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Some remarks and problems in number theory related to the work of Euler. (English) Zbl 0526.01014

The problems in number theory referred to in the title include: the prime number theorem; Erdős’ conjecture that if \(1\leq a_1< a_2< a_3\dots\) is a sequence of integers for which \(\sum_{n=1}^{\infty}(a_n)^{-1}=\infty\), then the sequence \(\{a_n\}\) contains arbitrarily long arithmetical progressions; the number \(p(n)\) of partitions of \(n\); the Euler function \(\varphi(n)\); the sum of reciprocal squares and the Zeta-function; and others. In each case the authors indicate Euler’s contribution to the subject and sketch later developments. In a number of cases they state their surprise that Euler did not pursue matters further, writing, for instance, that “with a little experimentation Euler could have discovered the prime number theorem”. The authors see these cases as evidence that, in number theory, Euler was not primarily interested in the functions that occur.
Reviewer: H.J.M.Bos

MSC:

01A50 History of mathematics in the 18th century
11-03 History of number theory
11N05 Distribution of primes
11B25 Arithmetic progressions
00A07 Problem books
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