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Ayşe Alaca

Publications and source records attributed to Ayşe Alaca.

5 recordsLinked to original sources
arXiv2016

The convolution sum $\sum_{al+bm=n} σ(l) σ(m)$ for $(a,b)=(1,28), (4,7), (1,14), (2,7), (1,7)$

We evaluate the convolution sum $\displaystyle W_{a,b}(n):= \sum_{al+bm=n} \hspace{-3mm} σ(l) σ(m)$ for $(a,b)=(1,28), (4,7), (2,7)$ for all positive integers $n$. We use a modular form approach. We also re-evaluate the known sums $W_{1,14}(n)$ and $W_{1,7}(n)$ with our method. We then use these evaluations to determine the number of representations of $n$ by the octonary quadratic form $x_1^2 + x_2^2 +x_3^2 + x_4^2 + 7(x_5^2 + x_6^2 + x_7^2 + x_8^2)$. Finally we compare our evaluations of the sums $W_{1,7}(n)$ and $W_{1,14}(n)$ with the evaluations of Lemire and Williams [10] and Royer [13] to express the modular forms $Δ_{4,7}(z)$, $Δ_{4,14, 1}(z)$ and $Δ_{4,14, 2}(z)$ (given in [10, 13]) as linear combinations of eta quotients.

math.NT↗
arXiv2016

Theta Products and Eta Quotients of Level $24$ and Weight $2$

We find bases for the spaces $M_2\Big(Γ_0(24),\Big(\frac{d}{\cdot}\Big)\Big)$ ($d=1,8,12, 24$) of modular forms. We determine the Fourier coefficients of all $35$ theta products $φ[a_1,a_2,a_3,a_4](z)$ in these spaces. We then deduce formulas for the number of representations of a positive integer $n$ by diagonal quaternary quadratic forms with coefficients $1$, $2$, $3$ or $6$ in a uniform manner, of which $14$ are Ramanujan's universal quaternary quadratic forms. We also find all the eta quotients in the Eisenstein spaces $E_2\Big(Γ_0(24),\Big(\frac{d}{\cdot}\Big)\Big)$ ($d=1,8,12,24$) and give their Fourier coefficients.

math.NT↗
arXiv2016

Evaluation of the Convolution Sum involving the Sum of Divisors Function for 14, 22 and 26

For all natural numbers $n$, we discuss the evaluation of the convolution sum, $\underset{\substack{{(l,m) \in \mathbb{N}_0^2} \\ {α\,l+β\,m=n} } }{\sum}σ(l)σ(m)$, where $αβ=14,22,26$. We generalize the extraction of the convolution sum using Eisenstein forms of weight $4$ for all pairs of positive integers $(α,β)$. We also determine formulae for the number of representations of a positive integer by the octonary quadratic forms $a\,(x_1^2 + x_2^2 + x_3^2 + x_4^2)+ b\,(x_5^2 + x_6^2 + x_7^2 + x_8^2)$, where $(a,b)= (1,1), (1,3), (2,3), (1,9)$. These numbers of representations of a positive integer are applications of the evaluation of certain convolution sums by J. G. Huard et al., A. Alaca et al. and D. Ye.

math.NT↗