dirichlet product
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2021 ◽  
Vol 6 (12) ◽  
pp. 13157-13169
Author(s):  
Zhengjin Bu ◽  
◽  
Zhefeng Xu

<abstract><p>In this paper we perform a further investigation for $ r $-gcd-sum function over $ r $-regular integers $ (\bmod\ n^{r}) $, and we derive two kinds of asymptotic formulas by making use of Dirichlet product, Euler product and some techniques. Moreover, we also establish estimates for the generalized $ r $-lcm-sum function over $ r $-regular integers $ (\bmod\ n) $.</p></abstract>


2020 ◽  
Vol 55 (2) ◽  
pp. 253-265
Author(s):  
Yoshinori Hamahata ◽  

We define the Dirichlet product for multiple arithmetic functions over function fields and consider the ring of the multiple Dirichlet series over function fields. We apply our results to absolutely convergent multiple Dirichlet series and obtain some zero-free regions for them.


2017 ◽  
Vol 13 (08) ◽  
pp. 2181-2193 ◽  
Author(s):  
Tomokazu Onozuka

First, we define the multiple Dirichlet product and study the properties of it. From those properties, we obtain a zero-free region of a multiple Dirichlet series and a multiple Dirichlet series expression of the reciprocal of a multiple Dirichlet series.


2016 ◽  
Vol 55 (1-2) ◽  
pp. 293-312
Author(s):  
Abderrahmane Nitaj ◽  
Willy Susilo ◽  
Joseph Tonien

1994 ◽  
Vol 04 (03) ◽  
pp. 313-328 ◽  
Author(s):  
ERIC VIOLARD

This paper presents the main concepts of the mathematical theory PEI for parallel programming and emphasizes its derivation power. The mathematical basis of this theory leads to a nice implementation in CENTAUR7 of an environment whose purpose is to transform parallel programs. It is illustrated by two similar examples: the convolution sum and the Dirichlet product. The second one uses non-affine dependencies that can be easily taken into account using PEI.


1973 ◽  
Vol 73 (3) ◽  
pp. 467-471 ◽  
Author(s):  
K. A. Jukes

Let (λm), (µn) (m, n = 0, 1, 2,…) satisfyrespectively. Let vp (p = 0, 1, 2, …) be the sequence (λm+µn) arranged in ascending order, equal sums λm+µn being considered as giving just one vp Then for given formal series Σam, Σbn the formal series C = Σ cp whereis called the general Dirichlet product of Σamand Σbn (see Hardy (2), p. 239). When λn = µn = n we have the Cauchy product. In the case λn = logm, µn = logn (m, n = 1, 2,…) we have vp =log p(p = 1, 2, …)and it is natural to call C the ordinary Dirichlet product.


1969 ◽  
Vol 66 (3) ◽  
pp. 563-567
Author(s):  
K. A. Jukes

Segal(1) in his paper ‘Summability by Dirichlet convolutions’ makes direct use (in the proofs of Theorems 4 and 6) of a result on the Dirichlet product of two series which may be stated as follows: is (C, k)-summable to the sum α (k≥0) and is (C, l)-summable to the sum β, (l ≥ 0), then the Dirichlet productis (C, k + l + 1)-summable to the sum αβ.’


1966 ◽  
Vol 9 (3) ◽  
pp. 287-296 ◽  
Author(s):  
T.M. K. Davison

Let A be the set of all functions from N, the natural numbers, to C the field of complex numbers. The Dirichlet product of elements f, g of A is given bywhere the summation condition means sum over all positive integers d which divide n.


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