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A shifted convolution sum for \mathrm{GL}(3) × \mathrm{GL}(2)

Forum Mathematicum, 2018
In this paper, we estimate the shifted convolution sum
Ping Xi
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Sums of Convolution Operators

SIAM Journal on Mathematical Analysis, 1972
Let $\Omega $ be an open set in $R_n $ and let $\mathcal{E}(\Omega )$ denote the space of infinitely differentiable functions on $\Omega $. Necessary and sufficient conditions are exhibited for a family $\{ \Omega _i \} _{i = 1}^N $ of open sets in $R_n$ and a family $\{ S_i \} _{i = 1}^N \subset \mathcal{E}'(R_n )$ in order that the convolution ...
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Sums with convolutions of Dirichlet characters

manuscripta mathematica, 2010
Let \(\chi_1\) and \(\chi_2\) be primitive Dirichlet characters with conductors \(q_1\) and \(q_2\), respectively, and let \[ S_{\chi_1,\chi_2}(X):=\sum_{ab\leq X}\chi_1(a)\chi_2(b). \] The authors prove that if \(X\geq q_2^{\frac 23}\geq q_1^{\frac 23}\) and \(\log X=q_2^{o(1)}\), then \[ \left| S_{\chi_1,\chi_2}(X)\right|\leq X^{\frac {13}{18}}q_1 ...
Banks, William D., Shparlinski, Igor E.
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On Convoluted Numbers and Sums

The American Mathematical Monthly, 1967
(1967). On Convoluted Numbers and Sums. The American Mathematical Monthly: Vol. 74, No. 3, pp. 235-246.
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The convolution sums of MacMahon’s q-series

The Ramanujan Journal
In his classical work on partitions and divisor functions, MacMahon introduced the two \(q\)-series \(A_k(q)\) and \(C_k(q)\), which have since been shown to be quasimodular forms and are closely linked to partition functions, modular forms, and infinite product identities.
Xia, Ernest X. W., Fan, Yan, Yao, Olivia X. M.   +2 more
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Problem of Minimizing a Sum of Differences of Weighted Convolutions

Computational Mathematics and Mathematical Physics, 2020
In this paper the problem of minimizing a sum of differences of weighted convolutions is formulated as the problem of optimal summing of elements of two sequences where indices play the role of variables. It is shown that the considered problem can be interpreted as the problem that minimizes the sum of squared distances between the elements of an ...
Kel'manov, A. V., Mikhailova, L. V., Ruzankin, P. S., Khamidullin, S. A.   +3 more
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Computing the convolution and the Minkowski sum of surfaces

Proceedings of the 21st Spring Conference on Computer Graphics, 2005
In many applications, such as NC tool path generation and robot motion planning, it is required to compute the Minkowski sum of two objects. Generally the Minkowski sum of two rational surfaces cannot be expressed in rational form. In this paper we show that for LN spline surfaces (surfaces with a linear field of normal vectors) a closed form ...
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CONVOLUTIONS OF RAMANUJAN SUMS AND INTEGRAL CIRCULANT GRAPHS

International Journal of Number Theory, 2012
There exist several generalizations of the classical Dirichlet convolution, for instance the so-called A-convolutions analyzed by Narkiewicz. We shall connect the concept of A-convolutions satisfying a weak form of regularity and Ramanujan sums with the spectrum of integral circulant graphs.
Le, T. A., Sander, J. W.
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Evaluation of some convolution sums

AIP Conference Proceedings, 2015
We evaluate the convolution sums∑l+50m=nσ(l)σ(m), ∑2l+25m=nσ(l)σ(m), ∑l+25m=nσ(l)σ(m),∑l+m=n,l≡a mod⁡5σ(l)σ(m), for a=0,1,2,3,4using the theory of quasimodular forms.
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Higher convolutions of Ramanujan sums

Journal of Number Theory
Letting \(c_q(n)\) to be the Ramanujan sum, in the paper under review, the authors provide higher convolutions of Ramanujan sums by computing the following limit \[ \lim_{x\to x}\frac{1}{x}\sum_{n\leq x}c_{q_1}(n+a_1)\cdots c_{q_k}(n+a_k). \] The result of the above limit is a multivariable multiplicative function, say \(f(q_1,\dots, q_k)\), for which ...
Goel, Shivani, Murty, M. Ram
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