CALCULUS Understanding
Its Concepts and Methods
CALCULUS Understanding
Its Concepts and Methods
Georg Friedrich Bernhard Riemann (1826--1866) --- Historical Sketch
Georg Friedrich Bernhard Riemann's name appears in every calculus course
because of the Riemann sum. But the man for
whom it is named did far more than that. A conjecture he made in 1859 is today
the most famous unsolved problem in mathematics, the
Riemann hypothesis. (There is a Clay
Mathematics
Institute
prize of a million dollars for the person who first solves it.)
Riemann was born in Germany in September 1826, his father Friedrich a Lutheran minister. He was second born in a family of six children. The father tutored the children, and when the young Bernhard was ten, a local teacher helped. Starting at age 14, he continued in public education and took a particular liking to mathematics. He read a book of some 900 pages on number theory in but six days.
At age 19, Bernhard entered the University of Göttingen, moving two years
later to the University of Berlin. Then in 1849, he moved back to
Göttingen. earning his doctorate under Carl Friedrich Gauss in 1851.
Eight years later, he was appointed to the chair of mathematics at
Göttingen, and within a few days was elected to the Berlin Academy of
Sciences, where he sent a report entitled On the number of
primes less than a given magnitude. In that paper appeared Riemann's
zeta
function
which
can also be written as the infinite
product
where
is the
th
prime. Dealing with this second representation, Riemann made a conjecture as
to where all of the roots of
lie in the complex plane, and it is this problem that was mentioned in the
first paragraph of this sketch.
Here are some problems having to do with prime numbers, which seems fitting in this sketch.
A prime number is called a Mersenne prime if it can be
written as
for
an integer. Name the first four Mersenne primes.
In the first four Mersenne primes, the exponent on
is itself a prime. Is this an accident?
A prime number is called a Fermat prime if it can be
written as
for
.
Pierre de Fermat conjectured that all numbers of that form are prime, but
Euler proved that
What
are the first four Fermat primes?
The
Clay Mathematics Institute was founded in 1998 by Mr. Landon T. Clay and his
wife Lavinia D. Clay. The aim of the Institute is "to increase and disseminate
mathematical knowledge."
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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.