International Journal of Number Theory

Let $k\ge 3$ be an integer, $s=2^k+1$, $(a_i,a_j)=1$, $1\le i<j\le s$, where $a_1,\dots ,a_s$ are nonzero integers, and let $n$ be an integer. Suppose that $a_1,\dots ,a_s$ satisfy some necessary congruent conditions. In this paper, it is proved that

• ⁽ⁱ⁾if $a_j$ are not all of the same sign, then the equation $a_1{p}_1^k+\cdots +a_s{p}_s^k=n$ has prime solutions satisfying $p_j\ll |n{|}^{1/k}+max{\{|a_j|\}}^{C(k)+𝜀}$,
• ⁽ⁱⁱ⁾if all $a_j$ are positive and $n\gg max{\{|a_j|\}}^{kC(k)+1+𝜀}$, then $a_1{p}_1^k+\cdots +a_s{p}_s^k=n$ is solvable in primes $p_j$,

where $C(k)=1+\frac{6\cdot 2^k+20}{3\cdot 2^{2k-1}+3\cdot 2^{k-1}-20}$.

Our result uses $C(3)=\frac{39}{22}\approx 1.7727\cdots$ as an improvement of the recent result $C(3)=2$ due to Zhao (2016), and largely improves the results $C(k)=3\cdot 2^{k-1}$ for $k\ge 4$ proved by Yang and Hu (2016).

We describe work of Faltings on the construction of étale cohomology classes associated to symplectic Shimura varieties and show that they satisfy certain trace compatibilities similar to the ones of Siegel units in the modular curve case. Starting from those, we construct a two-variable family of trace compatible classes in the cohomology of a unitary Shimura variety.

For a set $A$ of nonnegative integers, let $R_2(A,n)$ denote the number of solutions to $n=a+a^{\prime }$ with $a,a^{\prime }\in A,a<a^{\prime }$. Let $A_0$ be the set of all nonnegative integers $n$ with an even number of ones in the binary representation of $n$ and let $B_0=\mathbb{N}\backslash A_0$. Let $A\subseteq \mathbb{N}$ and $N$ be a positive integer such that $R_2(A,n)=R_2(\mathbb{N}\backslash A,n)$ for all $n\ge 2N-1$. In 2011, Chen proved that, if $|A\cap A_0|=+\infty$ and $|A\cap B_0|=+\infty$, then $R_2(A,n)\ge \frac{n}{40N(N+1)}-1$ for all $n\ge 1$. In this paper, we improve the lower bound by proving that $R_2(A,n)\ge \frac{n+3}{56N-52}-1$ for all $n\ge 1$.

A cyclic quadrilateral is called a Brahmagupta quadrilateral if the lengths of its four sides and two diagonals, and the area are all given by integers. In this paper, we consider the hitherto unsolved problem of finding two Brahmagupta quadrilaterals with equal perimeters and equal areas. We obtain two parametric solutions of the problem — the first solution generates examples in which each quadrilateral has two equal sides while the second solution gives quadrilaterals all of whose sides are unequal. We also show how more parametric solutions of the problem may be obtained.

Recently, many identities for the convolution sum $W_{a,b}(n):={\sum }_{a\ell +bm=n}\sigma (\ell )\sigma (m)$ of the divisor function $\sigma (n):={\sum }_{d|n}d$ have been obtained since Royer obtained by the theory of quasimodular forms. We also present new identities for $ab=17$, $29$, $41$, $47$, $59$ and $71$ by using quasimodular forms.

Using convexity properties of reciprocals of zeta functions, especially the reciprocal of the Riemann zeta function, we show that certain weighted Mertens sums are biased in favor of square-free integers with an odd number of prime factors. We study such type of bias for different ranges of the parameters and then consider generalizations to Mertens sums supported on semigroups of integers generated by relatively large subsets of prime numbers. We further obtain a wider range for the parameters both unconditionally and then conditionally on the Riemann Hypothesis. At the same time, we extend to certain semigroups, two classical summation formulas originating from the works of Landau concerning the behavior of derivatives of the reciprocal of the Riemann zeta function at $s=1$.

We introduce a symbolic representation of $r$-fold harmonic sums at negative indices. This representation allows us to recover and extend some recent results by Duchamp et al., such as recurrence relations and generating functions for these sums. This approach is also applied to the study of the family of extended Bernoulli polynomials, which appear in the computation of harmonic sums at negative indices. It also allows us to reinterpret the Raabe analytic continuation of the multiple zeta function as both a constant term extension of Faulhaber’s formula, and as the result of a natural renormalization procedure for Faulhaber’s formula.

An integer of the form $P_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ for an integer $x$ is called a generalized $m$-gonal number. For positive integers ${\alpha }_1,\dots ,{\alpha }_u$ and ${\beta }_1,\dots ,{\beta }_v$, a mixed sum $\mathrm{\Phi }={\alpha }_1{P}_4(x_1)+\cdots +{\alpha }_u{P}_4(x_u)+{\beta }_1{P}_8(y_1)+\cdots +{\beta }_v{P}_8(y_v)$ of generalized $4$- and $8$-gonal numbers is called universal if $\mathrm{\Phi }=N$ has an integer solution for every nonnegative integer $N$. In this paper, we prove that there are exactly 1271 proper universal mixed sums of generalized $4$- and $8$-gonal numbers. Furthermore, the “$61$-theorem” is proved, which states that an arbitrary mixed sum of generalized $4$- and $8$-gonal numbers is universal if and only if it represents the integers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$, $13$, $14$, $15$, $18$, $20$, $30$, $60$, and $61$.

An integer $n\ge 1$ is said to be practical if every natural number $m\le n$ can be expressed as a sum of distinct positive divisors of $n$. The number of practical numbers up to $x$ is asymptotic to $cx/logx$, where $c$ is a constant. In this paper, we show that $c=1.33607\dots .$

For $k\in \mathbb{N}$ and $p$ a prime number, define ${\mathrm{\Gamma }}^{\ast }(k,p)$ to be the smallest positive integer $n\in \mathbb{N}$ such that any diagonal form $f(x_1,\dots ,x_s)=a_1{x}_1^k+\cdots +a_s{x}_s^k$, with integer coefficients, has nontrivial zero over ${\mathbb{Q}}_p$ whenever $s\ge n$. A special case of a conjecture attributed to Artin states that ${\mathrm{\Gamma }}^{\ast }(k,p)\le k^2+1$. It is well known that the equality occurs when $p=k+1$. In this paper, we obtain the exact values of ${\mathrm{\Gamma }}^{\ast }(10,p)$ for all primes $p$ and, except for $p=11$, these values are much lower than those established in the conjecture, as might be expected.

In this paper, we establish a rate of convergence to zero of the measure of the set $x\in {\mathbb{Q}}_p$ for which the inequality $|P(x){|}_p<Q^{-w}$ for $w>n+1$ has a solution in the integer polynomials $P$ of degree $n$ and height bounded by $Q\in \mathbb{N}$.