Why Do Some Arguments Seem Wrong?
Or: how stupid am I?
Siegbert Tarrasch was a very strong player and teacher of chess in the late 19th century and into the early 20th century. He coined the term chess blindness—the failure of a chess player to make a normally obvious move or to see a normally obvious threat. He also stated the following rule, which is named for him:
Always put the rook behind the pawn Except when it is incorrect to do so.
Today I want to talk about a simple mathematical claim that I found to be hard to follow.
Is my trouble due to age? Or a version of chess blindness transferred to math? I am reminded of then-World Champion Vladimir Kramnik’s one-move blunder in 2006 against the chess program Deep Fritz 10 running on an ordinary quad-core PC.
Kramnik played 34…Qe3??? just as Ken was leaving to teach his intro graduate theory class, and when he arrived he put the match broadcast on the classroom Internet console just to be sure that this and 35.Qh7 checkmate had really happened. This still stands as the last time a human player has faced a computer in a match on even terms.
I have been working in the theory salt-mines for many decades now—too long perhaps. But I believe that I still can understand basic arguments, create new ones, and generally understand things. Recently I read an argument in a number theory text that was a “throw-away” easy remark. Just obvious. No proof supplied. None needed. Yet I was puzzled, the remark seemed wrong; it seemed too simple to be true.
The statement and it’s claimed “obviousness” made me think about all those statements in our papers, in our talks, and in our lectures, statements that may not be so obvious to all. How many times in the past did I claim something to another—in talking or in writing—that was not obvious? How often?
The statement concerns the Riemann Zeta function and one of its near cousins. Recall, since many of you may know it, that the zeta function is defined for complex with real part greater than to be the value of the series
This function can be extended to all complex values except for via technical means that are not needed here today. See this for more and a pretty picture. Of course the behavior of this function is well known to capture the global structure of the prime numbers, and this makes its behavior very “complex”—so difficult that the question of where its zeroes lie is one of the six outstanding Clay Millennium Prize Problems.
A cousin of the zeta function is a function that was used by Dirichlet to prove his famous arithmetic progression theorem. Let be a character (more on this in a moment), then define to be:
A character is an arithmetic function that is both multiplicative and periodic, but that is really not important. The only character that is needed is the following basic one: Define to be zero except when and are co-prime. For example, and .
One comment that has nothing to do with my confusion is this is a perfect example of math’s often inconsistent notation. Note that the zeta function is just , yet I believe I have never seen it written in this way. Oh well.
A natural question is how do we relate with the original zeta function? It seems plausible that they should be related—the question is just how. Here is the statement when is a prime:
That is it. Pretty simple relationship. No?
This is the statement that puzzled me. The one that I could not believe is true. It just seemed too simple. Is it clear to you? If so I am sorry for wasting your time, but for me it just seemed wrong. Let me try and unravel why I was troubled by it.
The equation really says this:
What confused me was that I worried, was there some double counting happening? This would be a problem.
The proof that the equation above is correct is embarrassingly simple. Here is the key insight: Define to be the set
and define to be the set
The key insight is that these sets are the same: . If is in , then has a common factor with the prime and so for some . This shows that is in . If is in , then and have common factor greater than . So is in .
The equation is:
Now we can rewrite the right hand side:
This follows since either or —pretty deep, color me something. Then the last two terms are just the same since is equal to :
The Blind Spot?
Ken thinks I was fooled because one sum used n to index Term(n) while the other used n to index Term(qn). He thinks it is similar to the fallacy
in the formal-languages part of a theory course where squaring involves concatenation, or where L is a set of numbers and it’s just multiplication. His advice to students is to rewrite any set definition of the form
where is a function or compound expression, into the “atomic set” form
This he says gives the reader an extra chance to realize in the case of that “…can be written as …” is wrong and “…can be written as where and ” is correct.
In my Riemann case, the advice becomes to rewrite the compound use of the index as into the form
Well here the involves non-standard notation, but this churns out the same clarification as my use of sets above.
Do you sometimes run into math statements like this? How often does this happen? Can we do a better job at explaining things?