Brundaban Sahu

Designation

Professor

Joining Date

02/08/2010

Research Area

Number Theory

Education

Ph. D., 2008, Harish-Chandra Research Institute

M. Sc., 2001, Sambalpur University

B. Sc., 1999, Gangadhar Meher College

Research Interest

Supercongruences: The numbers which occur in Ap\'{e}ry's proof of the irrationality of zeta(2) and zeta(3) have many interesting congruence properties.Work started with F. Beukers and D. Zagier, then extended by G. Almkvist, W. Zudilin and S. Cooper recently has complemented the Ap\'{e}ry numbers with set of sequences know as Ap\'{e}ry-like numbers which share many of the remarkable properties of the Ap\'{e}ry numbers. We study supercongruences properties of Ap\'{e}ry-like numbers.

Differential Operators: There are many interesting connections between differential operates and modular forms. Using Rankin-Cohen type differential operators on Jacobi forms/ Siegel modular forms we study certain arithmetic of Fourier coefficients.

Convolution sums and applications: We compute convolution sums of divisor function using the theory of modular forms and quasi modular forms and apply those to find number of representations of an integer by certain quadratic forms.

Representations of numbers by quadartic forms: We find number of representations of an integer by certain quadratic forms by computing modular forms/ theta series.

Publications

On the number of representations of a natural number by certain quaternary quadratic forms, Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics and Statistics, vol-340, pp 173--198, 2020 (with B. Ramakrishnan and Anup Kumar Singh)

Rankin-Cohen type operators for Hilbert-Jacobi forms, Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics and Statistics, vol-340, pp 143--156, 2020 (with Moni Kumari)

Rankin-Cohen brackets on Jacobi forms of several variables and special values of certain Dirichlet series,
Int. J. Number Theory (15), 2019, no-5, 925-933 (with Abhash Kumar Jha)

On the number of representations by certain quadratic forms in 8 variables, Automorphic Forms and Related Topics, 215-224, Contemp. Math., Vol-732, American Mathematical Society, Providence, RI, 2019 (with B. Ramakrishnan and Anup Kumar Singh)

Differential operators on Jacobi forms and special values of certain Dirichlet series, Automorphic Forms and Related Topics, 91-99, Contemp. Math., Vol-732, American Mathematical Society, Providence, RI, 2019 (with Abhash Kumar Jha)

Rankin-Cohen brackets on Hilbert modular forms and special values of certain Dirichlet series, Functiones et Approximatio Commentarii Mathematici, 58 (2018), 2, 257-268 (with Moni Kumari)

On the number of representations of certain quadratic forms and a formula for the Ramanujan Tau function, Functiones et Approximatio Commentarii Mathematici, 58 (2018), 2, 233-244 (with B. Ramakrishnan and Anup Kumar Singh)

Representations of an integer by some quaternary and octonary quadratic forms, In: Akbary, Amir, Gun, Sanoli (eds.), Geometry, Algebra, Number Theory and Their Information Technology Applications (GANITA) in honor of Professor V. Kumar Murty, Springer Proceedings in Mathematics and Statistics, vol 251, pp: 383-409

On the number of representations by certain octonary quadratic forms with coefficients 1, 2, 3, 4 and 6, Int. J. Number Theory, 14 (2018), 3, 751- 812 (with B. Ramakrishnan and Anup Kumar Singh)

On the representations of a positive integer by ceratin classes of quadratic forms in eight variables, In: Andrews G., Garvan F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series in honor of Professor K. Alladi, Springer Proceedings in Mathematics and Statistics, vol 221. Springer, Cham, pp 641-664 (with B. Ramakrishnan and Anup Kumar Singh)

Rankin-Cohen brackets on Siegel modular forms and Special values of Certain Dirichlet series, The Ramanujan Journal, 44 (2017), 1, 63-73 (with Abhash Kumar Jha)

On the number of representations of ceratin quadratic forms in 20 and 24 variables, Funct. Approx. Comment. Math., 54 (2016), 2, 151-161 (with B. Ramakrishnan).

Identities for the Ramanujan Tau function and certain convolution sum identities for the divisor functions, Number Theory, Lecture Notes Series in Ramanujan Mathematicial Society, No. 23, 2016, 63-75 (with B. Ramakrishnan)

Evaluation of convolution sums and some remarks on cusp forms of weight 4 and level 12, Math. J. Okayama Univ. 59 (2017), 71-79 (with B. Ramakrishnan)

Rankin-Cohen brackets on Jacobi Forms and the adjoint of some linear maps, The Ramanujan Journal, 39 (2016), 3, 533-544 (with Abhash Kumar Jha)

Supercongruences for sporadic sequences, Proc. Edinb. Math. Soc., 59 (2016), 2, 503-518 (with R. Osburn, A. Straub)

On the number of representations of certain quadratic forms of sixteen variables Int. J. Number Theory, 10 (2014), 8, 1929-1937(with B. Ramakrishnan),

A supercongruence for generalized Domb numbers, Funct. Approx. Comment. Math., 48 (2013), 1, 29-36 (with R. Osburn)

Evaluation of the convolution sums ∑_{l+15m=n} σ(l)σ(m) and ∑_{3l+5m=n}σ(l)σ(m) and an application, Int. J. Number Theory, 9 (2013), 3, 799-809 (with B. Ramakrishnan)

Supercongruences for Apéry-like numbers, Advances in Applied Mathematics, 47 (2011), 631-638 (with R. Osburn)

Congruences via modular forms, Proc. Amer. Math. Soc., 139 (2011), 7, 2375-2381 (with R. Osburn)

Rankin's method and Jacobi forms of several variables, Journal of the Australian Math. Soc., 88 (2010), 1, 131-143 (with B. Ramakrishnan)

Distribution of Residues Modulo p, Acta Arith. 129 (2007), 325-333 (with S. Gun, Florian Luca, P. Rath, R. Thangadurai)

On the Fourier expansions of Jacobi forms of half integral weight, Int. J. Math. Math. Sci. Vol 2006 (with B. Ramakrishnan)

Distribution of quadratic non-residues which are not primitive roots, Math. Bohemica, 130 (2005), 4, 387-396 (with S. Gun, R.Thangadurai, B. Ramakrishnan)

Preprints

Rankin-Cohen Brackets and van der Pol-Type Identities for the Ramanujan's Tau Function (with B. Ramakrishnan), arXiv:0711.3512v1

Teaching

M463: Finite Fields, Fall 2021

Project/Thesis Guidance

Abhash Kumar Jha, PhD Thesis 2016

Moni Kumari, PhD Thesis 2018

NISER

Brundaban Sahu

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