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| [The following remarks refer mainly to the early hardback editions of the book. Most errata were cleared up by the fifth printing, and if any remain in the paperback edition, I shall record them in the paperback web page. So the paperback web page includes a continuation of these errata.] I did my best with proof-reading, but in a book of this kind, written on a tight schedule (because we knew that two other books on the same topic were being produced), there will be some typos and minor errors of fact. So I set up this page to log errata, for correction in some hoped-for future edition. However, there weren't as many errata as I had feared, and I started to get interesting comments from readers, including some of the professionals mentioned in the book. I have therefore expanded this into an "Errata and Comments" page. If there is a future revised edition, the errata will of course be corrected, and I shall incorporate as many of the other comments as my publisher permits. If, when reading the book, you spot any errors that are not on the list below, or if you have any interesting observations or constructive criticisms, please send me an email. Thanks! [One point that is more of an omission than an erratum: Several readers have asked me for the correct pronunciation of Bernhard Riemann's name. It never occurred to me to give this in the book, though of course I should have. Bernhard: the "h" and both the "r"s are pronounced. Riemann: "REE-mahn,: the "a" a teeny bit longer than in English "man." After-note on the above, added 11/6/04: That is how I have heard it from German-speakers. I am reliably told, however, that the "a" in "Riemann" is in fact closer to a schwa – that is, the "neutral" unstressed English vowel in "about." Since most of the German-speakers I know are in fact Swiss or Austrian, I may have been hearing nonstandard German; or perhaps my hearing needs attention.] Added after 5th printing: Most of the simple typos--the ones I have marked with an asterisk below--were corrected on the hardback's fifth printing in July 2003.
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| Page | Line(s) | Error | |
| 15 | 20-21 | "Modern analysis does not admit..." There is actually a "non-standard" analysis built around a rigorous definition of the concept "infinitesimal," and associated most particularly with the work of Abraham Robinson in the 1960s (though some of the concepts go back to Hilbert). Non-standard analysis, while interesting and perfectly sound, has, however, had little impact on the everyday work of mathematicians in the fields I am writing about. Furthermore, I have my work cut out just explaining traditional analysis to non-specialist readers, and didn't want to add distractions of this sort. Possibly I should have said "modern standard analysis"... but even that would have muddied the waters a bit. Perhaps a footnote is needed here... | |
| 18 | 13 | "Trigonometry, for example,..." That second comma should be moved forward to come after the parentheses. | |
| 39 | 7 | "33.6247" should be "33.5069." | |
| (same) | 16 | "The word 'exponential' is one of those that has..." The word "has" should be "have." | |
| 44 | 28 | "34.5378" should be "34.5387"; "33.6247" should be "33.5069"; and "2.7156" should be "3.0794." | |
| 74 | 1 | The comma after "trillionth" should be removed. | |
| 80 | 2 | "Expression 5-3" should be "Expression 5-2." | |
| * | (same) | last | "x" should be "s." |
| 89 | 6 | (First line of quoted passage.) "defeat" should be "defeats." | |
| 93 | 12 | "...20 miles east of Cologne..." Düren is actually 20 miles southwest of Cologne. | |
| 100 | 17, 27 | The number 104 was repeated in the first number list--should have gone "104 105." This error was picked up by the second list, where "104" should be "105." | |
| 109 | 21 | "It is, for example transparent..." Should be a comma after "example." | |
| * | 111 | 4 | "...so long as x is not equal to -1..." Should be "N," not "x." |
| 125 | 12 | Prof. Erik Hemmingsen of Syracuse University notes that while Selberg was at the Institute when his paper was published, the relevant work was done at Syracuse. For more on the Selberg-Erdős controversy, and a full extract from Prof. Hemmingsen's letter, see my "FAQ" page note to p. 125. | |
| * | 141 | 16 | "west" and "east are the wrong way round. |
| * | 143 | 3 | "x" should be "s." |
| * | 147 | 14 | The values given for (-1/4)! and (1/2)! are the wrong way round; e.g. (1/2)! is Gamma(3/2), which is half root pi. |
| 148 | 10 | "...if x = ½" That "x" should be an "s." | |
| 161 | 29 | "Until quite recently it was thought..." Analytic number theorist Sid Graham of Central Michigan University notes that there were much earlier results that cast doubt on my Theorem 15-1, notably a 1942 result of Ingham's ("On Two Conjectures in the Theory of Numbers," Amer. J. Math. 64, 313-319.) The result by Odlyzko and te Riele that I mention was based on Ingham's work. Sid: "Even though the Mertens Conjecture was not disproved until 1985, people were skeptical about it long before." | |
| * | 162 | 18 | "George" should be "Georges." (Same in the index, p.417.) |
| * | 165 | 24 | The initial "E" of the name "Emile Picard" should have an acute accent. |
| 171 | 14 | "Natural numbers tend to be written like this" Should be a colon after "this." | |
| 171 | 15 | Same problem. | |
| 171 | 20 | Same problem (colon instead of comma here). | |
| 171 | 21 | This sentence leaves the impression that a vulgar fraction must be improper. According to the OED, any fraction written as two integers with a fraction bar is vulgar. | |
| * | (same) | 28 | The colon at the end of this line should be outside the quotes. |
| 172 | 1 | "Complex numbers look like this," That comma should be a colon. | |
| 173 | 14 | "...any infinite sequence of numbers in R has a limit in R..." Ouch! I mean, of course, any convergent infinite sequence. | |
| * | 203 | 22-23 | "If ez = w, then w = log z." This should read: "If ez = w, then z = log w." |
| 207 | Last line | "...about a function;..." That semicolon should be a colon. | |
| 219 | 29 | "...the argument ant has a twin brother..." A reader points out that worker ants are technically female, so this should be "sister." | |
| 228 | 13 | "The second..." should be "The third..." | |
| (same) | 30 | "...has been best publicized..." That "best" should be "better." | |
| 241 | 7, 9 | The function shown is 0.1x2, not x2; the text should reflect this. | |
| * | 259 | 8 | "50" should be "100" |
| 272 | 17-23 | The five sentences in this paragraph are out of order. The fifth sentence ought to be the third; the third and fourth should be fourth and fifth. | |
| * | 274 | 10 | The second matrix should have "-5" in the bottom left corner, not "5." |
| 275 | 13 | Semicolon should be colon. | |
| 316 | 16 | "Michael Berry argued that..." I got a very
generous & funny letter from Sir Michael, thanking me for the book and
adding a couple to my stock of mathematician anecdotes. He did make
one point of criticism, though, which I think belongs here. I
reproduce it from his letter verbatim, with his permission.
Here it is:
“...what you write,
although correct, misses the crucial point provided by the quantum
analogy, namely the prediction and detailed description1,2 of
deviations from GUE statistics in the correlations between widely
separated zeros. Coincidentally,
Andrew Odlyzko had noticed these deviations (in the number variance of the
zeros), and had wondered whether they were the result of a flaw in his
programs. He very kindly made
his data available to me, and the deviations fitted the ‘quantum’
theory accurately, apart from some fine-scale oscillations that have now
been explained by Jon Keating and Eugene Bogomolny3.
This explanation of the deviations is in my view the strongest
evidence supporting the Riemann Hypothesis and locates the elusive
operator in the class of quantum systems with classical chaos, rather than
the class of random matrices.” 1
Berry, M.V., “Semiclassical formula for the number variance of
the Riemann Zeros” in Nonlinearity 1 (1988), pp.399-407. 2
Berry, M.V. and Keating, J.P., “The Riemann Zeros and Eigenvalue
Asymptotics” in the SIAM Review, Vol. 41, No. 2 (1999),
pp.236-266. [“SIAM”
stands for “Society for Industrial and Applied Mathematics.”] 3
Bogomolny, E. and Keating, J.P., “Asymptotics of the pair
correlation of Riemann zeros,” (1999). |
|
| * | 325 | 23 | "...all infinitely many zeros..." The word "all" should not be there. |
| * | 327 | 5 | The Greek letter "zeta" should appear at the end of this line, before the period. |
| * | 332 | 25 | "...as an infinite product, F(x) =..." Should be "F(z)," not "F(x)." |
| * | 339 | 19 | The "i" is missing from that 4th zero (should be 1/2 - 21.022040i). |
| 345 | 27 | "Since the primary terms..." I call them the "principal" terms everywhere else. | |
| 369 | 34 | At the time of writing, I did not know of Julian Havil's book Gamma: Exploring Euler's Constant, which came out at about the same time as Prime Obsession. This is an entire book devoted to the mysterious gamma. It is beautifully done and full of interesting math — though at a somewhat higher level than my own book. I recommend it to any reader who wants to learn why the number 0.577215664901532860606512... is so darned important. | |
| 384 | 23 | Titchmarsh's book was published in a revised (by Roger Heath-Brown) edition in 1986. | |
| 385 | 10 | Sir Michael Atiyah was playing an old violin here: the algebra=time / geometry=space notion goes back at least as far as Hamilton (i.e. the 1840s). | |
| 386 | 2-3 | "...'Ensemble' (this usage of the word, by the way, is due to Albert Einstein..." This is probably wrong. A physicist has pointed out to me that one of the chapter headings in Willard Gibbs's book Elementary Principles in Statistical Mechanics is: "On the motion of Systems and Ensembles of Systems Through Long Periods of Time." This book was published in 1902; that is, three years before Einstein came blazing on the scene with his three papers in the Annalen der Physik. Gibbs seems to be the most likely originator of this usage. However, if anyone can nail the origin down more precisely, I'd be grateful to hear about it. | |
| 391 | 1 | "...otherwise unknown to me." Sid Graham: "Eugene Ng got his Ph.D. at UCLA under the direction of Ron Miech, in either 1980 or 1981. ... Ng was on the faculty of the University of Texas at El Paso for a while, but left in 1991. I haven't heard anything from him since then... AMS membership listing gives his current [mid-2003] address as Alhambra, California." | |
| * | 397 | 12 | "...what can be said about the size of..." The expression immediately following should not have the "modulus brackets" (i.e. straight vertical lines) around it. Subsequent occurrences of the expression on this page should have the "modulus brackets" (and do). |
| * | 421 | 31 | Index entry for "Weierstrass, Karl": page number 152 should be included. |