Dirichlet 1830
Euler extended the harmonic series to an infinite number of infinite
series,
one for each natural number, corresponding to the sum of reciprocals of
squares, cubes. etc.
Derichlet extended the discrete
collection of infinite series to a continuum of infinite series.
(s) = 
(1/ns) where s
could be any number greater than 1.
He proved in 1837 that in any arithmetic
progression with first term coprime
to the difference there are infinitely many primes. This had been
conjectured
by Gauss.
Note:
Proposed in 1837 the modern definition of a function.
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