The great prime number race. (English) Zbl 1462.11005
Student Mathematical Library 92. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-6257-4/pbk; 978-1-4704-6279-6/ebook). xii, 138 p. (2020).
A version of the prime number theorem states that the number \(\pi(x)\)
of prime numbers \(p \le x\) is asymptotically equal to
Li\((x) = \int_0^x \frac{dt}{\log t}\). J. E. Littlewood [C. R. Acad. Sci., Paris 158, 1869–1872 (1914; JFM 45.0305.01)]
proved that \(\pi(x) - \mathrm{Li}(x)\) changes sign infinitely often.
For small values of \(x\), we have \(\pi(x) < \mathrm{Li}(x)\), and one may
ask for the smallest value of \(x\) such that \(\pi(x) > \mathrm{Li}(x)\).
It is difficult to come up with more boring questions in number theory –
BUT: the mathematics needed for understanding this question and for
getting closer to an answer is very beautiful, and the author does an
outstanding job in explaining the necessary background: Riemann’s
zeta function, its Euler product expansion, the functional equation,
Riemann’s formula, Zagier’s exposition of Newman’s proof of the prime
number formula (see [D. Zagier, Am. Math. Mon. 104, No. 8, 705–708 (1997; Zbl 0887.11039)]), and finally
Littlewood’s theorem. The book is accessible to anyone with a solid background in complex analysis and is highly recommended.
Reviewer: Franz Lemmermeyer (Jagstzell)
MSC:
11-03 | History of number theory |
11A41 | Primes |
11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
11N05 | Distribution of primes |
Online Encyclopedia of Integer Sequences:
Denominator of Bernoulli number B_n.a(n) = smallest k such that li(k) - pi(k) >= n, where li(k) is the logarithmic integral and pi(x) is the number of primes <= x.