International Journal of Number Theory最新文献

This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers $n$ such that $n,n+h$ and $n+k$ are all sums of two squares where $h$ and $k$ are two arbitrary integers, and as an immediate corollary obtain, in parametric terms, three consecutive integers that are sums of two squares. Similarly we obtain $n$ in parametric terms such that all the four integers $n,n+1,n+2,n+4$ are sums of two squares. We also find infinitely many integers $n$ such that all the five integers $n,n+1,n+2,n+4,n+5$ are sums of two squares, and finally, we find infinitely many arithmetic progressions, with common difference $4$, of five integers all of which are sums of two squares.

Finding the Frobenius number and the genus of any numerical semigroup $S$ is a well-known open problem. Similarly, it has been studied how to express the Frobenius number and the genus of a quotient of a numerical semigroup. In this paper, by enumerating the Hilbert series of each type of numerical semigroup, we show an expression for the genus of a quotient of numerical semigroups generated by one of the following series: arithmetic progression, geometric series, and Pythagorean triple.

Recently, Andrews introduced separable integer partition classes and analyzed some well-known theorems. In this paper, we investigate partitions with parts separated by parity introduced by Andrews with the aid of separable integer partition classes with modulus $2$. We also extend separable integer partition classes with modulus $1$ to overpartitions, called separable overpartition classes. We study overpartitions and the overpartition analogue of Rogers–Ramanujan identities, which are separable overpartition classes.

By assuming Vinogradov–Korobov-type zero-free regions and the generalized Ramanujan–Petersson conjecture, we establish nontrivial upper bounds for almost all short sums of Fourier coefficients of Hecke–Maass cusp forms for $SL(n,\mathbb{Z})$. As applications, we obtain nontrivial upper bounds for the averages of shifted sums involving coefficients of the Hecke–Maass cusp forms for $SL(n,\mathbb{Z})$. Furthermore, we present a conditional result regarding sign changes of these coefficients.

Keating and Rudnick [The variance of the number of prime polynomials in short intervals and in residue classes, Int. Math. Res. Not.2014(1) (2014) 259–288] derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. We follow their approach but apply this involution, in addition, to the arithmetic progressions. This creates dual arithmetic progressions in the case when the modulus $Q$ is a polynomial in ${𝔽}_q[T]$ such that $Q(0)\ne 0$. The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition.

Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus $g>1$ must be integral or half integral. Actually he proved that for a system $v(M)$ of complex numbers of absolute value 1

$$\begin{array}{ll}\hfill v(M)det{(CZ+D)}^r(r\in ℝ)(0.1)& \end{array}$$

can be an automorphy factor only if $2r$ is integral. We give a different proof for this. It uses Mennicke’s result [Zur Theorie der Siegelschen Modulgruppe, Math. Ann.159 (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for $S{L}_n(n\ge 3)$ and $S{p}_{2n}(n\ge 2)$, Publ. Math. Inst. Hautes Études Sci.33 (1967) 59–137] of Bass–Milnor–Serre.

In this work, we establish two interesting partition identities involving the minimal odd excludant, which has attracted great attention in recent years. In particular, we find a strong refinement of Euler’s celebrated theorem that the number of partitions of an integer into odd parts equals the number of partitions of that integer into distinct parts.

We define a three-character analogue of the generalized Dedekind–Rademacher sum introduced by Hall, Wilson, and Zagier and prove its reciprocity formula which contains all of the reciprocity formulas in the literature for generalized Dedekind–Rademacher sums attached (and not attached) to Dirichlet characters as special cases. Additionally, we prove related polynomial reciprocity formulas which contain all of the polynomial reciprocity formulas in the literature as special cases, such as those given by Carlitz, Beck & Kohl, and the present author.

Let $\tau$ denote the divisor function and ${\mathscr{H}}=\{h_1,\dots ,h_k\}$ be an admissible set. We prove that there are infinitely many $n$ for which the product ${\prod }_{i=1}^k(n+h_i)$ is square-free and ${\sum }_{i=1}^k\tau (n+h_i)\le \lfloor {\rho }_k\rfloor$, where ${\rho }_k$ is asymptotic to $\frac{2126}{2853}{k}^2$. It improves a previous result of Ram Murty and Vatwani, replacing $3/4$ by $2126/2853$. The main ingredients in our proof are the higher rank Selberg sieve and Irving–Wu–Xi estimate for the divisor function in arithmetic progressions to smooth moduli.

In this paper, we investigate the set $S(n)$ of positive integer solutions of the title Diophantine equation. In particular, for a given $n$ we prove boundedness of the number of solutions, give precise upper bound on the common value of ${\sigma }_2({\overline{X}}_n)$ and ${\sigma }_n({\overline{X}}_n)$ together with the biggest value of the variable $x_n$ appearing in the solution. Moreover, we enumerate all solutions for $n\le 16$ and discuss the set of values of $x_n/x_{n-1}$ over elements of $S(n)$.