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Customer Review

on July 18, 2012
In his chapter on prime numbers, Mackenzie commits a serious error in his discussion of the Riemann Hypothesis. On page 131, he says, "In 1859, Bernard Riemann took another amazing step forward, which explains why the zeta function (discussed in Chapter 12) is now named after him rather than Chebyshev." Three sentences later, he continues, "...you need to know the infinitely many places in the plane where the Riemann zeta function takes the value zero." The zeta function discussed in Chapter 12 is the Euler zeta function, not the Riemann zeta function. There is no value of x, real or complex, for which the zeta function presented in Chapter 12 is zero! Mackenzie shows only the Euler zeta function (page 102). Any serious reader who attempts to find a value of x for which it is zero will become very frustrated, for there are no such values. The Riemann zeta function is an analytic continuation of the Euler zeta function and is significantly different in form. It does have zeros. I appreciate the fact that this book is not aimed at an audience literate in advanced mathematics, but there is no excuse for conflating these two very different functions; this makes nonsense of his discussion of the Riemann Hypothesis on pages 132 and 133. A simple solution would be to display the Riemann zeta function and note its similarity, in part, to the Euler zeta function. Mackenzie could then say that while the Euler zeta function has no zeros, the Riemann zeta function does. His discussion of the Riemann Hypothesis and the importance of the (nontrivial) zeros of the Riemann zeta function would then make sense. Unhappily, I should add that Mackenzie is not the first popular expositor of the Riemann Hypothesis to commit this serious error.
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