Number Of Divisors Research Articles

Tropical sequences associated with Somos sequences

Since the seminal note published by M. Somos in 1989, a great deal of attention of specialists in number theory and adjacent areas are attracted by nonlinear sequences that satisfy a quadratic recurrence relation. At the same time, special attention is paid to the construction of Somos integer sequences and their Laurent property with respect to initial values and coefficients of a recurrence. In the fundamental works of Robinson, Fomin and Zelevinsky the Laurent property of the Somos-k sequence for k = 4, 5, 6, 7 was proved. In the works of Hone, representations for Somos-4 and 5 sequences were found via the Weierstrass sigma function on elliptic curves, and for k = 6 via the Klein sigma function on hyperelliptic curve of genus It should also be noted that the Somos sequences naturally arise in the construction of cryptosystems on elliptic and hyperelliptic curves over a finite field. This is explained by the reason that addition theorems hold for the sequences mentioned above, and they naturally arise when calculating multiple points on elliptic and hyperelliptic curves. For k = 4, 5, 6, 7, the Somos sequences are Laurent polynomials of k initial variables and ordinary polynomials in the coefficients of the recurrence relation. Therefore, these Laurent polynomials can be written as an irreducible fraction with an ordinary polynomial in the numerator with initial values and coefficients as variables. In this case, the denominator can be written as a monomial of the initial variables. Using tropical functions, we prove that the degrees of the variables of the above monomial can be represented as quadratic polynomials in the order index of the element of the Somos sequence, whose free terms are periodic sequences of rational numbers. Moreover, in each case these polynomials and the periods of their free terms are written explicitly

The reciprocal sum of divisors of Mersenne numbers

We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers $2^n-1$. Conditional on the Elliott-Halberstam Conjecture and the Generalized Riemann Hypothesis, we determine $\max_{n\le x} \sum_{p \mid 2^n-1} 1/p$ to within $o(1)$ and $\max_{n\le x} \sum_{d\mid 2^n-1}1/d$ to within a factor of $1+o(1)$, as $x\to\infty$. This refines, conditionally, earlier estimates of Erdős and Erdős-Kiss-Pomerance. Conditionally (only) on GRH, we also determine $\sum 1/d$ to within a factor of $1+o(1)$ where $d$ runs over all numbers dividing $2^n-1$ for some $n\le x$. This conditionally confirms a conjecture of Pomerance and answers a question of Murty-Rosen-Silverman. Finally, we show that both $\sum_{p\mid 2^n-1} 1/p$ and $\sum_{d\mid 2^n-1}1/d$ admit continuous distribution functions in the sense of probabilistic number theory.

Relations of Combinatoric Convolution Sums for restricted divisor functions and Bernoulli polynomials

Relations of Combinatoric Convolution Sums for restricted divisor functions and Bernoulli polynomials

The variance of the number of sums of two squares in [inline-graphic 01] [T] in short intervals

Consider the number of integers in a short interval that can be represented as a sum of two squares. What is an estimate for the variance of these counts over random short intervals? We resolve a function field variant of this problem in the large $q$ limit, finding a connection to the $z$-measures first investigated in the context of harmonic analysis on the infinite symmetric group. A similar connection to $z$-measures is established for sums over short intervals of the divisor functions $d_z(n)$. We use these results to make conjectures in the setting of the integers which match very well with numerically produced data. Our proofs depend on equidistribution results of N. Katz and W. Sawin.

Hyperelliptic Sigma Functions and Adler-Moser Polynomials

В работе В. М. Бухштабера и Д. В. Лейкина, опубликованной в 2004 г. в журнале «Функциональный анализ и его приложения», для каждого $g>0$ была построена система из $2g$ многомерных уравнений теплопроводности в неголономном репере. Сигма-функция универсальной гиперэллиптической кривой рода $g$ является решением этой системы. В нашей предыдущей работе, опубликованной в журнале «Функциональный анализ и его приложения», были получены явные выражения для операторов Шрeдингера, определяющих уравнения рассматриваемой системы в гиперэллиптическом случае. В данной работе на основе этих результатов показано, что если начальное условие является полиномом, то решение рассматриваемой системы определено однозначно с точностью до постоянного множителя. Это находит важные приложения в широко известной задаче разложения в ряд гиперэллиптической сигма-функции. Дано явное описание связи таких решений с известными полиномами Бурхналла-Чаунди и Адлера-Мозера. Найдена система линейных дифференциальных уравнений второго порядка, определяющая соответствующий полином Адлера-Мозера.

Arithmetic properties of the sum of divisors

The divisor function σ(n) denotes the sum of the divisors of the positive integer n. For a prime p and m∈N, the p-adic valuation of m is the highest power of p which divides m. Formulas for νp(σ(n)) are established. For p=2, these involve only the odd primes dividing n. These expressions are used to establish the bound ν2(σ(n))≤⌈log2⁡(n)⌉, with equality if and only if n is the product of distinct Mersenne primes, and for an odd prime p, the bound is νp(σ(n))≤⌈logp⁡(n)⌉, with equality related to solutions of the Ljunggren-Nagell diophantine equation.

A note on Titchmarsh divisor problem under the generalized Riemann hypothesis

Let Λ(n) be the von Mangoldt function and τ(n) the divisor function. We focus on the summation T ( x ) = ∑ 1 < n ≤ x Λ ( n ) τ ( n − 1 ) . Under the Riemann hypothesis related to Dirichlet L-functions, it is proved that T ( x ) = x P 1 ( log x ) + O ( x 1 − 𝜃 ) holds for 𝜃<16×10−4. Here P1(t) is a polynomial of degree 1.

Resurgent expansion of Lambert series and iterated Eisenstein integrals

We consider special Lambert series as generating functions of divisor sums and determine their complete transseries expansion near rational roots of unity. Our methods also yield new insights into the Laurent expansions and modularity properties of iterated Eisenstein integrals that have recently attracted attention in the context of certain period integrals and string theory scattering amplitudes.

Average behavior of the triple divisor function over values of quadratic form

In this paper, we apply the circle method to study the average behavior of the triple divisor function over values of quadratic form $$ {n}_1^2+{n}_2^2+\cdots +{n}_l^2 $$ with l ⩾ 3. We improve and generalize previous results.

Bi-unitary multiperfect numbers, IV(a)

A divisor d of a positive integer n is called a unitary divisor if \gcd(d, n/d)=1; and d is called a bi-unitary divisor of n if the greatest common unitary divisor of d and n/d is unity. The concept of a bi-unitary divisor is due to D. Surynarayana (1972). Let \sigma^{**}(n) denote the sum of the bi-unitary divisors of n. A positive integer n is called a bi-unitary multiperfect number if \sigma^{**}(n)=kn for some k \geq 3. For k=3 we obtain the bi-unitary triperfect numbers. Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. The present paper is Part IV(a) in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we found all bi-unitary triperfect numbers of the form n=2^{a}u, where 1\leq a \leq 6 and u is odd. There exist exactly ten such numbers. In this part we solve partly the case a=7. We prove that if n is a bi-unitary triperfect number of the form n=2^{7}.5^{b}.17^{c}.v, where (v, 2.5.17)=1, then b\geq 2. We then confine ourselves to the case b=2. We prove that in this case we have c=1 and further show that n=2^{7}.3^{2}.5^{2}.7.13.17=44553600 is the only bi-unitary triperfect number of this form.

A remark on the average number of divisors of a quadratic polynomial

In recent work, we use Dudek’s method together with a result of Zagier to establish an asymptotic formula for the average number of divisors of an irreducible quadratic polynomial of the form $$x^{2}-bx+c$$ with b, c integers. In this note, we remark that one can adopt the work of Hooley to derive a more precise asymptotic formula for the case $$x^{2}-bx+c$$ with $$b^{2}-4c$$ not a square, and as a consequence, re-establish the weaker asymptotic formula given in our recent work by different arguments.

The Ising correlation C(M, N) for ν = −k

We present Painlevé VI sigma form equations for the general Ising low and high temperature two-point correlation functions C(M, N) with M ⩽ N in the special case ν = −k where ν = sinh 2E h /k B T/sinh 2E v /k B T. More specifically four different non-linear ODEs depending explicitly on the two integers M and N emerge: these four non-linear ODEs correspond to distinguish respectively low and high temperature, together with M + N even or odd. These four different non-linear ODEs are also valid for M ⩾ N when ν = −1/k. For the low-temperature row correlation functions C(0, N) with N odd, we exhibit again for this selected ν = −k condition, a remarkable phenomenon of a Painlevé VI sigma function being the sum of four Painlevé VI sigma functions having the same Okamoto parameters. We show in this ν = −k case for T < T c and also T > T c, that C(M, N) with M ⩽ N is given as an N × N Toeplitz determinant.

Factorization theorems for relatively prime divisor sums, GCD sums and generalized Ramanujan sums

We build on and generalize recent work on so-termed factorization theorems for Lambert series generating functions. These factorization theorems allow us to express formal generating functions for special sums as invertible matrix transformations involving partition functions. In the Lambert series case, the generating functions at hand enumerate the divisor sum coefficients of $$q^n$$ as $$\sum _{d|n} f(d)$$ for some arithmetic function f. Our new factorization theorems provide analogs to these established expansions generating corresponding sums of the form $$\sum _{d: (d,n)=1} f(d)$$ (type I sums) and the Anderson–Apostol sums $$\sum _{d|(m,n)} f(d) g(n/d)$$ (type II sums) 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 conclude the last section of the article by directly expanding new formulas for an arithmetic function g by the type II sums using discrete, and discrete time, Fourier transforms (DFT and DTFT) for functions over inputs of greatest common divisors.

Quadratic Chabauty for (bi)elliptic curves and Kim’s conjecture

We explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. We remove the assumption of semistability in the description of the quadratic Chabauty sets $\mathcal{X}(\mathbb{Z}_p)_2$ containing the integral points $\mathcal{X}(\mathbb{Z})$ of an elliptic curve of rank at most $1$. Motivated by a conjecture of Kim, we then investigate theoretically and computationally the set-theoretic difference $\mathcal{X}(\mathbb{Z}_p)_2\setminus \mathcal{X}(\mathbb{Z})$. We also consider some algorithmic questions arising from Balakrishnan--Dogra's explicit quadratic Chabauty for the rational points of a genus-two bielliptic curve. As an example, we provide a new solution to a problem of Diophantus which was first solved by Wetherell. Computationally, the main difference from the previous approach to quadratic Chabauty is the use of the $p$-adic sigma function in place of a double Coleman integral.

Sigma Functions and Lie Algebras of Schrödinger Operators

In a 2004 paper by V. M. Buchstaber and D. V. Leikin, published in “Functional Analysis and Its Applications,” for each $$g > 0$$ , a system of $$2g$$ multidimensional Schrodinger equations in magnetic fields with quadratic potentials was defined. Such systems are equivalent to systems of heat equations in a nonholonomic frame. It was proved that such a system determines the sigma function of the universal hyperelliptic curve of genus $$g$$ . A polynomial Lie algebra with $$2g$$ Schrodinger operators $$Q_0, Q_2, \dots, Q_{4g-2}$$ as generators was introduced. In this work, for each $$g > 0,$$ we obtain explicit expressions for $$Q_0$$ , $$Q_2$$ , and $$Q_4$$ and recurrent formulas for $$Q_{2k}$$ with $$k>2$$ expressing these operators as elements of a polynomial Lie algebra in terms of the Lie brackets of the operators $$Q_0$$ , $$Q_2$$ , and $$Q_4$$ . As an application, we obtain explicit expressions for the operators $$Q_0, Q_2, \dots, Q_{4g-2}$$ for $$g = 1,2,3,4$$ .

Analogues of the Robin–Lagarias criteria for the Riemann hypothesis

Robin’s criterion states that the Riemann hypothesis is equivalent to [Formula: see text] for all integers [Formula: see text], where [Formula: see text] is the sum of divisors of [Formula: see text] and [Formula: see text] is the Euler–Mascheroni constant. We prove that the Riemann hypothesis is equivalent to the statement that [Formula: see text] for all odd numbers [Formula: see text]. Lagarias’s criterion for the Riemann hypothesis states that the Riemann hypothesis is equivalent to [Formula: see text] for all integers [Formula: see text], where [Formula: see text] is the [Formula: see text]th harmonic number. We establish an analogue to Lagarias’s criterion for the Riemann hypothesis by creating a new harmonic series [Formula: see text] and demonstrating that the Riemann hypothesis is equivalent to [Formula: see text] for all odd [Formula: see text]. We prove stronger analogues to Robin’s inequality for odd squarefree numbers. Furthermore, we find a general formula that studies the effect of the prime factorization of [Formula: see text] and its behavior in Robin’s inequality.

Sums of higher divisor functions

Let dk(n) denote the k-th divisor function. In this paper, we study the asymptotic formula of the sum∑1≤n1,n2,…,nl≤xdk(n12+n22+⋯+nl2), where n1,n2,…,nl∈Z+, k≥4 and l≥3 are integers. Previously only the cases of k=2,3 are studied.

Inequalities between sums over prime numbers in progressions

We investigate a new type of tendency between two progressions of prime numbers which is in support of the claim that prime numbers that are congruent to 3 modulo 4 are favored over prime numbers that are congruent to 1 modulo 4. In particular, we show that the Riemann hypothesis for the corresponding L-function is equivalent to the occurrence of such a tendency. A generalization to teams of progressions of prime numbers is given, where the teams are formed by grouping according to the values of a quadratic character. In this way, it is shown that there is a tendency favoring prime numbers belonging to progressions arising from the quadratic nonresidues modulo a prime number congruent to 3 or 5 modulo 8. The scope of the tendency is extended conditionally, either by assuming the Riemann hypothesis for certain Dirichlet L-functions or by the presence of Siegel zeros. Our approach requires numerical verifications over certain ranges of the parameters, and in this respect, we freely benefit from computer software to carry out such tasks. Lastly, the divisor function is seen to be favorable over its average value along semigroups by comparing partial sums of the associated Dirichlet series.

Highly composite polynomials and the maximum order of the divisor function in pmb {mathbb {F}}_q[t

We investigate the analogues, in mathbb {F}_q[t], of highly composite numbers and the maximum order of the divisor function, as studied by Ramanujan. In particular, we determine a family of highly composite polynomials which is not too sparse, and we use it to compute the logarithm of the maximum of the divisor function at every degree up to an error of a constant, which is significantly smaller than in the case of the integers, even assuming the Riemann Hypothesis.

Nondiagonal terms in the second moment of automorphic -functions

We investigate the exact formula for the second moment of central -values associated to primitive cusp forms of level one and weight , , which was proved by Kuznetsov in 1994. Nondiagonal terms in this formula are expressed in terms of several convolution sums of divisor functions. We investigate the main terms of the convolution sums and show that they cancel out. Bibliography: 8 titles.