Topic
Divisor
About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.
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TL;DR: The smart method of Gelfond and Shnirelman-Nair as discussed by the authors allows one to obtain a lower bound for the prime counting function in an elementary way in terms of integrals of suitable integer polynomials.
Abstract: The smart method of Gelfond–Shnirelman–Nair allows one to obtain a lower bound for the prime counting function \({\pi(x)}\) in an elementary way in terms of integrals of suitable integer polynomials. In this paper we carry on the study of the sets of integer polynomials relevant for the method.
2 citations
TL;DR: In this paper, the authors evaluate the convolution sums ∑ak+bl+cm =nσ(k)σ(l)σσ(m) for all positive integers a,b,c,n with lcm(a,b.c.n) = 7, 8 or 9 using theory of modular form.
Abstract: It is known that the generating functions of divisor functions are quasimodular forms of weight 2. Hence their product is a quasimodular form of higher weight. In this paper, we evaluate the convolution sums ∑ak+bl+cm=nσ(k)σ(l)σ(m) for all positive integers a,b,c,n with lcm(a,b,c) = 7, 8 or 9 using theory of modular form.
2 citations
TL;DR: For a tropical curve and a finite subgroup of the isometry group of $\Gamma$ as discussed by the authors, it was shown that the complete linear system associated to a $K$-invariant effective divisor on the tropical curve is finitely generated.
Abstract: For a tropical curve $\Gamma$ and a finite subgroup $K$ of the isometry group of $\Gamma$, we prove, extending the work by Haase, Musiker and Yu ([6]), that the $K$-invariant part of the complete linear system associated to a $K$-invariant effective divisor on $\Gamma$ is finitely generated
2 citations
TL;DR: In this article, the authors studied geometric structures on the complement of a toric mirror arrangement associated with a root system and proved that these connections are torsion free and flat.
Abstract: We study geometric structures on the complement of a toric mirror arrangement associated with a root system. Inspired by those root system hypergeometric functions found by Heckman–Opdam, and in view of the work of Couwenberg–Heckman–Looijenga on the geometric structures on projective arrangement complements, we consider a family of connections on a total space, namely, a $${\mathbb {C}}^{\times }$$
-bundle on the complement of a toric mirror arrangement (=finite union of hypertori, determined by a root system). We prove that these connections are torsion free and flat, and hence define a family of affine structures on the total space, which is equivalent to a family of projective structures on the toric arrangement complement. We then determine a parameter region for which the projective structure admits a locally complex hyperbolic metric. In the end, we show that the space in question can be biholomorphically mapped onto a divisor complement of a ball quotient if the Schwarz conditions are invoked.
2 citations
TL;DR: In this article, an effective method of checking the estimated quotient figure and of checking an estimate before writing the product is described, regardless of the size of the divisor, and this method can be done mentally and in the majority of cases quite easily.
Abstract: Ability in estimating the quotient figure in long division plays a major role in facilitating the computation The time wasted and the accompany ing irritations of the all-too-common "try, erase, and try again" procedure are quite unnecessary Careful estimates, based on scientific principles, can quite easily be made Efficiency in long division is thereby materially in creased Such a method of estimating the quotient figure was described by the writer in the March, 1945 issue of the Journal of Educational Research A further increase in efficiency is possible and entirely practical This increase may be effected by checking an estimate before writing the product The purpose of this article is to describe an effective method of checking the estimated quotient figure and to show that the efficiency of this method is high This method of checking the estimated quotient figure is as follows: Multiply the first two figures of the divisor by the estimated quotient figure This product, when compared with the appropriate part of the dividend, will, in almost every instance, show how nearly correct the estimate is Since only the first two figures of a divisor are included, this method of checking is entirely independent of the size of the divisor Furthermore, this type of checking, regardless of the size of the divisor, can be done mentally and, in the majority of cases, quite easily The demonstration example at the left will serve to illustrate the method of estimating the quotient figure and of checking the estimate 61 To estimate, approximate the divisor by rounding the 6274)382714 divisor to the first figure Think of the divisor as 6000 37644 The estimate is 6 6274 Is 6 correct? Check before writing the product Only 6274 the first two figures of the divisor need be multiplied 6 x 62=372 372 can be subtracted from 382, and it appears that 6 is probably the correct quotient figure However, if, in mul tiplying the whole divisor by the quotient figure, the product due to the third figure of the divisor (6 x 7), effects too great an increase in the prod 52
2 citations