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Focus on complex formulas in Lost Notebook

Krishnaswami Alladi

Srinivasa Ramanujan’s impact discussed at Mathfest event in the U.S.

The impact of the work of Srinivasa Ramanujan was discussed in Madison, Wisconsin, between July 31 and August 2 at the Mathfest event, which was the annual summer meeting of the Mathematical Association of America (MAA).

One of the highlights of Mathfest was an invited special session on “Ramanujan’s impact on number theory — then and now,” organised by Professor James Sellers of Pennsylvania State University. The four speakers at this special session were Professor George Andrews (Pennsylvania State University), Professor Bruce Berndt (University of Illinois at Urbana), Professor Krishnaswami Alladi (University of Florida), and Professor Ken Ono (University of Wisconsin).

Ramanujan’s Lost Notebook contains a vast collection of deep identities on q-hypergeometric series and related topics, including a significant number on mock theta functions. Professor George Andrews and Professor Bruce Berndt are now editing the Lost Notebook. Over the past few decades, most of the formulas in the Lost Notebook have been proved and explained by leading researchers, most notably George Andrews. But there are still several that are not fully understood. Professor Andrews discussed some of these in his lecture.

In his famous paper on modular equations and approximations to pi, Ramanujan recorded 17 hypergeometric-like series representations for the reciprocal of pi. Their proofs were completed only in 1987 by Professor Jonathan and Professor Peter Borwein. The Borwein brothers showed how one of these formulas of Ramanujan could be used to calculate several million digits of pi with great efficiency.

Over the past two decades, several leading researchers including Bruce Berndt and his collaborators, the brothers David and Gregory Chudnovsky, and Bill Gosper, have found new hypergeometric series representations for the reciprocal of pi, often by returning to Ramanujan’s paper and utilising his ideas. In his lecture, Professor Berndt provided a historical survey of such formulas stemming from Ramanujan’s work, and explained Ramanujan’s ideas on the Eisenstein series as well.

In 1917, Hardy and Ramanujan wrote an important paper of the number of prime factors of the integers. Professor Alladi, who spoke on this topic, said that it was indeed surprising that even though prime numbers were investigated since Greek antiquity, the first systematic discussion of the number of prime factors was done only in the early 20th century by Hardy and Ramanujan. The probabilistic underpinnings of the results of this important paper was realised only two decades later by Hungarian mathematicians Paul Turan, Paul Erdos, and Mark Kac, and led to the creation of Probabilistic Number Theory, a subject that is active even today. After tracing some of the landmark results in Probabilistic Number Theory arising from the Hardy-Ramanujan paper, Professor Alladi pointed out how results on the distribution of the number and size of the prime factors are utilised nowadays in establishing algorithms for testing primality.

Ramanujan’s tau function has served as a prototype for some of the deepest work in number theory in the 20th century. Ramanujan conjectured a bound for the tau function, which turned out to be an important example of the Weil Conjectures proved by Pierre Deligne, for which Deligne received the Fields Medal in 1978. Ramanujan’s incredible congruence for the tau function was useful to Andrew Wiles in the course of proving Fermat’s Last Theorem. Although the values of the tau function rise quite rapidly, the famous conjecture of Lehmer that the tau function is never zero is still unresolved. After describing the enormous impact of the tau function on number theory, Professor Ono discussed some deep questions which are still open, such as the Lehmer conjecture.

(The writer is with the University of Florida, Gainesville)

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