We obtain asymptotic formulas with remainder terms for the hyperbolic summations $\sum_{mn\le x} f((m,n))$ and $\sum_{mn\le x} f([m,n])$, where $f$ belongs to certain classes of arithmetic functions, $(m,n)$ and $[m,n]$ denoting the gcd and lcm of the integers $m,n$. In particular, we investigate the functions $f(n)=τ(n), \log n, ω(n)$ and $Ω(n)$. We also define a common generalization of the latter three functions, and prove a corresponding result.
In this paper, we consider the divisibility problem of LCM matrices by GCD matrices in the ring $M_n(\mathbb{Z})$ proposed by Hong in 2002 and in particular a conjecture concerning the divisibility problem raised by Zhao in 2014. We present some certain gcd-closed sets on which the LCM matrix is not divisible by the GCD matrix in the ring $M_n(\mathbb{Z})$. This could be the first theoretical evidence that Zhao's conjecture might be true. Furthermore, we give the necessary and sufficient conditions on the gcd-closed set $S$ with $|S|\leq 8$ such that the GCD matrix divides the LCM matrix in th…
In this paper, we study the eigenvalues of the GCD matrix $(S_n)$ and the LCM matrix $[S_n]$ defined on $S_n=\{1,2,\ldots,n\}$. We present upper and lower bounds for the smallest and the largest eigenvalues of $(S_n)$ and $[S_n]$ in terms of particular arithmetical functions.
We study the distribution of the generalized gcd and lcm functions on average. The generalized gcd function, denoted by $(m,n)_b$, is the largest $b$-th power divisor common to $m$ and $n$. Likewise, the generalized lcm function, denoted by $[m,n]_b$, is the smallest $b$-th power multiple common to $m$ and $n$. We derive asymptotic formulas for the average order of the arithmetic, geometric, and harmonic means of $(m,n)_b$. Additionally, we also deduce asymptotic formulas with error terms for the means of $(n_1,n_2,\cdots, n_k)_b$, and $[n_1,n_2,\cdots, n_k]_b$ over a set of lattice points, th…
Let $S=\{x_1,x_2,\dots,x_n\}$ be a set of distinct positive integers, and let $f$ be an arithmetical function. The GCD matrix $(S)_f$ on $S$ associated with $f$ is defined as the $n\times n$ matrix having $f$ evaluated at the greatest common divisor of $x_i$ and $x_j$ as its $ij$ entry. The LCM matrix $[S]_f$ is defined similarly. We consider inertia, positive definiteness and $\ell_p$ norm of GCD and LCM matrices and their unitary analogs. Proofs are based on matrix factorizations and convolutions of arithmetical functions.
We use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd's and lcm's of $k$ positive integers, where $k\ge 2$ is fixed. We refine and generalize an asymptotic formula of Bordellès (2007), and extend certain related results of Hilberdink and Tóth (2016). We also formulate some conjectures and open problems.
Let $R$ be a gcd-domain (for example let $R$ be a unique factorization domain), and let $(a_n)_{n\geqslant1}$ be a sequence of nonzero elements in $R$. We prove that $\gcd(a_n,a_m)=a_{\gcd(n,m)}$ for all $n,m\geqslant1$ if and only if $$a_n=\prod\limits_{d\mid n} c_d\quad\mbox{for} \ n\geqslant1, $$ where $c_1=a_1$ and $c_n=\mbox{lcm}(a_1,a_2,\dots,a_n)/\mbox{lcm}(a_1,a_2,\dots,a_{n-1})$ for $n\geqslant2$. All equalities with gcd and lcm are determined up to units of $R$.
This paper is devoted to the study of statistical properties of the greatest common divisor and the least common multiple of random samples of positive integers.
For integers $x$ and $y$, $(x, y)$ and $[x, y]$ stand for the greatest common divisor and the least common multiple of $x$ and $y$ respectively. Denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix having the $a$th power of $(x_i,x_j)$ (resp. $[x_i,x_j]$) as its $(i,j)$-entry. For any $x\in S$, define $G_{S}(x):=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S) \Rightarrow y\in \{d,x\}\}$. In this paper, we show th…
Given two sets $A$ and $B$ of integers, we consider the problem of finding a set $S \subseteq A$ of the smallest possible cardinality such the greatest common divisor of the elements of $S \cup B$ equals that of those of $A \cup B$. The particular cases of $B = \emptyset$ and $\#B = 1$ are of special interest and have some links with graph theory. We also consider the corresponding question for the least common multiple of the elements. We establish NP-completeness and approximation results for these problems by relating them to the Minimum Cover Problem.
We obtain asymptotic formulas for the sums $\sum_{n_1,\ldots,n_k\le x} f((n_1,\ldots,n_k))$ and $ \sum_{n_1,\ldots,n_k\le x} f([n_1,\ldots,n_k])$ involving the gcd and lcm of the integers $n_1,\ldots,n_k$, where $f$ belongs to certain classes of additive arithmetic functions. In particular, we consider the generalized omega function $Ω_{\ell}(n)= \sum_{p^ν\mid\mid n} ν^{\ell}$ investigated by Duncan (1962) and Hassani (2018), and the functions $A(n)=\sum_{p^ν\mid\mid n} νp$, $A^*(n)= \sum_{p \mid n} p$, $B(n)=A(n)-A^*(n)$ studied by Alladi and Erdős (1977). As a key auxiliary result we use an …
We prove limit theorems for the greatest common divisor and the least common multiple of random integers. While the case of integers uniformly distributed on a hypercube with growing size is classical, we look at the uniform distribution on sublevel sets of multivariate symmetric polynomials, which we call hyperbolic regions. Along the way of deriving our main results, we obtain some asymptotic estimates for the number of integer points in these hyperbolic domains, when their size goes to infinity.
In this note we prove a new estimate on so-called GCD sums (also called Gál sums), which, for certain coefficients, improves significantly over the general bound due to de la Bretèche and Tenenbaum. We use our estimate to prove new results on the equidistribution of sequences modulo 1, improving over a result of Aistleitner, Larcher, and Lewko on how the metric poissonian property relates to the notion of additive energy. In particular, we show that arbitrary subsets of the squares are metric poissonian.
Let $\gcd(j,k)$ be the greatest common divisor of the integers $j$ and $k$. In this paper, we give several interesting asymptotic formulas for weighted averages of the $\gcd$-sum function $f(\gcd(j,k)) $ and the function $\sum_{d|k, d^{s}|j}(f*μ)(d) $ for any positive integers $j$ and $k$, namely $$ \sum_{k\leq x}\frac{1}{k^{r+1}}\sum_{j=1}^{k}j^{r}f(\gcd(j,k)) \quad \text{and} \quad \sum_{k\leq x}\frac{1}{k^{s(r+1)}}\sum_{j=1}^{k^s}j^{r} \sum_{\substack{d|k d^{s}|j}}(f*μ)(d), $$ with any fixed integer $s> 1$ and any arithmetical function $f$. We also establish mean value formulas for the erro…
Let $\gcd(k,j)$ be the greatest common divisor of the integers $k$ and $j$. For any arithmetical function $f$, we establish several asymptotic formulas for weighted averages of gcd-sum functions with weight concerning logarithms, that is $$\sum_{k\leq x}\frac{1}{k} \sum_{j=1}^{k}f(\gcd(k,j)) \log j.$$ More precisely, we give asymptotic formulas for various multiplicative functions such as $f=id$, $ϕ$, $id_{1+a}$ and $ϕ_{1+a}$ with $-1<a<0$. We also establish some formulas of Dirichlet series having coefficients of the sum function $\sum_{j=1}^{k}s_{k}(j)\log j$ where $s_{k}(j)$ is Anderson--Ap…
In 1876 H. J. S. Smith defined an LCM matrix as follows: let S = {x_1, x_2, ..., x_n} be a set of positive integers. The LCM matrix [S] is the n $\times$ n matrix with lcm(x_i , x_j) as its ij entry. During the last 30 years singularity of LCM matrices has interested many authors. In 1992 Bourque and Ligh ended up conjecturing that if the GCD closedness of the set S (which means that gcd(x_i, x_j) $\in$ S for all i, j $\in$ {1, 2, . . . , n}), suffices to guarantee the invertibility of the matrix [S]. However, a few years later this conjecture was proven false first by Haukkanen et al. and the…
We prove that \[ \sum_{k,{\ell}=1}^N\frac{(n_k,n_{\ell})^{2α}}{(n_k n_{\ell})^α} \ll N^{2-2α} (\log N)^{b(α)} \] holds for arbitrary integers $1\le n_1<\cdots < n_N$ and $0<α<1/2$ and show by an example that this bound is optimal, up to the precise value of the exponent $b(α)$. This estimate complements recent results for $1/2\le α\le 1$ and shows that there is no "trace" of the functional equation for the Riemann zeta function in estimates for such GCD sums when $0<α<1/2$.
Let $\gcd(k,j)$ denote the greatest common divisor of the integers $k$ and $j$, and let $r$ be any fixed positive integer. Define $$ M_r(x; f) := \sum_{k\leq x}\frac{1}{k^{r+1}}\sum_{j=1}^{k}j^{r}f(\gcd(j,k)) $$ for any large real number $x\geq 5$, where $f$ is any arithmetical function. Let $ϕ$, and $ψ$ denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of $M_r(x; {\rm id})$, $M_r(x;ϕ)$ and $M_r(x;ψ)$. Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic for…
It is shown that the maximum of $|ζ(1/2+it)|$ on the interval $T^{1/2}\le t \le T$ is at least $\exp\left((1/\sqrt{2}+o(1)) \sqrt{\log T \log\log\log T/\log\log T}\right)$. Our proof uses Soundararajan's resonance method and a certain large GCD sum. The method of proof shows that the absolute constant $A$ in the inequality \[ \sup_{1\le n_1<\cdots < n_N} \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N \exp\left(A\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right), \] established in a recent paper of ours, cannot be taken smaller than $1$.
Let $S=\{x_1,x_2,\ldots,x_n\}$ be a finite set of distinct positive integers. Throughout this article we assume that the set $S$ is GCD closed. The LCM matrix $[S]$ of the set $S$ is defined to be the $n\times n$ matrix with $\mathrm{lcm}(x_i,x_j)$ as its $ij$ element. The famous Bourque-Ligh conjecture used to state that the LCM matrix of a GCD closed set $S$ is always invertible, but currently it is a well-known fact that any nontrivial LCM matrix is indefinite and under the right circumstances it can be even singular (even if the set $S$ is assumed to be GCD closed). However, not much more …
It is proved that \[ \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N\exp\left(C\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right) \] holds for arbitrary integers $1\le n_1<\cdots < n_N$. This bound is essentially better than that found in a recent paper of Aistleitner, Berkes, and Seip and can not be improved by more than possibly a power of $1/\sqrt{\log\log\log N}$. The proof relies on ideas from classical work of Gál, the method of Aistleitner, Berkes, and Seip, and a certain completeness property of extremal sets of square-free numbers.
In this paper, we establish the following two identities involving the Gamma function and Bernoulli polynomials, namely $$ \sum_{k\leq x}\frac{1}{k^s} \sum_{j=1}^{k^s}\logΓ\left(\frac{j}{k^s}\right) \sum_{\substack{d|k \\ d^{s}|j}}f*μ(d) \quad {\rm and } \quad \sum_{k\leq x}\frac{1}{k^s}\sum_{j=0}^{k^{s}-1} B_{m}\sum_{\substack{d|k \\ d^{s}|j}} f*μ(d) $$ with any fixed integer $s> 1$ and any arithmetical function $f$. We give asymptotic formulas for them with various multiplicative functions $f$. We also consider several formulas of the Dirichlet series associated with the above identities. Th…
For a nonempty finite set $A$ of positive integers, let $\gcd\left(A\right)$ denote the greatest common divisor of the elements of $A$. Let $f\left(n\right)$ and $Φ\left(n\right)$ denote, respectively, the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\right) = 1$ and the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\cup\left\{n\right\}\right) =1$. Let $D\left(n\right)$ be the divisor sum of $f\left(n\right)$. In this article, we obtain partial sums of $f\left(n\right)$, $Φ\left(n\right)$ and $D\left(n\right)$. We also obtain…
We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these established expansions generating sums of the form $\sum_{d: (d,n)=1} f(d)$ (type I) and the Anderson-Apostol sums $\sum_{d|(m,n)} f(d) g(n/d)$ (type II) for any arithmetic functions $f$ and $g$. Our treatment of the type II sums includes a matrix-based factorization method relating the partition function $p(n)$ to arbitrary arithmetic functions $f$. We also conclude…
In recent years the question of maximizing GCD sums regained interest due to its firm link with large values of $L$-functions. In the present paper we initiate the study of minimizing for positive weights~$w$ of normalized $L^1$- norm the sum $\sum_{m_1 , m_2 \leqslant N} w({m_1})w({m_2})\frac{(m_1,m_2)}{\sqrt{m_1m_2}} $. We consider as well the intertwined question of minimizing a weighted version of the usual multiplicative energy. We give three applications of our results. Firstly we obtain a logarithmic refinement of Burgess' bound on character sums $\displaystyle{\sum_{M<n\leqslant M+N}χ(…
In 2021, Guyer and Mbirika gave two equivalent formulas that computed the greatest common divisor (GCD) of all sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In this current paper, we extend their results to the GCD of all sums of $k$ consecutive squares of these numbers. Denoting these GCD values by the symbol $\mathcal{G}_{G_0, G_1}^2\!(k)$, we prove $\mathcal{G}_{G_0, G_1}^2\!(k) = \gcd\left(G_k G_{k+1} - G_0 G_1,\; G_{k+1}…
Using GCD sums, we show that the set of the primes has small common multiplicative energy with an arbitrary exponentially big integer set $S$ and, in particular, size of any arithmetic progression in $S$ having the beginning at zero, is at most $O(\log |S| \cdot \log \log |S|)$. This result can be considered as an integer analogue of Vinogradov's question about the least quadratic non--residue. The proof rests on a certain repulsion property of the function $f(x)=\log x$. Also, we consider the case of general $k$--convex functions $f$ and obtain a new incidence result for collections of the cu…
Given two polynomials $P(\underline x)$, $Q(\underline x)$ in one or more variables and with integer coefficients, how does the property that they are coprime relate to their values $P(\underline n), Q(\underline n)$ at integer points $\underline n$ being coprime? We show that the set of all $\gcd (P(\underline n), Q(\underline n))$ is stable under gcd and under lcm. A notable consequence is a result of Schinzel: if in addition $P$ and $Q$ have no fixed prime divisor (i.e., no prime dividing all values $P(\underline n)$, $Q(\underline n)$), then $P$ and $Q$ assume coprime values at "many" inte…
We explore the sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In particular, we give precise values for the greatest common divisor (GCD) of all sums of $k$ consecutive terms of $\left(G_n\right)_{n \geq 0}$. When $G_0 = 0$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Fibonacci numbers, and when $G_0 = 2$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Lucas numbers. Denoting the GCD of all sum…
Let $\gcd(d_{1},\ldots,d_{k})$ be the greatest common divisor of the positive integers $d_{1},\ldots,d_{k}$, for any integer $k\geq 2$, and let $τ$ and $μ$ denote the divisor function and the Möbius function, respectively. For an arbitrary arithmetic function $g$ and for any real number $x>5$ and any integer $k\geq 3$, we define the sum $$ S_{g,k}(x) :=\sum_{n\leq x}\sum_{d_{1}\cdots d_{k}=n} g(\gcd(d_{1},\ldots,d_{k})) $$ In this paper, we give asymptotic formulas for $S_{τ,k}(x)$ and $S_{μ,k}(x)$ for $k\geq 3$.
Let $S_{\rm lcm}(n)$ denote the set of permutations $π$ of $[n]=\{1,2,\dots,n\}$ such that ${\rm lcm}[j,π(j)]\le n$ for each $j\in[n]$. Further, let $S_{\rm div}(n)$ denote the number of permutations $π$ of $[n]$ such that $j\midπ(j)$ or $π(j)\mid j$ for each $j\in[n]$. Clearly $S_{\rm div}(n)\subset S_{\rm lcm}(n)$. We get upper and lower bounds for the counts of these sets, showing they grow geometrically. We also prove a conjecture from a recent paper on the number of "anti-coprime" permutations of $[n]$, meaning that each $\gcd(j,π(j))>1$ except when $j=1$.
We prove that the reciprocal sum $S_k(x)$ of the least common multiple of $k\geq 3$ positive integers in $\N\cap [1,x]$ satisfies $$ S_k(x)=P_{2^k-1}(\log x)+O(x^{-θ_k+ε}) $$ where $P$ is a polynomial of degree $2^k-1$ and $θ_k=\frac{2^k}{(k+1)^{\frac{k+1}2}}\cdot \frac3{2^k+6k-5}$. This was conjectured in Hilberdink, Luca, and Tóth~\cite[Remark 2.4]{HLT}. We also prove asymptotic formulas for similar sums conjectured there.
The explicit formulas for the sums of positive powers of the integers $s_i$ unrepresentable by the triple of integers $d_1,d_2,d_3\in {\mathbb N}, \gcd(d_1,d_2,d_3)=1$, are derived.
Menon's identity is a classical identity involving gcd sums and the Euler totient function $ϕ$. In a recent paper, Zhao and Cao derived the Menon-type identity $\sum\limits_{\substack{k=1}}^{n}(k-1,n)χ(k) = ϕ(n)τ(\frac{n}{d})$, where $χ$ is a Dirichlet character mod $n$ with conductor $d$. We derive an identity similar to this replacing gcd with a generalization it. We also show that some of the arguments used in the derivation of Zhao-Cao identity can be improved if one uses the method we employ here.
We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and lower bounds less efficient (but nontrivial) for quadratic sequences whose general term has the form $u_n = a n (n + t) + b$ with $(a, t, b) \in {\mathbb{Z}}^3, a \geq 5, t \geq 0, \rm{gcd}(a, b) = 1$. From this, we deduce for instance the lower bound: $\mathrm{lcm}\{1^2 + 1, 2^2 + 1, ..., n^2 + 1\} \geq 0,32 (1,442)^n$ (for all $n \geq 1$). In the last part…
Upper bounds for GCD sums of the form [\sum_{k,{\ell}=1}^N\frac{(\gcd(n_k,n_{\ell}))^{2α}}{(n_k n_{\ell})^α}] are proved, where $(n_k)_{1 \leq k \leq N}$ is any sequence of distinct positive integers and $0<α\le 1$; the estimate for $α=1/2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $α=1/2$. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to …
We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of the Ramanujan sums with weights concerning logarithms, values of arithmetic functions for gcd's, the Gamma function, the Bernoulli polynomials, and binomial coefficients.
Let $q$ be a positive integer, $χ$ a nontrivial character mod $q$, $\mathcal{I}$ an interval of length not exceeding $q.$ In this paper we shall study the character sum analogue of the well-known Kloosterman sum,\[\sum_{\substack{a\in\mathcal{I} \gcd(a,q)=1}}χ(ma+n\overline{a}),\] where $\overline{a}$ is the multiplicative inverse of $a\bmod q$. The mean square values and bilinear forms for such sums are proved.
For the Gauss sums which are defined by S_n(a,q) := \sum_{x (mod q)} e(ax^n/q), Stechkin (1975) conjectured that the quantity A := \sup_{n,q\ge 2} \max_{\gcd(a,q)=1} |S_n(a,q)|/q^(1-1/n) is finite. Shparlinski (1991) proved that A is finite, but in the absence of effective bounds on the sums S_n(a,q) the precise determination of A has remained intractable for many years. Using recent work of Cochrane and Pinner (2011) on Gauss sums with prime moduli, in this paper we show that with the constant given by A = |S_6(4787,4606056)|/4606056^(5/6) = 4.709236... one has the sharp inequality |S_n(a,q)|…
Let $q$ be an integer $\geq 2$ and let $S_q(n)$ denote the sum of digits of $n$ in base $q$. For \[ α=[0;\overline{1,m}],\ m\geq 2, \] let $S_α(n)$ denote the sum of digits in the Ostrowski $α$-representation of $n$. Let $m_1,m_2\geq 2$ be integers with $$\gcd(q-1,m_1)=\gcd(m,m_2)=1.$$ We prove that there exists $δ>0$ such that for all integers $a_1,a_2$, \begin{eqnarray*} &&|\{0\leq n<N: S_{q}(n)\equiv a_1\pmod{m_1},\ S_α(n)\equiv a_2\pmod{m_2}\}| &=&\frac{N}{m_1m_2}+O(N^{1-δ}). \end{eqnarray*} The asymptotic relation implied by this equality was proved by Coquet, Rhin & Toffin and the equali…
We present an explicit evaluation of the double Gauss sum $\displaystyle G(a,b,c;S;p^n):=\sum_{x,y=0}^{p^n-1} e^{2πi S(ax^2+bxy+cy^2)/p^n}$, where $a, b, c$ are integers such that $\gcd(a,b,c)=1$, $p$ is a prime, $n$ is a positive integer, and $S$ is an integer coprime to $p$.
This paper is devoted to establish nontrivial effective lower bounds for the least common multiple of consecutive terms of a sequence ${(u_n)}_{n \in \mathbb{N}}$ whose general term has the form $u_n = r {[n]}_q + u_0$, where $q , r$ are positive integers and $u_0$ is a non-negative integer such that $\mathrm{gcd}(u_0 , r) = \mathrm{gcd}(u_1 , q) = 1$. For such a sequence, we show that for all positive integer $n$, we have $\mathrm{lcm}\{u_1 , u_2 , \dots , u_n\} \geq c_1 \cdot c_2^n \cdot q^{\frac{n^2}{4}}$, where $c_1$ and $c_2$ are positive constants depending only on $q , r$ and $u_0$. Thi…
Let $n\ge 1, e\ge 1, k\ge 2$ and $c$ be integers. An integer $u$ is called a unit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(u, n)=1$. A unit $u$ is called an exceptional unit in the ring $\mathbb{Z}_n$ if $\gcd(1-u,n)=1$. We denote by $\mathcal{N}_{k,c,e}(n)$ the number of solutions $(x_1,...,x_k)$ of the congruence $x_1^e+...+x_k^e\equiv c \pmod n$ with all $x_i$ being exceptional units in the ring $\mathbb{Z}_n$. In 2017, Mollahajiaghaei presented a formula for the number of solutions $(x_1,...,x_k)$ of the congruence $x_1^2+...+x_k^2\equiv c\pmod n$ with all $x_i$ be…
If $k$ is a sufficiently large positive integer, we show that the Diophantine equation $$n (n+d) \cdots (n+ (k-1)d) = y^{\ell}$$ has at most finitely many solutions in positive integers $n, d, y$ and $\ell$, with $\operatorname{gcd}(n,d)=1$ and $\ell \geq 2$. Our proof relies upon Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial.
Let $S(a,b)$ denote the normalized Dedekind sum. We study the range of possible values for $S(a,b)=\frac{k}{q}$ with $\gcd(k,q)=1$. Girstmair proved local restrictions on $k$ depending on $q\pmod{12}$ and whether $q$ is a square and conjectured that these are the only restrictions possible. We verify the conjecture in the cases $q$ even, $q$ a square divisible by $3$ or $5$, and $2\le q\le 200$ (the latter by computer), and provide progress towards a general approach.
Menon's identity is a classical identity involving gcd sums and the Euler totient function $ϕ$. A natural generalization of $ϕ$ is the Klee's function $Φ_s$. In this paper we derive a Menon-type identity using Klee's function and a generalization of the gcd function. This identity generalizes an identity given by Lee and Kim in [\textit{J. Number Theory 175, 42--50(2017)}].
We deduce asymptotic formulas for the alternating sums $\sum_{n\le x} (-1)^{n-1} f(n)$ and $\sum_{n\le x} (-1)^{n-1} \frac1{f(n)}$, where $f$ is one of the following classical multiplicative arithmetic functions: Euler's totient function, the Dedekind function, the sum-of-divisors function, the divisor function, the gcd-sum function. We also consider analogs of these functions, which are associated to unitary and exponential divisors, and other special functions. Some of our results improve the error terms obtained by Bordellès and Cloitre. We formulate certain open problems.
