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1
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The hybrid mean value of Dedekind sums and two-term exponential sums

100%
Leran C., Xiaoxue L.
Open Mathematics
|
2016
|
tom 14
|
nr 1
436-442
EN
In this paper, we use the mean value theorem of Dirichlet L-functions, the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the two-term exponential sums, and give an interesting identity and asymptotic formula for it.
2
Content available remote

Evaluation of the convolution sums ∑ al + bm = n lσ(l)σ(m) withab≤ 9

76%
Park Y. K.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1389-1399
EN
The generating functions of divisor functions are quasimodular forms of weight 2 and their products belong to a space of quasimodular forms of higher weight. In this article, we evaluate the convolution sums ∑al+bm=nlσ(l)σ(m) $$\begin{array}{} \displaystyle\sum\limits_{al+bm=n}\,l\sigma(l)\sigma(m) \end{array} $$ for all positive integers a, b and n with ab ≤ 9 and gcd(a, b) = 1.
3
Content available remote

Upper bound estimate of incomplete Cochrane sum

76%
Ma Y., Peng W., Zhang T.
Open Mathematics
|
2017
|
tom 15
|
nr 1
852-858
EN
By using the properties of Kloosterman sum and Dirichlet character, an optimal upper bound estimate of incomplete Cochrane sum is given.
4
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Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

52%
Ntienjem E.
Open Mathematics
|
2017
|
tom 15
|
nr 1
446-458
EN
The convolution sum, [...] ∑(l,m)∈N02αl+βm=nσ(l)σ(m), $ \begin{array}{} \sum\limits_{{(l\, ,m)\in \mathbb{N}_{0}^{2}}\atop{\alpha \,l+\beta\, m=n}} \sigma(l)\sigma(m), \end{array} $ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms [...] a(x12+x22+x32+x42)+b(x52+x62+x72+x82), $a\,(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})+b\,(x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}),$ where (a, b) = (1, 11), (1, 13).
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