Number Of Divisors Research Articles

On Wigert's type divisor problem

In this paper we study a special kind of divisor function which was first introduced by Wigert while investigating a series involving k-th power divisor function. We will give a sharper estimate of the error term appeared in the asymptotic formula for the above mentioned divisor function. Furthermore, we will give an account of the mean value of the error term as well as mean square of the error term.

Algebraic Construction of the Sigma Function for General Weierstrass Curves

The Weierstrass curve X is a smooth algebraic curve determined by the Weierstrass canonical form, yr+A1(x)yr−1+A2(x)yr−2+⋯+Ar−1(x)y+Ar(x)=0, where r is a positive integer, and each Aj is a polynomial in x with a certain degree. It is known that every compact Riemann surface has a Weierstrass curve X, which is birational to the surface. The form provides the projection ϖr:X→P as a covering space. Let RX:=H0(X,OX(∗∞)) and RP:=H0(P,OP(∗∞)). Recently, we obtained the explicit description of the complementary module RXc of RP-module RX, which leads to explicit expressions of the holomorphic form except ∞, H0(P,AP(∗∞)) and the trace operator pX such that pX(P,Q)=δP,Q for ϖr(P)=ϖr(Q) for P,Q∈X\{∞}. In terms of these, we express the fundamental two-form of the second kind Ω and its connection to the sigma function for X.

The sigma function over a family of curves with a singular fiber

The sigma function over a family of curves with a singular fiber

Sums of the kth divisor function of the ternary quadratic with prime variables

Sums of the kth divisor function of the ternary quadratic with prime variables

On divisors of sums of polynomials

Let A and B be sets of polynomials of degree n over a finite field. We show, that if A and B are large enough, then A+B has an irreducible divisor of large degree for some A∈A and B∈B.

Bi-unitary multiperfect numbers, IV(c)

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(c) in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we determined all bi-unitary triperfect numbers of the form n=2^{a}u, where 1\leq a \leq 6 and u is odd. In part V we fixed the case a=8. The case a=7 is more difficult. In Parts IV(a-b) we solved partly this case, and in the present paper (Part IV(c)) we continue the study of the same case (a=7).

Solving the Quispel–Roberts–Thompson maps using Kajiwara–Noumi–Yamada’s representation of elliptic curves

It is well known that the dynamical system determined by a Quispel–Roberts–Thompson map (a QRT map) preserves a pencil of biquadratic polynomial curves on . In most cases this pencil is elliptic, i.e. its generic member is a smooth algebraic curve of genus one, and the system can be solved as a translation on the elliptic fiber to which the initial point belongs. However, this procedure is rather complicated to handle, especially in the normalization process. In this paper, for a given initial point on an invariant elliptic curve, we present a method to construct the solution directly in terms of the Weierstrass sigma function, using Kajiwara–Noumi–Yamada’s parametric representation of elliptic curves.

On certain sums of number theory

We study sums of the shape [Formula: see text] where [Formula: see text] is either the von Mangoldt function or the Dirichlet–Piltz divisor functions. We improve previous estimates when [Formula: see text] and [Formula: see text], and provide new results when [Formula: see text] with [Formula: see text], breaking the [Formula: see text]-barrier in each case. The functions [Formula: see text], [Formula: see text] and [Formula: see text] are also investigated.

A paucity problem associated with a shifted integer analogue of the divisor function

Suppose that θ is irrational. Then almost all elements ν∈Z[θ] that may be written as a k-fold product of the shifted integers n+θ(n∈N) are thus represented essentially uniquely.

On Ramanujan-type congruences for multiplicative functions

The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we investigate the existence and classification of Ramanujan-type congruences for functions in multiplicative number theory.

The infrared spectra of amides. Part 1. The stretching vibrations of primary carboxamides

The infrared spectra of more than 150 primarycarboxamides in the NH stretching and the carbonyl region have been measured. Comparison is made with Bellamys analysis of the stretching modes of amines. The offset to higher frequencies of amides compared with amines is attributed to the widening of the bond angle. The symmetric and antisymmetric stretching of the NH2 bonds is analysed via a rearrangement of the Linnett equations. It is shown thereby that changes of bond strength vary the symmetric and antisymmetric frequencies approximately proportionately whilst deviations from that relationship are due either to changes of the bond angle, steric hindrance or to hydrogen bonding. The spectral changes of m- and p-substituted amides are correlated with Hammett sigma functions, both in the carbonyl and the NH2 stretching region. Intensities and the influence of steric effects are also discussed.

Moment estimates for the exponential sum with higher divisor functions

We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc bounds for the exponential sum with $\tau_k$, and asymptotics for high moments of the Dirichlet kernel.

Sums of the higher divisor function of diagonal homogeneous forms

Sums of the higher divisor function of diagonal homogeneous forms

Polar harmonic Maaß forms and holomorphic projection

Recently, Mertens, Ono and the third author studied mock modular analogues of Eisenstein series. Their coefficients are given by small divisor functions, and have shadows given by classical Shimura theta functions. Here, we construct a class of small divisor functions [Formula: see text] and prove that these generate the holomorphic part of polar harmonic Maaß forms of weight [Formula: see text]. To this end, we essentially compute the holomorphic projection of mixed harmonic Maaß forms in terms of Jacobi polynomials, but without assuming the structure of such forms. Instead, we impose translation invariance and suitable growth conditions on the Fourier coefficients. Specializing to a certain choice of characters, we obtain an identity between [Formula: see text] and Hurwitz class numbers, and ask for more such identities. Moreover, we prove [Formula: see text]-adic congruences of our small divisor functions when [Formula: see text] is an odd prime. If [Formula: see text] is non-trivial we rewrite the generating function of [Formula: see text] as a linear combination of Appell–Lerch sums and their first two normalized derivatives. Lastly, we offer a connection of our construction to meromorphic Jacobi forms of index [Formula: see text] and false theta functions.

On the Solution of a Conformal Mapping Problem by Means of Weierstrass Functions

The conformal mapping problem for the section of a channel filled with porous material under a rectangular dam onto the upper half-plane is considered. Similar problems arise in computing of fluid flow in hydraulic structures. As a solution method, the representation of Christoffel–Schwartz elliptic integral in terms of Weierstrass functions is used. The calculation is based on Taylor series for the sigma function, the coefficients of which are determined recursively. A simple formula for a conformal mapping is obtained, which depends on four parameters and uses the sigma function. A numerical experiment was carried out for a specific area. The degeneration of the region, which consists in the dam width tending to zero, is considered, and it is shown that the resulting formula has a limit that implements the solution of the limiting problem. A refined proof of Weierstrass recursive formula for the coefficients of Taylor series of the sigma function is presented.

Phylogenetic and Phenotypic Analyses of a Collection of Food and Clinical Listeria monocytogenes Isolates Reveal Loss of Function of Sigma B from Several Clonal Complexes.

To understand the molecular mechanisms that contribute to the stress responses of the important foodborne pathogen Listeria monocytogenes, we collected 139 strains (meat, n = 25; dairy, n = 10; vegetable, n = 8; seafood, n = 14; mixed food, n = 4; and food processing environments, n = 78), mostly isolated in Ireland, and subjected them to whole-genome sequencing. These strains were compared to 25 Irish clinical isolates and 4 well-studied reference strains. Core genome and pan-genome analysis confirmed a highly clonal and deeply branched population structure. Multilocus sequence typing showed that this collection contained a diverse range of strains from L. monocytogenes lineages I and II. Several groups of isolates with highly similar genome content were traced to single or multiple food business operators, providing evidence of strain persistence or prevalence, respectively. Phenotypic screening assays for tolerance to salt stress and resistance to acid stress revealed variants within several clonal complexes that were phenotypically distinct. Five of these phenotypic outliers were found to carry mutations in the sigB operon, which encodes the stress-inducible sigma factor sigma B. Transcriptional analysis confirmed that three of the strains that carried mutations in sigB, rsbV, or rsbU had reduced SigB activity, as predicted. These strains exhibited increased tolerance to salt stress and displayed decreased resistance to low pH stress. Overall, this study shows that loss-of-function mutations in the sigB operon are comparatively common in field isolates, probably reflecting the cost of the general stress response to reproductive fitness in this pathogen. IMPORTANCE The bacterial foodborne pathogen Listeria monocytogenes frequently contaminates various categories of food products and is able to cause life-threatening infections when ingested by humans. Thus, it is important to control the growth of this bacterium in food by understanding the mechanisms that allow its proliferation under suboptimal conditions. In this study, intraspecies heterogeneity in stress response was observed across a collection consisting of mainly Irish L. monocytogenes isolates. Through comparisons of genome sequence and phenotypes observed, we identified three strains with impairment of the general stress response regulator SigB. Two of these strains are used widely in food challenge studies for evaluating the growth potential of L. monocytogenes. Given that loss of SigB function is associated with atypical phenotypic properties, the use of these strains in food challenge studies should be re-evaluated.

The partition-frequency enumeration matrix

We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite number-theoretic matrix. We call this matrix the partition-frequency enumeration (PFE) matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-theoretic functions. The calculus is extended to arbitrary generating functions, and functions with Weierstrass products. As a by-product, we recover (and extend) some well-known recurrence relations for many number-theoretic functions, including the sum of divisors function, Ramanujan’s $$\tau $$ function, sums of squares and triangular numbers, and for $$\zeta (2n)$$ , where n is a positive integer. These include classical results due to Euler, Ewell, Ramanujan, Lehmer, and others. As one application, we embed Ramanujan’s famous congruences $$p(5n+4)\equiv 0\; (\mathrm{mod}\,5)$$ and $$\tau (5n+5)\equiv 0\; (\mathrm{mod}\,5)$$ into an infinite family of such congruences.

Parametrization of Kloosterman sets and SL3-Kloosterman sums

We stratify the SL3 big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the SL3 long word Kloosterman sum is decomposed into finer parts, and we write it as a finite sum of a product of two classical Kloosterman sums. The fine Kloosterman sums end up being the correct pieces to consider in the Bruggeman-Kuznetsov trace formula on the congruence subgroup Γ0(N)⊆SL3(Z). Another application is a new explicit formula, expressing the triple divisor sum function in terms of a double Dirichlet series of exponential sums, generalizing Ramanujan's formula.

Arithmetic convolution sums derived from eta quotients related to divisors of 6

Abstract The aim of this paper is to find arithmetic convolution sums of some restricted divisor functions. When divisors of a certain natural number satisfy a suitable condition for modulo 12, those restricted divisor functions are expressed by the coefficients of certain eta quotients. The coefficients of eta quotients are expressed by the sine function and cosine function, and this fact is used to derive formulas for the convolution sums of restricted divisor functions and of the number of divisors. In the sine function used to find the coefficients of eta quotients, the result is obtained by utilizing a feature with symmetry between the divisor and the corresponding divisor. Let N , r N,r be positive integers and d d be a positive divisor of N N . Let e r ( N ; 12 ) {e}_{r}\left(N;\hspace{0.33em}12) denote the difference between the number of 2 N d − d \frac{2N}{d}-d congruent to r r modulo 12 and the number of those congruent to − r -r modulo 12. The main results of this article are to find the arithmetic convolution identities for ∑ a 1 + ⋯ + a j = N ( ∏ i = 1 j e ˆ ( a i ) ) {\sum }_{{a}_{1}+\cdots +{a}_{j}=N}({\prod }_{i=1}^{j}\hat{e}\left({a}_{i})) with e ˆ ( a i ) = e 1 ( a i ; 12 ) + 2 e 3 ( a i ; 12 ) + e 5 ( a i ; 12 ) \hat{e}\left({a}_{i})={e}_{1}\left({a}_{i};\hspace{0.33em}12)+2{e}_{3}\left({a}_{i};\hspace{0.33em}12)+{e}_{5}\left({a}_{i};\hspace{0.33em}12) and j = 1 , 2 , 3 , 4 j=1,2,3,4 . All results are obtained using elementary number theory and modular form theory.

Asymptotic identities for additive convolutions of sums of divisors

AbstractIn a 1916 paper, Ramanujan studied the additive convolution $S_{a, b}(n)$ of sum-of-divisors functions $\sigma_a(n)$ and $\sigma_b(n)$ , and proved an asymptotic formula for it when a and b are positive odd integers. He also conjectured that his asymptotic formula should hold for all positive real a and b. Ramanujan’s conjecture was subsequently proved by Ingham, and then by Halberstam with a power saving error term.In this paper, we give a new proof of Ramanujan’s conjecture that obtains lower order terms in the asymptotics for most ranges of the parameters. We also describe a connection to a counting problem in geometric topology that was studied in the second author’s thesis and which served as our initial motivation in studying this sum.