B. Ramakrishnan

Research Summary:

1. (with S. Gun): Lacunarity of certain Dedekind η-Products: A formal power series
   x^&nu&sum_n=0^&infin a(n) xⁿ is called lacunary if the number of non-zero Fourier coefficients a(n) with n &le t is o(t). In other words, the arithmetic density of its non-zero coefficients is zero.
   
   In 1985 J. -P. Serre [ Glasgow Math. J. 27, 203--221] classified all even powers of the Dedekind eta-function which are lacunary. More precisely, he proved that &eta^r(z) is lacunary if and only if r=2,4,6,8,10,14 or 26.
   
   B. Gordon and S. Robins [ Glasgow Math. J. 37 (1995), 1--14] used Serre's method to classify all two eta products of the form &eta^r(z) &eta^s(2z), where r+s is even and rs &ne 0.
   
   In this work, we classify all lacunary forms associated to the eta-products of the form &eta^r(z) &eta^s(3z), where r+s is even and rs &ne 0. In particular, we prove the following theorem
   
   Theorem: Suppose that r+s is even and rs &ne 0. Then &eta^r(z) &eta^s(3z) is lacunary if and only of (r,s) is one of the following 18 pairs:
   k=1;   (1,1)   (-1,3)   (3,-1)
   k=2;   (-1,5)   (5,-1)   (1,3)   (3,1)   (2,2)
   k=3;   (3,3)
   k=4;   (-1,9)   (9,-1)   (-2,10)   (10,-2)
             (-3,11)   (11,-3)   (5,3)   (3,5)
   k=7;   (7,7).

2. (with S. Gun) On the representation of integers as sums of odd number of squares.
   Short Summary: A classical problem in number theory is to give an explicit formula for the number of ways one can represent a non-negative integer n as a sum of kquares, where k is a positive integer. The study of r_k(n) has a long history. For k= 2, 4, 6, 8, elegant formulae for r_k(n) were found by Jacobi. A general formula for r_k(n), when k is even was stated by Ramanujan . It was proved by Mordell. It is also known that r_k(n) can be expressed in terms of coefficients of Eisenstein series and cusp forms and Rankin showed that the cusp form part is non-trivial for k > 8. Recently, combining a variety of methods, S. Milne obtained formulas for r_4s²(n) and r_4s²+4s(n) for every positive integer s. Using Zagier's work on the Kac-Wakimoto conjecture, K. Ono [ J.Number Theory 95 (2002), 253--258.] obtained formulas (simpler than Milne's) for r_4s²(n) and r_4s²+4s(n), which are sums of products of divisor functions. For odd values of k a general formula is not known, though formulas for particular values of k are known. For example, the formulas for r_k(n) or r_k(n²) are known for k = 1, 3, 5, 7, 9, 11, 13. These formulas were obtained either by an elementary method or by using the theory of modular forms.
   
   In a recent work, S. Cooper [ J. Number Theory 103 (2003), 135--162] conjectured a formula for r_2k+1(p²). Using the Shimura Correspondence we obtain an explicit formula for r_2k+1(n²), n &ge 1. As a consequence, we prove that the values at 1-k of the Dirichlet L-functions twisted by the character &chi₄ are rational, which can be written down explicitly using our formula. Here &chi₄ is the trivial character or the non-trivial character modulo 4 depending on whether k is even or odd.

3. (with Shaun Cooper, Michael D. Hirschhorn and S. Gun) Multiplicative Relations for Certain Powers of Euler's Product.
   Short Summary: Certain arithmetic relations for the coefficients in the expansions of (q)_&infin^r, (q)_&infin^r(q^t)_&infin^s, t=2,3,4 were studied by M. Newman, S. Cooper, M. D. Hirschhorn, R. Lewis, S. Ahlgren and R.Chapman. In this work, we prove similar identities for certain multi-product expansions using an elementary method. This result is equivalent to show that certain Dedekind eta-products are lacunary.

4. (with S. Gun, B. Sahu and R. Thangadurai) Distribution of Quadratic non-residues which are not primitive roots.
   Short Summary: In this paper we study the distribution of quadratic non-residues which are not primitive roots modulo p^h or 2p^h where p is an odd prime and h &ge 1 is an integer using elementary and combinatorial methods.

5. (with M. Manickam) An estimate for certain average of the special values of character twists of Hecke L-functions: We prove the Lindelöf hypothesis in weight aspect for the average of the special values of the twisted L-functions of newforms, which generalizes the work of W. Kohnen and J. Sengupta.

Publications:

1. An Introduction to Modular Forms and Hecke Operators, (with M. Manickam), in Elliptic Curves, Modular Forms and Cryptography, Hindustan Book Agency (2003), 223--245.
2. Elliptic Curves, Modular Forms and Cryptography, Edited by A. K. Bhandari, D. S. Nagaraj, B. Ramakrishnan, and T. N. Venkataramana, Hindustan Book Agency 2003.

Conference/Workshops Attended:

1. Attended and gave a 40 minutes talk in the 18th Annual Workshop on Automorphic Forms held at the Department of Mathematics, University of California, Santa Barbara, USA during March 21--24, 2004.

Visits to other Institutes:

1. Visited Department of Mathematics, Vivekananada College, Chennai during December 2003 -- January 2004.