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: In this article, the authors consider the classes of quasimultiplicative, semimultiplicative and Selberg multiplicative functions and apply them to Raamanujan's sum and its analogue with respect to regular integers.
Abstract: We consider the classes of quasimultiplicative, semimultiplicative and Selberg multiplicativefunctions as extensions of the class of multiplicative functions. We apply these concepts to Ra-manujan’s sum and its analogue with respect to regular integers (mod r). Mathematics Subject Classi cation: 11A25, 11L03Keywords: quasimultiplicative function, semimultiplicative function, Selberg multiplicative function,Ramanujan’s sum, regular integer 1 Introduction An arithmetical function f: N !C is said to be multiplicative if f(mn) = f(m)f(n) for all m;n2Nwith (m;n) = 1. These functions play a central role in number theory. The works of E. T. Bell andR. Vaidyanathaswamy are prominent in the history of multiplicative functions, see e.g. [4, 24].Many of the classical arithmetical functions are multiplicative, e.g. the Mobius function, Euler’stotient function and the divisor functions. On the other hand, multiplicative functions have some weakpoints, e.g., they are destroyed by compositions such as cf(n);f(kn);f(k=n);f(n=k);f([k;n]), where[k;n] is the lcm of kand n. This has led to certain extensions of the class of multiplicative functions. Inthis paper we introduce quasimultiplicative, semimultiplicative and Selberg multiplicative functions,see [11, 15, 19]. As a motivation of these concepts we also consider multiplicative properties ofRamanujan’s sum and its analogue with respect to regular integers [9].There are also important subclasses of the class of multiplicative functions in the number theoreticliterature, e.g., the class of rational arithmetical functions, see [12]. We do not consider these classesin this paper.
6 citations
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TL;DR: In this paper, an asymptotic formula for the average divisor sum in a convenient form, and an explicit upper bound for this sum with the correct main term were provided.
Abstract: Consider the divisor sum $\sum_{n\leq N}\tau(n^2+2bn+c)$ for integers $b$ and $c$. We extract an asymptotic formula for the average divisor sum in a convenient form, and provide an explicit upper bound for this sum with the correct main term. As an application we give an improvement of the maximal possible number of $D(-1)$-quadruples.
6 citations
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TL;DR: In this article, the average behavior of the coefficients of Dedekind zeta function over square numbers was studied in Galois fields of degree d, where l ⩾ 1 is an integer and e > 0 is an arbitrarily small constant.
6 citations
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09 Apr 1991
TL;DR: In this paper, a quotient digit selecting device selects one from all quotient digits obtainable under an applied radix based on the signs and the upper digit values of the divisor and the partial remainder represented in the two's complement representation or, alternatively, on the upper and lower digit values represented in a redundant binary representation.
Abstract: A divider unit is provided for a high-radix division using a partial remainder. A quotient digit selecting device selects one from all quotient digits obtainable under an applied radix based on the signs and the upper digit values of the divisor and the partial remainder represented in the two's complement representation or, alternatively, on the upper digit values of the divisor and the partial remainder represented in the redundant binary representation. A number of divisor's multiple generating devices each generate at least one of 0 and a value obtained by multiplying the divisor with 2j (j=integer). At least one adding and subtracting device provides at least three inputs to generate a first product corresponding to any desired multiple of the divisor by adding or subtracting the outputs from the multiple generating devices and to generate another partial remainder by adding or subtracting the first product with a second product corresponding to a value obtained by multiplying the partial remainder by the radix.
6 citations
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TL;DR: In this paper, it was shown that two integral circulant graphs with multiplicative divisor sets are isomorphic if and only if their spectral vectors coincide, i.e., they have the same eigenvalues.
Abstract: Each integral circulant graph ICG(n,D) is characterised by its order n and a
set D of positive divisors of n in such a way that it has vertex set Z=nZ and
edge set {(a,b) : a, b Z=nZ, gcd(a - b,n) D}. According to a conjecture
of So two integral circulant graphs are isomorphic if and only if they are
isospectral, i.e. they have the same eigenvalues (counted with
multiplicities). We prove a weaker form of this conjecture, namely, that two
integral circulant graphs with multiplicative divisor sets are isomorphic if
and only if their spectral vectors coincide.
6 citations