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Number Theory by George E. Andrews, which is apparently accessible to liberal arts majors, contains this beautiful equation:

$$1+\sum_{n=1}^{\infty}{\frac{(b+a)(b+aq)...(b+aq^{n-1})z^n}{(1-q)(1-q^2)...(1-q^n)}}=\prod_{n=0}^{\infty}{\frac{1+azq^n}{1-bzq^n}}$$

while Analytic Number Theory by Apostol, which is apparently accessible to "sophisticated high school students," contains the following

$$\sum_{p\le x\text{, }p \equiv h (\text{mod k})}{\frac{\text{log }p}{p}}=\frac{1}{\phi(k)}\text{log }x+\frac{1}{\phi(k)}\sum^{\phi(k)}_{r=2}{\bar{\chi}_r(h)\sum^{ }_{p \le x}{\frac{\chi_r(p) \text{log } p}{p}}}+O(1)$$

Both books are fairly old. Have our standards for education changed? Are the authors crazy? Any thoughts would be appreciated :)

Dr. Momo
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  • there are very crazy books that contain the adjective “introductory” or “introduction to___” in the title. I’ve learnt to ignore that particular word. – peek-a-boo Jul 14 '23 at 16:31
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    Same remark with the word “clearly” in any math book :) Sometimes it’s actually clear, sometimes it’s not (and sometimes, it may seem clear initially but then you think about it and it’s actually not). – peek-a-boo Jul 14 '23 at 16:35
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  • These descriptions seem to be given by the authors themselves, and therefore should be taken with a grain of salt (or ignored). 2. The context of the quote from Apostol's book clarifies that "high school students" is mostly referring to course prerequisites, i.e. don't need to know calculus; if you understand chapters 1 to n, then in principle you are equipped to understand chapter n+1. This says nothing about the mathematical maturity needed to parse/understand the book though, which I guess is why he adds "sophisticated."
    • – angryavian Jul 14 '23 at 16:42
  • Who is making these claims for the books? (I was reading Apostol in high school, but I was far from a usual case, because I also took a functional analysis class in high school.) in any event, attacking claims about books without giving us the source of the claims seems not useful. – Thomas Andrews Jul 14 '23 at 16:50
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    Also, the prerequisites of books doesn't mean that, by the time you reach these results, the readers' sophistication will still be elementary. If you start any math book on page $200,$ you are likely to be mystified. The whole point of most math books is to take you from the simple to the more complex. – Thomas Andrews Jul 14 '23 at 16:53
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    @ThomasAndrews For the Andrews book, the quote is from the publisher's description of the book. For the Apostol book, the quote is from the preface by the author. – angryavian Jul 14 '23 at 17:09
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    "Have our standards for education changed?" I think so, but your question is very badly posed. To add to the previous comments, with which I fully agree: it's entirely different to have an exercise that asks to prove these formulas from scratch without help, and to have a long, guided problem to get to them. For instance in high school a teacher once gave me a problem that led to $\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}$. Each step was rather straightforward. You can't judge the difficulty of a book by a random formula taken from it. – Jean-Claude Arbaut Jul 14 '23 at 17:57