Search ResultsShowing 1-12 of 12
-
arXiv:2302.00964 (Published 2023-02-02)
Figurate numbers, forms of mixed type and their representation numbers
Comments: Any comments or suggestions are welcome. arXiv admin note: substantial text overlap with arXiv:1904.06369Categories: math.NTIn this article, we consider the problem of determining formulas for the number of representations of a natural number $n$ by a sum of figurate numbers with certain positive integer coefficients. To achieve this, we prove that the associated generating function gives rise to a modular form of integral weight under certain condition on the coefficients when even number of higher figurate numbers are considered. In particular, we obtain modular property of the generating function corresponding to a sum of even number of triangular numbers with coefficients. We also obtain modularity property of the generating function of mixed forms involving figurate numbers (including the squares and triangular numbers) with coefficients and forms of the type $m^2+mn+n^2$ with coefficients. In particular, we show the modularity of the generating function of odd number of squares and odd number of triangular numbers (with coefficients). As a consequence, explicit formulas for the number of representations of these mixed forms are obtained using a basis of the corresponding space of modular forms of integral weight. We also obtain several applications concerning the triangular numbers with coefficients similar to the ones obtained in \cite{ono}. In \cite{xia}, Xia-Ma-Tian considered some special cases of mixed forms and obtained the number of representations of these 21 mixed forms using the $(p,k)$ parametrisation method. We also derive these 21 formulas using our method and further obtain as a consequence, the $(p,k)$ parametrisation of the Eisenstein series $E_4(\tau)$ and its duplications. It is to be noted that the $(p,k)$ parametrisation of $E_4$ and its duplications were derived by a different method in \cite{{aw},{aaw}}. We illustrate our method with several examples.
-
arXiv:1904.06369 (Published 2019-04-12)
On triangular numbers, forms of mixed type and their representation numbers
Comments: 37 pages, 17 tablesCategories: math.NTIn \cite{ono}, K. Ono, S. Robins and P.T. Wahl considered the problem of determining formulas for the number of representations of a natural number $n$ by a sum of $k$ triangular numbers and derived many applications, including the one connecting these numbers with the number of representations of $n$ as a sum of $k$ odd square integers. They also obtained an application to the number of lattice points in the $k$-dimensional sphere. In this paper, we consider triangular numbers with positive integer coefficients. First we show that if the sum of these coefficients is a multiple of $8$, then the associated generating function gives rise to a modular form of integral weight (when even number of triangular numbers are taken). We then use the theory of modular forms to get the representation number formulas corresponding to the triangular numbers with coefficients. We also obtain several applications concerning the triangular numbers with coefficients similar to the ones obtained in \cite{ono}. In the second part of the paper, we consider more general mixed forms (as done in Xia-Ma-Tian \cite{xia}) and derive modular properties for the corresponding generating functions associated to these mixed forms. We provide sample formulas for these representation numbers, where the number of variables is $4, 6$. Our sample formulas include all the 21 formulas proved in \cite[Theorem 1.1]{xia}. Finally, we show that our method of deriving the 21 formulas together with the $(p,k)$ parametrization of the generating functions of the three mixed forms imply the $(p,k)$ parametrization of the Eisenstein series $E_4(\tau)$ and its duplications. It is to be noted that the $(p,k)$ parametrization of $E_4$ and its duplications were derived by a different method in \cite{{aw},{aaw}}.
-
arXiv:1801.04392 (Published 2018-01-13)
Certain quaternary quadratic forms of level 48 and their representation numbers
Comments: 16 pages, 12 tablesCategories: math.NTIn this paper, we find a basis for the space of modular forms of weight $2$ on $\Gamma_1(48)$. We use this basis to find formulas for the number of representations of a positive integer $n$ by certain quaternary quadratic forms of the form $\sum_{i=1}^4 a_i x_i^2$, $\sum_{i=1}^2 b_i(x_{2i-1}^2 + x_{2i-1}x_{2i}+x_{2i}^2)$ and $a_1x_1^2 + a_2 x_2^2 + b_1(x_3^2+x_3x_4+x_4^2)$, where $a_i$'s belong to $\{1,2,3,4,6,12\}$ and $b_i$'s belong to $\{1,2,4,8,16\}$.
-
arXiv:1708.04266 (Published 2017-08-05)
Representations of an integer by some quaternary and octonary quadratic forms
Comments: 20 pages, 4 tables. arXiv admin note: text overlap with arXiv:1607.03809Categories: math.NTIn this paper we consider certain quaternary quadratic forms and octonary quadratic forms and by using the theory of modular forms, we find formulae for the number of representations of a positive integer by these quadratic forms.
-
arXiv:1702.01249 (Published 2017-02-04)
On the number of representations of certain quadratic forms and a formula for the Ramanujan Tau function
Comments: 11 pagesCategories: math.NTIn this paper, we find the number of representations of the quadratic form $x_1^2+ x_1x_2 + x_2^2 + \ldots + x_{2k-1}^2 + x_{2k-1}x_{2k} + x_{2k}^2,$ for $k=7,9,11,12,14$ using the theory of modular forms. By comparing our formulas with the formulas obtained by G. A. Lomadze, we obtain the Fourier coefficients of certain newforms of level $3$ and weights $7,9,11$ in terms of certain finite sums involving the solutions of similar quadratic forms of lower variables. In the case of $24$ variables, comparison of these formulas gives rise to a new formula for the Ramanujan Tau function.
-
arXiv:1607.04764 (Published 2016-07-16)
On the representations of a positive integer by certain classes of quadratic forms in eight variables
Comments: 18 pages, 7 tables. arXiv admin note: substantial text overlap with arXiv:1607.03809Categories: math.NTIn this paper we use the theory of modular forms to find formulas for the number of representations of a positive integer by certain class of quadratic forms in eight variables, viz., forms of the form $a_1x_1^2 + a_2 x_2^2 + a_3 x_3^2 + a_4 x_4^2 + b_1(x_5^2+x_5x_6 + x_6^2) + b_2(x_7^2+x_7x_8 + x_8^2)$, where $a_1\le a_2\le a_3\le a_4$, $b_1\le b_2$ and $a_i$'s $\in \{1,2,3\}$, $b_i$'s $\in \{1,2,4\}$. We also determine formulas for the number of representations of a positive integer by the quadratic forms $(x_1^2+x_1x_2+x_2^2) + c_1(x_3^2+x_3x_4+x_4^2) + c_2(x_5^2+x_5x_6+x_6^2) + c_3(x_7^2+x_7x_8+x_8^2)$, where $c_1,c_2,c_3\in \{1,2,4,8\}$, $c_1\le c_2\le c_3$.
-
arXiv:1607.03809 (Published 2016-07-13)
On the number of representations by certain octonary quadratic forms with coefficients 1, 2, 3, 4 and 6
Comments: 38 pages, 10 TablesCategories: math.NTIn this paper, we find formulas for the number of representations of certain diagonal octonary quadratic forms with coefficients $1,2,3,4$ and $6$. We obtain these formulas by constructing explicit bases of the space of modular forms of weight $4$ on $\Gamma_0(48)$ with character.
-
arXiv:1312.4672 (Published 2013-12-17)
Non-vanishing of $L$-functions associated to cusp forms of half-integral weight
Comments: 8 pages, Accepted for publication in Oman conference proceedings (Springer)Categories: math.NTTags: conference paperIn this article, we prove non-vanishing results for $L$-functions associated to holomorphic cusp forms of half-integral weight on average (over an orthogonal basis of Hecke eigenforms). This extends a result of W. Kohnen to forms of half-integral weight.
-
On the Restriction Map for Jacobi Forms
In this article we give the description of the kernel of the restriction map for Jacobi forms of index 2 and obtain the injectivity of $D_0\oplus D_2$ on the space of Jacobi forms of weight 2 and index 2. We also obtain certain generalization of these results on certain subspace of Jacobi forms of square-free index $m$.
-
Evaluation of the convolution sums $\sum_{l+15m=n} σ(l) σ(m)$ and $\sum_{3l+5m=n} σ(l) σ(m)$ and some applications
Comments: To appear in IJNTCategories: math.NTWe evaluate the convolution sums $\sum_{l,m\in {\mathbb N}, {l+15m=n}} \sigma(l) \sigma(m)$ and $\sum_{l,m\in {\mathbb N}, {3l+5m=n}} \sigma(l) \sigma(m)$ for all $n\in {\mathbb N}$ using the theory of quasimodular forms and use these convolution sums to determine the number of representations of a positive integer $n$ by the form $$ x_1^2 + x_1x_2 + x_2^2 + x_3^2 + x_3x_4 + x_4^2 + 5 (x_5^2 + x_5x_6 + x_6^2 + x_7^2 + x_7x_8 + x_8^2). $$ We also determine the number of representations of positive integers by the quadratic form $$ x_1^2 + x_2^2+x_3^2+x_4^2 + 6 (x_5^2+x_6^2+x_7^2+x_8^2), $$ by using the convolution sums obtained earlier by Alaca, Alaca and Williams \cite{{aw3}, {aw4}}.
-
arXiv:0808.2395 (Published 2008-08-18)
Rankin's method and Jacobi forms of several variables
Comments: 11 pagesCategories: math.NTFollowing Rankin's method, D. Zagier computed the $n$-th Rankin-Cohen bracket of a modular form $g$ of weight $k_1$ with the Eisenstein series of weight $k_2$ and then computed the inner product of this Rankin-Cohen bracket with a cusp form $f$ of weight $k = k_1+k_2+2n$ and showed that this inner product gives, upto a constant, the special value of the Rankin-Selberg convolution of $f$ and $g$. This result was generalized to Jacobi forms of degree 1 by Y. Choie and W. Kohnen. In this paper, we generalize this result to Jacobi forms defined over ${\mathcal H} \times {\mathbb C}^{(g, 1)}$.
-
arXiv:0711.3512 (Published 2007-11-22)
Rankin-Cohen Brackets and van der Pol-Type Identities for the Ramanujan's Tau Function
Comments: 14 pagesCategories: math.NTWe use Rankin-Cohen brackets for modular forms and quasimodular forms to give a different proof of the results obtained by D. Lanphier and D. Niebur on the van der Pol type identities for the Ramanujan's tau function. As consequences we obtain convolution sums and congruence relations involving the divisor functions.