On the Sums $$\sum_{k=-\infty}^{\infty} (4k+1)^{-n}$$

Year of Award: 2004

Publication Information: The American Mathematical Monthly, August-September 2003, pp. 561-573.

Summary: This paper interprets the sum in the title as a volume of an n-dimensional polytope which can in turn be related to combinatorial problems.  The sum is relevant to the Euler and Bernoulli numbers that arise in the classical study of the zeta function.

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About the Author: (from The American Mathematical Monthly, (2003))

Noam Elkies is a number theorist, most of whose work concerns Diophantine geometry, computational number theory, and connections with other fields such as sphere packing and error-correcting codes. He also publishes occasionally in enumerative combinatorics and combinatorial games. He twice represented the United States at the International Mathematical Olympiad, winning gold medals both times, and was a Putnam Fellow in each of the three years he took the Putnam examination. He has been at Harvard since coming there as a graduate student in 1985; after earning his Ph.D. under Barry Mazur, he was a Junior Fellow, then Associate Professor, and was granted tenure in 1993 at age 26, the youngest in Harvard's history. His work has also been recognized by awards such as a Packard Fellowship and the Prix Peccot of the College de France. Elkies' main interest outside mathematics is music, mainly classical piano and composition. Recently performed works include a full-length opera, Yossele Solovey, and several orchestral compositions, one of which had Elkies playing the solo piano part in Boston's Symphony Hall. He still has some time for chess, where he specializes in composing and solving problems; he won the world championship for solving chess problems in 1996, and earned the Solving Grandmaster title in 2001.

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Noam Elkies
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Noam Elkies
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Publication Date:
Tuesday, September 23, 2008
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Summary:

This paper interprets the sum in the title as a volume of an n-dimensional polytope which can in turn be related to combinatorial problems. The sum is relevant to the Euler and Bernoulli numbers that arise in the classical study of the zeta function.