Educational Affiliations


Research Interests

Impact on the Field



Dr. Subbarao’s doctoral thesis was on functional analysis, and in the beginning of his career, he wrote a few papers in functional analysis and topology; however, for most of his mathematical research career he devoted himself to the field of number theory, one of the most ancient and important specializations in mathematics.  Solving problems in number theory requires the researcher to be proficient in a wide variety of other mathematical specializations, such as algebra, complex function theory, algebraic geometry, and probability theory.   Though once concerned the “purest” or pure mathematical pursuits, today research findings in number theory are applied to solve practical problems in such fields as cryptology and dynamical systems.

Dr. Subbarao concentrated mainly on the areas of arithmetical functions and partition functions.  Specific problems that he investigated include:  

·        the distribution of additive and q-additive functions

·        the mean values of multiplicative and q-multiplicative functions

·        the characterization of the power function with complex exponent among the multiplicative functions

·        the distribution of generalized K-free integers

·        the Scholz-Brauer problem in addition chains

On the partition function p(n), Dr. Subbarao made the following important conjecture:

      For every positive integer m, on every arithmetic progression r (mod m); 0 ≤ r < m – 1, p(n) assumes both even and odd values infinitely often.

For the cases m = 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 40, the conjecture had been established by the efforts of many mathematicians. In fact, the works of Subbarao, Hirschhorn, and Subbarao and Hirschhorn among others like Frank Garvan and D. Stanton established these cases by elegant combinatorial methods.

This conjecture generated a lot of research by others in the field.   A major achievement toward proving this conjecture was made by K. Ono, whose result is as follows: following result of K.

For every positive integer m, in every arithmetic progression r (mod m), 0 ≤ r < m – 1, p(n) assumes even values infinitely often. Also, if p(n) assumes odd value for a single n in an arithmetical progression, then p(n) assumes odd values for infinitely many n in that arithmetical progression.

Quantitative versions of the above result were obtained later by J. P. Serre and S. Ahlgren. The odd case of the conjecture is still open, but has been verified for all m ≤ 105.  Dr. Subbarao offered a $500 prize for a complete proof of his conjecture. He had an analogous conjecture for product partitions.

Many other conjectures and unsolved problems appear in his papers with Erdös, Strauss, Katai, Hardy et al.  Here is another example:

If p1, . . . , pr are any distinct primes and a1, . . . , ar positive integers, then ∏ (piai - 1) divides (( ∏ piai) – 1) only if r = 1.

In one of his papers in 1966, Dr. Subbarao investigated the integer valued additive functions, and characterized the power function (with positive integer exponent among them) with some congruence condition, and formulated some interesting conjectures. This paper also inspired extensive research, the results of which were formulated in more than 20 research papers.

Formulating open research problems was a typical character of his research activity.  By this means, he successfully activated the research of other mathematicians, and thus contributed to the growth of the field as a whole.