let B(n) be the largest height of a polynomial in $\mathbb{Z}[x]$ dividing $x^n-1$. In this paper we investigate the maximal height of divisors of…

In this paper, we study numbers $n$ that can be factored in four different ways as $n = A B = (A + a_1) (B - b_1) = (A + a_2) (B - b_2) = (A + a_3) (B - b_3)$ with $B \le A$, $1 \le a_1 < a_2 < a_3 \le C$ and $1 \le b_1 < b_2 < b_3 \le C$.…

We consider a binary quadratic variant of the Titchmarsh divisor problem and give an asymptotic formula for $\sum_{p^2+q^2\leq N} \tau(p^2+q^2+1)$, where $p,q$ are primes.

The twisted $T$-adic exponential sum associated to a polynomial in one variable is studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the…

We study the sum of divisors of the quadratic form $m_1^2+m_2^2+m_3^2$. Let $$S_3(X)=\sum_{1\le m_1,m_2,m_3\le X}\tau(m_1^2+m_2^2+m_3^2).$$ We obtain the asymptotic formula $$S_3(X)=C_1X^3\log X+ C_2X^3+O(X^2\log^7 X),$$ where $C_1,C_2$ are…

We calculate Gaussian averages of arbitrary exponentials of the matrix variable $X$ with the help of superintegrability, which provides explicit expressions for Schur averages. As in the simpler cases the answer is expressed in terms of…

In this paper we compute the exact divisibility of exponential sums associated to binomials $F(X)=aX^{d_1} +b X^{d_2}$. In particular, for the case where $\max\{d_1,d_2\}\leq\sqrt{p-1}$, the exact divisibility is computed. As a byproduct of…

The subset of quadratic primes {p = an^2 + bn + c : n => 1} generated by an irreducible polynomial f(x) = ax^2 + bx + c over the integers is widely believed to be an unbounded subset of prime numbers. This note provides the details of a…

We study upper bounds for sums of Dirichlet characters. We prove a uniform upper bound of the character sum over all proper generalized arithmetic progressions, which generalizes the classical Polya and Vinogradov inequality. Our argument…

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…

Let $q > 2$ be a prime number and define $\lambda_q := \left( \frac{\tau}{q} \right)$ where $\tau(n)$ is the number of divisors of $n$ and $\left( \frac{\cdot}{q} \right)$ is the Legendre symbol. When $\tau(n)$ is a quadratic residue modulo…

This short note provides a sharper upper bound of a well known inequality for the sum of divisors function. This is a problem in pure mathematics related to the distribution of prime numbers. Furthermore, the technique is completely…