amy glen : euler’s contribution to number theory

From the time of Fermat (1601–1665) to that of Euler, the development of analysis was of primary interest. Consequently, there was very little progress in number theory during this period. Euler not only revived number theory, but applied analytic methods to his studies.

Perhaps the result that brought Euler the most fame in his earlier years was his solution to what had become known as the Basel problem. This

was to find a closed form for the sum of the infinite series of the form

Many of the top mathematicians, including the Bernoulli family, had tried unsuccessfully to find this sum. However, in 1735, Euler succeeded where they had failed by verifying not only numerically, but exactly, the value of

for the above sum. Euler's investigations did not stop there, and he proceeded to consider the sums of the reciprocals of higher powers. In 1736, he discovered that

,
where the B2k are the Bernoulli numbers defined by

After discovering this result on

, Euler's attention turned to the zeta function:

considered as a function of s.

He considered the following infinite product over all primes p:

.
Using similar reasoning about infinite products for partitions, he asserted that a term of the above product will be of the form

where

. Furthermore, by unique factorization, every n appears only once, and therefore Euler deduced that

.
Known today as the Euler Product Formula, it is essentially an analytic way of stating unique factorization.