## Exponential Sums and Differential Equations by Nicholas M. Katz

By Nicholas M. Katz

This e-book is worried with parts of arithmetic, in the beginning sight disjoint, and with a number of the analogies and interactions among them. those components are the speculation of linear differential equations in a single complicated variable with polynomial coefficients, and the speculation of 1 parameter households of exponential sums over finite fields. After reviewing a few effects from illustration idea, the booklet discusses effects approximately differential equations and their differential galois teams (G_) and one-parameter households of exponential sums and their geometric monodromy teams (G). the ultimate a part of the booklet is dedicated to comparability theorems concerning G and G_ of certainly "corresponding" events, which supply a scientific clarification of the extraordinary "coincidences" discovered "by hand" within the hypergeometric case.

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Qxd 3/22/05 12:05 PM Page 31 conjectures • 31 is likely to be true, although they cannot prove it. Such deep conjectures have contributed enormously to the progress of mathematics. Fermat’s Last Theorem, labeled a “theorem” only because Fermat claimed to have proved it, was for centuries a plausible conjecture until it was finally proved by Andrew Wiles. Today’s most famous and hardest mathematical problem is by common consent the Riemann hypothesis, a conjecture about the distribution of the prime numbers.

Fermat’s Last Theorem, labeled a “theorem” only because Fermat claimed to have proved it, was for centuries a plausible conjecture until it was finally proved by Andrew Wiles. Today’s most famous and hardest mathematical problem is by common consent the Riemann hypothesis, a conjecture about the distribution of the prime numbers. Conjectures about prime numbers have another feature that can be both intriguing and infuriating. Because the primes are quite frequent among the “small” integers, there are many tempting conjectures that fail as soon as we get out a modern electronic calculator or a powerful computer.

Conway and Guy 1996, 107–10) • G. J. Fee and S. Plouffe have computed B200,000, which has about 800,000 digits. Bertrand’s postulate Joseph Bertrand (1822–1900) was a precocious student who published his first paper, on electricity, at the age of seventeen, but then became more notable as a teacher than as an original mathematician. Bertrand’s postulate states that if n is an integer greater than 3, then there is at least one prime between n and 2n − 2. (This is the precise theorem. ) Strangely, although it continues to be called a postulate, it is actually a theorem: it was proved by Tchebycheff in 1850 after Bertrand in 1845 had verified it for n less than 3,000,000.