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Some mathematicians spend their careers on research problems that only a few hundred people can read. Others spend it helping children feel at home with numbers. Mangala Narlikar did both. This post covers what is on the record about her life, what her research was about, a worked example of the sieve idea behind it, and what a student can borrow from her approach.
Who was Mangala Narlikar?
The Indian Express obituary of 17 July 2023 describes her as an eminent mathematician who carried out research in pure mathematics, with core interests in real and complex analysis, analytic geometry, number theory, algebra and topology. It says she worked first at the Tata Institute of Fundamental Research in Mumbai and then as a lecturer at the universities of Bombay and Pune. She died in Pune, aged 80. She was married to the astrophysicist Jayant Narlikar, whose work is in our post on Jayant Narlikar and cosmology.
The same report quotes Arvind Paranjpye, director of the Nehru Planetarium in Mumbai: "She dedicated her life to teaching mathematics and science in India. She has written several books for school students on mathematics in English, Hindi and Marathi." He added that in Pune she taught mathematics to school and college students.
What did she research?
Her papers sit in analytic number theory, which uses tools from calculus to study whole numbers. Crossref records several of them:
On a theorem of Erdos and Szemeredi, Hardy-Ramanujan Journal, 1980, with her affiliation given as the Tata Institute of Fundamental Research.
Contributions to the Erdos-Szemeredi theory of sieved integers, with K. Ramachandra, Acta Arithmetica, 1980.
On the mean square value of Hurwitz zeta function, Proceedings of the Indian Academy of Sciences, 1981.
On orders solely of abelian groups II, with S. Srinivasan, Bulletin of the London Mathematical Society, 1988.
The last title asks a question you can understand with school algebra: for which whole numbers n is every group of size n abelian, meaning its operation does not depend on order? For n = 6 the answer is no, because the six symmetries of an equilateral triangle form a group in which the order of moves matters. For n = 15 every group turns out to be abelian.
Worked example: what is a sieve?
"Sieved integers" are the numbers left after you strike out multiples of chosen primes. You have already used one: the sieve of Eratosthenes. Here is a small version.
Write the numbers 1 to 30.
Strike out every multiple of 2, then of 3, then of 5.
What remains is 1, 7, 11, 13, 17, 19, 23 and 29: eight numbers.
Now predict that count without listing. Half the numbers survive the 2-sieve, two-thirds of those survive the 3-sieve, and four-fifths of those survive the 5-sieve: 30 x 1/2 x 2/3 x 4/5 = 8. The prediction is exact here because 30 = 2 x 3 x 5. For other ranges it is only approximately right, and measuring how far off it can be is the kind of question sieve theory studies.
A common mistake is to subtract the fractions, as in 30 x (1 - 1/2 - 1/3 - 1/5), which double-counts numbers like 6 and 10 and gives the wrong answer. Our post on primes for Class 6 starts from the same sieve.
Why does writing maths for students matter?
Research mathematicians often know exactly where a topic connects to the next one, and that makes them well placed to explain the first steps. Paranjpye's tribute stresses this side of her work: generations of students in Pune learned from her, and her books reached readers in three languages.
For parents, our guide on helping with maths you have forgotten offers ways to support a child even if your own school maths is rusty.
What can a student take from her career?
Big ideas start small. The sieve you learn early grows into research mathematics.
Explain in more than one language. Writing for students in English, Hindi and Marathi meant finding plain words for every step.
A career can have more than one shape. Research, university teaching and books for children all count as mathematics.
When a step in a number theory problem is unclear, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer.
Try it: sieve the numbers 1 to 42 with the primes 2, 3 and 7, predict the count with fractions, then check by listing.
