Dr. Anup Kumar Singh

@allduniv.ac.in

Assistant Professor, Department of Mathematics, Faculty of Science
University of Allahabad

Dr. Anup Kumar Singh


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https://researchid.co/anupsingh

RESEARCH, TEACHING, or OTHER INTERESTS

Algebra and Number Theory
11

Scopus Publications

35

Scholar Citations

4

Scholar h-index

Scopus Publications

  • Explicit evaluation of triple convolution sums of the divisor functions
    B. Ramakrishnan, Brundaban Sahu, Anup Kumar Singh
    International Journal of Number Theory, 2024
    In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums [Formula: see text] for odd integers [Formula: see text] and [Formula: see text], where [Formula: see text] is the sum of the [Formula: see text]th powers of the positive divisors of [Formula: see text]. We consider four cases, namely (i) [Formula: see text], (ii) [Formula: see text]; [Formula: see text] (iii) [Formula: see text]; [Formula: see text] and (iv) [Formula: see text], and give explicit expressions for the respective convolution sums. We provide several examples of these convolution sums in each case and further use these formulas to obtain explicit formulas for the number of representations of a positive integer [Formula: see text] by certain positive definite quadratic forms. The existing formulas for [Formula: see text] (in [20]), [Formula: see text] (in [7]), [Formula: see text] (in [35]), [Formula: see text], [Formula: see text] (in [30]) and [Formula: see text] (in [31]), which were all obtained by using the theory of quasimodular forms, follow from our method.
  • A simple extension of Ramanujan–Serre derivative map and some applications
    B. Ramakrishnan, Brundaban Sahu, Anup Kumar Singh
    Ramanujan Journal, 2023
  • REPRESENTATIONS OF SQUARES BY CERTAIN DIAGONAL QUADRATIC FORMS IN AN ODD NUMBER OF VARIABLES
    Balakrishnan Ramakrishnan, Brundaban Sahu, Anup Kumar Singh
    Rocky Mountain Journal of Mathematics, 2023
    We consider diagonal quadratic forms a1x12+a2x22+⋯+aℓxℓ2, where ℓ≥5 is an odd integer and ai≥1 are integers. By using the extended Shimura correspondence, we obtain explicit formulas for the number of representations of |D|n2 by such quadratic forms, where D is either a squarefree integer or a fundamental discriminant such that (−1)(ℓ−1)∕2D>0. We demonstrate our method with many examples, in particular recovering results of Cooper, Lam and Ye (2013): all their formulas (when ℓ=5) for n2 for quinary quadratic forms and all the representation formulas for septenary quadratic forms when n is even. (Those formulas were originally derived by combining certain theta function identities with a method of Hurwitz.) Our method works with arbitrary coefficients ai. As a consequence of some of our formulas, we obtain identities among the representation numbers and also congruences involving the Fourier coefficients of certain newforms of weights 6 and 8 and divisor functions.
  • Some identities for the partition function
    Ankush Goswami, Abhash Kumar Jha, Anup Kumar Singh
    Journal of Mathematical Analysis and Applications, 2022
  • Ramanujan—Mordell Type Formulas Associated to Certain Quadratic Forms of Discriminant 20 k or 32 k
    Anup Kumar Singh, Dongxi Ye
    Indian Journal of Pure and Applied Mathematics, 2020
  • On the Number of Representations of a Natural Number by Certain Quaternary Quadratic Forms
    B. Ramakrishnan, Brundaban Sahu, Anup Kumar Singh
    Springer Proceedings in Mathematics and Statistics, 2020
  • On the number of representations of certain quadratic forms in 8 variables
    B. Ramakrishnan, Brundaban Sahu, Anup Singh
    Contemporary Mathematics, 2019
    . In this paper, we find the number of representations of integers by certain quadratic forms in 8 variables by using the theory of modular forms. By expressing these formulas in terms of certain convolution sums of the divisor function and using our formulas, we deduce formulas for the convolution sums W j, 7 ( n ) for j = 1 , 2 , 3 , 4.
  • On the number of representations of certain quadratic forms and a formula for the ramanujan tau function
    B. Ramakrishnan, Brundaban Sahu, Anup Kumar Singh
    Functiones Et Approximatio Commentarii Mathematici, 2018
    In 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.
  • On the number of representations by certain octonary quadratic forms with coefficients 1, 2, 3, 4 and 6
    B. Ramakrishnan, Brundaban Sahu, Anup Kumar Singh
    International Journal of Number Theory, 2018
    In this paper, we find formulas for the number of representations of certain diagonal octonary quadratic forms with coefficients [Formula: see text] and [Formula: see text]. We obtain these formulas by constructing explicit bases of the space of modular forms of weight [Formula: see text] on [Formula: see text] with character.
  • Representations of an Integer by Some Quaternary and Octonary Quadratic Forms
    B. Ramakrishnan, Brundaban Sahu, Anup Kumar Singh
    Springer Proceedings in Mathematics and Statistics, 2018
  • On the representations of a positive integer by certain classes of quadratic forms in eight variables
    B. Ramakrishnan, Brundaban Sahu, Anup Kumar Singh
    Springer Proceedings in Mathematics and Statistics, 2017

RECENT SCHOLAR PUBLICATIONS

  • Explicit evaluation of triple convolution sums of the divisor functions
    B Ramakrishnan, B Sahu, AK Singh
    International Journal of Number Theory 20 (04), 1073-1098 , 2024
    2024
    Citations: 3
  • A simple extension of Ramanujan–Serre derivative map and some applications
    B Ramakrishnan, B Sahu, AK Singh
    The Ramanujan Journal 61 (4), 1379-1410 , 2023
    2023
    Citations: 4
  • REPRESENTATIONS OF SQUARES BY CERTAIN DIAGONAL QUADRATIC FORMS IN AN ODD NUMBER OF VARIABLES
    B Ramakrishnan, B Sahu, AK Singh
    Rocky Mountain Journal of Mathematics 53 (4), 1219-1244 , 2023
    2023
  • Some identities for the partition function
    A Goswami, AK Jha, AK Singh
    Journal of Mathematical Analysis and Applications 508 (1), 125864 , 2022
    2022
    Citations: 8
  • Ramanujan—Mordell Type Formulas Associated to Certain Quadratic Forms of Discriminant 20 k or 32 k
    AK Singh, D Ye
    Indian Journal of Pure and Applied Mathematics 51 (3), 1083-1096 , 2020
    2020
    Citations: 1
  • Shimura and Shintani liftings of certain cusp forms of half-integral and integral weights
    MK Pandey, B Ramakrishnan, AK Singh
    Tsukuba Journal of Mathematics 43 (2), 191-210 , 2019
    2019
  • On the number of representations of a natural number by certain quaternary quadratic forms
    B Ramakrishnan, B Sahu, AK Singh
    International conference on number theory, 173-198 , 2018
    2018
  • On the number of representations of certain quadratic forms and a formula for the Ramanujan Tau function
    B Ramakrishnan, B Sahu, AK Singh
    Functiones et Approximatio Commentarii Mathematici 58 (2), 233-244 , 2018
    2018
  • On the number of representations by certain octonary quadratic forms with coefficients 1, 2, 3, 4 and 6
    B Ramakrishnan, B Sahu, AK Singh
    International Journal of Number Theory 14 (03), 751-812 , 2018
    2018
    Citations: 8
  • Certain quaternary quadratic forms of level 48 and their representation numbers
    B Ramakrishnan, B Sahu, AK Singh
    arXiv preprint arXiv:1801.04392 , 2018
    2018
  • Representations of an integer by some quaternary and octonary quadratic forms
    B Ramakrishnan, B Sahu, AK Singh
    Conference on Geometry, Algebra, Number Theory, and their Information … , 2016
    2016
    Citations: 2
  • On the representations of a positive integer by certain classes of quadratic forms in eight variables
    B Ramakrishnan, B Sahu, AK Singh
    Gainesville International Number Theory Conference, 641-664 , 2016
    2016
    Citations: 3
  • On the number of representations of certain quadratic forms in 8 variables
    B Ramakrishnan, B Sahu, AK Singh
    Automorphic forms and related topics 732, 215-224 , 2016
    2016
    Citations: 6

MOST CITED SCHOLAR PUBLICATIONS

  • Some identities for the partition function
    A Goswami, AK Jha, AK Singh
    Journal of Mathematical Analysis and Applications 508 (1), 125864 , 2022
    2022
    Citations: 8
  • On the number of representations by certain octonary quadratic forms with coefficients 1, 2, 3, 4 and 6
    B Ramakrishnan, B Sahu, AK Singh
    International Journal of Number Theory 14 (03), 751-812 , 2018
    2018
    Citations: 8
  • On the number of representations of certain quadratic forms in 8 variables
    B Ramakrishnan, B Sahu, AK Singh
    Automorphic forms and related topics 732, 215-224 , 2016
    2016
    Citations: 6
  • A simple extension of Ramanujan–Serre derivative map and some applications
    B Ramakrishnan, B Sahu, AK Singh
    The Ramanujan Journal 61 (4), 1379-1410 , 2023
    2023
    Citations: 4
  • Explicit evaluation of triple convolution sums of the divisor functions
    B Ramakrishnan, B Sahu, AK Singh
    International Journal of Number Theory 20 (04), 1073-1098 , 2024
    2024
    Citations: 3
  • On the representations of a positive integer by certain classes of quadratic forms in eight variables
    B Ramakrishnan, B Sahu, AK Singh
    Gainesville International Number Theory Conference, 641-664 , 2016
    2016
    Citations: 3
  • Representations of an integer by some quaternary and octonary quadratic forms
    B Ramakrishnan, B Sahu, AK Singh
    Conference on Geometry, Algebra, Number Theory, and their Information … , 2016
    2016
    Citations: 2
  • Ramanujan—Mordell Type Formulas Associated to Certain Quadratic Forms of Discriminant 20 k or 32 k
    AK Singh, D Ye
    Indian Journal of Pure and Applied Mathematics 51 (3), 1083-1096 , 2020
    2020
    Citations: 1
  • REPRESENTATIONS OF SQUARES BY CERTAIN DIAGONAL QUADRATIC FORMS IN AN ODD NUMBER OF VARIABLES
    B Ramakrishnan, B Sahu, AK Singh
    Rocky Mountain Journal of Mathematics 53 (4), 1219-1244 , 2023
    2023
  • Shimura and Shintani liftings of certain cusp forms of half-integral and integral weights
    MK Pandey, B Ramakrishnan, AK Singh
    Tsukuba Journal of Mathematics 43 (2), 191-210 , 2019
    2019
  • On the number of representations of a natural number by certain quaternary quadratic forms
    B Ramakrishnan, B Sahu, AK Singh
    International conference on number theory, 173-198 , 2018
    2018
  • On the number of representations of certain quadratic forms and a formula for the Ramanujan Tau function
    B Ramakrishnan, B Sahu, AK Singh
    Functiones et Approximatio Commentarii Mathematici 58 (2), 233-244 , 2018
    2018
  • Certain quaternary quadratic forms of level 48 and their representation numbers
    B Ramakrishnan, B Sahu, AK Singh
    arXiv preprint arXiv:1801.04392 , 2018
    2018