How many Pythagorean triples are there?

Abstract

In this chapter we determine an asymptotic formula for the number of primitive right triangles with bounded hypotenuse.

Appendices

Exercises

• 13.1 Prove Lemma 13.2.

• 13.2 Prove Corollary 13.6.

• 13.3 An arithmetic function is a function $f: \mathbb N\rightarrow \mathbb C$. For arithmetic functions f, g, we define the arithmetic function $f*g$ by

  $$ (f*g)(n) = \sum _{d \mid n} f(d) g(\frac{n}{d}). $$

  Show that for all arithmetic functions f, g, h we have the following properties:

  • a. $f*(g*h) = (f*g)*h$;

  • b. $f*g = g*f$;

  • c. If $e(n) = \delta _{n0}$, Kronecker’s delta, then $f*e = e*f = f$. Note that

    $$ e(n) = {\left\{ \begin{array}{ll} 1 &{} n =1; \\ 0 &{} n \ne 1. \end{array}\right. } $$

• 13.4 Prove the claim in Remark 13.4.

• 13.5 ($\maltese$) Investigate the error term in Lemma 13.1.

• 13.6 ($\maltese$) Numerically verify the assertion of Theorem 13.5 and Corollary 13.6. Investigate the error terms in these results.

• 13.7 Define a function $\mathbf {1}$ by $\mathbf {1}(n)=1$ for all n. Show that $\mathbf {1} * \mu = e$. Prove the Möbius Inversion Formula : If $f(n) = \sum _{d \mid n} g(d)$, then $g(n) = \sum _{d\mid n} \mu (d) f(\frac{n}{d})$.

• 13.8 Show that $\sum _{d\mid n} \varphi (d) = n$. Use this relation to derive a formula for the $\varphi$-function.

• 13.9 An arithmetic function f is called multiplicative if for every m, n with $\gcd (m, n) = 1$ we have $f(mn) = f(m)f(n)$. Show that if f, g are multiplicative, then so is $f*g$.

• 13.10 For a natural number n set $\sigma (n) = \sum _{d \mid n} d$. Find a formula for $\sigma (n)$ in terms of the prime factorization of n.

• 13.11 Show that for all $a, b \in \mathbb N$ with $a, b > 1$ we have

  $$ \frac{\sigma (a)}{a}< \frac{\sigma (ab)}{ab} < \frac{\sigma (a)\sigma (b)}{ab}. $$

• 13.12 Show that for $a, b >1$,

  $$ \sigma (ab) > 2 \sigma (a)^{1/2} \sigma (b)^{1/2}. $$

• 13.13 Show that for all $a, b \in \mathbb N$,

  $$ \sigma (a)\sigma (b) = \sum _{d \mid \gcd (a, b)} d \sigma \left( \frac{ab}{d^2}\right) . $$

  In particular, $\sigma$ is a multiplicative function.

• 13.14 Find an asymptotic formula for

  $$ \sum _{a, b \le X \atop \gcd (a, b) = 1} ab $$

  as $X \rightarrow \infty$.

• 13.15 Find an asymptotic formula for

  $$ \sum _{n \le X} \varphi (n) $$

  as $X \rightarrow \infty$.

• 13.16 Prove the following statement: Let $(c_n)_n$ be a sequence of complex numbers, and $f:[1, \infty ) \rightarrow \mathbb C$ a function with continuous derivative. Then

  $$ \sum _{n \le x} c_n f(n) = \left( \sum _{n \le x} c_n\right) f(x) - \int _1^x \left( \sum _{n \le t} c_n\right) f'(t)\, dt. $$

• 13.17 Show

  $$ \sum _{d \le x} \frac{1}{d} =\log x + O(1). $$

• 13.18 Recall the notion of average order from Definition 9.3.

  • a. Let d(n) be the number of divisors of n. Show that

    $$ \sum _{k \le n} d(k) = \sum _{k \le n} \left[ \frac{n}{k}\right] . $$

    Conclude that d(n) has average order $\log x$;

  • b. Let $\phi (n)$ be the Euler totient function. Show that the average order of $\phi (n)$ is $\zeta (2) x$;

  • c. Let $\omega (n)$ be the number of distinct prime divisors of n. Show that the average order of $\omega (n)$ is $\log \log x$.

• 13.19 Find a multiplicative function f such that

  $$ \sum _{d \mid n} \frac{\mu (d)d^2 f(n/d)}{\phi (d)} = \sigma (n) f(n), \quad n \in \mathbb N. $$

• 13.20 Use Parseval’s formula [41, Theorem 8.16] applied to the function $f(x)=x$ on the interval [0, 1] to give another proof for Euler’s identity, $\zeta (2) = \pi ^2/6$.

• 13.21 Pick two natural numbers at random. What is the probably that they are coprime?

• 13.22 Prove that for each natural number r,

  $$ B_r = - \sum _{k=0}^{r-1} \left( {\begin{array}{c}r\\ k\end{array}}\right) \frac{B_k}{r-k+1}. $$

  Use this relation to find the first few Bernoulli numbers.

• 13.23 Show that all Bernoulli numbers are rational.

• 13.24 Show that for each natural number r, $B_{2r+1} =0$.

• 13.25 Find an asymptotic formula for the number of primitive right triangles with perimeter bounded by X as $X \rightarrow \infty$.

Notes

1.1 Lehmer’s theorem and Manin’s conjecture

Lehmer [83] published a different proof of Corollary 13.6 in 1900. The argument we present here shows that any power saving improvement in the error term of Lemma 13.1 would improve the error terms in Theorem 13.5 and Corollary 13.6 to $O(\sqrt{B})$. The quantity considered in Corollary 13.6 appears in the Online Encyclopedia of Integer Sequences:

http://oeis.org/A156685

The question of counting integral solutions with bounded size to algebraic equations with infinitely many solutions is a very active area of research of current interest. Theorem 13.5 has now been greatly generalized. Yuri Manin has formulated several conjectures that connect the arithmetic features of some classes of equations where one expects a lot of solutions to the geometry of the resulting solution sets; see [104] for various questions and conjectures.

1.2 A family of number theorists

The Lehmer of Corollary 13.6 is Derrick Norman Lehmer (July 27, 1867–September 8, 1938). He was the father of Derrick Henry Lehmer (February 23, 1905–May 22, 1991) who was a mathematician credited with many contributions to number theory. D. H. Lehmer was married to Emma Markovna Lehmer (née Trotskaia) (November 6, 1906–May 7, 2007) who was a number theorist herself with over 50 publications to her name, [84]. There have been several other families of mathematicians in history, most notably the Bernoulli family. And here is a joke: What was the most influential mathematician family in history? Clearly Gauss’s family, because it doesn’t matter what the rest of his family did.

1.3 The Riemann zeta function

The complex function

$$ \zeta (s) = \sum _{n=1}^\infty \frac{1}{n^s} $$

is called the Riemann zeta function . This series converges absolutely for $\mathfrak {R}s > 1$. Riemann was certainly not the first person to study this function. In fact, by the time of the publication of Riemann’s work in 1859 various mathematicians, Euler in particular, had studied the values of the zeta function for integer values of s for at least two centuries; see [109] for a survey. The problem of computing $\zeta (2)$ which we discussed in this chapter was posed by Pietro Mengoli in 1650 and solved by Euler in 1735. Riemann, in a spectacular paper [93], proved the analytic continuation of the zeta function, proved the functional equation, discussed the connection to the distribution of prime numbers, and formulated a conjecture about prime numbers, nowadays known as the Riemann Hypothesis.

First a word about analytic continuation. Suppose we have a function f(s) which is holomorphic on an open subset U of complex numbers, and suppose V is an open set in $\mathbb C$ containing U. We call a function g, holomorphic on V, the analytic continuation of f if the restriction of g to U is equal to f. It is not terribly hard to show that for $\mathfrak {R}s > 1$ we have

$$ \zeta (s) = s \int _1^\infty \frac{[x]}{x^{s+1}} \, dx = \frac{s}{s-1} - s \int _1^\infty \frac{\{x\}}{x^{s+1}} \, dx. $$

The expression on the right-hand side is meromorphic on $\mathfrak {R}s > 0$ with a simple pole at $s=1$, however, and this provides an analytic continuation for $\zeta (s)$ to a larger domain. But this is not where the analytic continuation stops. In fact, if we set

$$ \xi (s) = s(s-1)\pi ^{-s/2} \Gamma \left( \frac{s}{2}\right) \zeta (s), $$

then Riemann showed that $\xi (s)$ is holomorphic on $\mathfrak {R}s > 0$ and

$$\begin{aligned} \xi (1-s) = \xi (s). \end{aligned}$$

(13.5)

Since $\xi (s)$ is holomorphic for $\mathfrak {R}s > 0$, and $\xi (1-s)$ is holomorphic for $\mathfrak {R}(1-s) >0$, i.e., $\mathfrak {R}s < 1$, we obtain the holomorphy of $\xi (s)$ on the entire set of complex numbers. This further shows that $\zeta (s)$ has an analytic continuation to the entire complex plane to a meromorphic function with a unique simple pole at $s=1$ with residue 1. Since we already have computed the value of $\zeta (s)$ for even positive integers 2k, we can use the functional equation (13.5) to compute the values of the analytic continuation of $\zeta (s)$ for odd negative numbers. In fact, for $n \in \mathbb N$,

$$ \zeta (1-2n) = - \frac{B_{2n}}{2n}. $$

For example, $\zeta (-1) = -1/12$. One can similarly compute the value of $\zeta (0)$ to be $-1/2$. Again, we should emphasize that these are the values of the analytically continued function, and they should not be taken to mean

$$ 1+ 1 + 1 + \dots = -\frac{1}{2}, $$

or

$$ 1+2 + 3 + \dots = - \frac{1}{12}. $$

Let us illustrate what is happening here with an easy example. Suppose $U = \{s \in \mathbb C\mid |s| < 1\}$ and $f(s) = \sum _{k=0}^\infty s^k$. The series defining f is absolutely convergent on U and defines a holomorphic function there. By general properties of geometric series, for $|s|<1$, we have

$$ f(s) = \frac{1}{1-s}. $$

The function $g(s)=1/(1-s)$ is holomorphic on the much larger domain $V = \{ s \in \mathbb C\mid s \ne 1\}$. Note that outside the open set U the function g(s) is not given by the original series defining f(s). This important point is the source of many paradoxes in the theory of infinite series. For example, the value of the function g(s) at $s=2$ is equal to $-1$. If we set $s=2$ in the formula for f(s) we formally get

$$ 1 + 2 + 4 + 8 + 16 + 32 + 64 + \dots $$

Does this then mean

$$ 1 + 2 + 4 + 8 + 16 + 32 + 64 + \dots = -1? $$

Absolutely not! In fact the series defining f(s) is not even defined for $s=2$.

We now turn to the connections between the zeta function and the distribution of prime numbers. Euler observed the product formula that now bears his name: For $\mathfrak {R}s > 1$ we have

$$ \zeta (s) = \prod _{p \text { prime}} \frac{1}{1-p^{-s}}. $$

If we use this formula to compute $(d/ds) \log \zeta (s)$ we obtain

$$\begin{aligned} - \frac{\zeta '(s)}{\zeta (s)} = \sum _{k\ge 1}\sum _{p \text { prime}} \frac{\log p}{p^{ks}} = \sum _{n=1}^\infty \frac{\varLambda (n)}{n^s}, \end{aligned}$$

(13.6)

with $\varLambda (n)$ being the von Mangoldt function defined by

$$ \varLambda (n) = {\left\{ \begin{array}{ll} \log p &{} n = p^k, p \text { prime}; \\ 0 &{} \text { otherwise}. \end{array}\right. } $$

An idea that Riemann brought into this subject was contour integration. For a complex function f(s) and a real number c let us define

$$ \int _{(c)} f(s)\, ds = \lim _{R \rightarrow \infty } \int _{c - i R}^{c+iR} f(s) \, ds. $$

Fix a real number $c>1$. A contour integration computation shows that for $x>1$, non-integer,

$$ \sum _{n < x} \varLambda (n) =\frac{1}{2\pi i} \int _{(c)} \left( -\frac{\zeta '(s)}{\zeta (s)}\right) \frac{x^s}{s} \, ds. $$

The function $-\zeta '(s)/\zeta (s)$ has a simple pole at $s=1$ with residue 1. Suppose we can shift the contour back to $(c')$, for a number $c'<1$. Then we would obtain

$$\begin{aligned} \sum _{n < x} \varLambda (n) =x + \frac{1}{2\pi i} \int _{(c')} \left( -\frac{\zeta '(s)}{\zeta (s)}\right) \frac{x^s}{s} \, ds. \end{aligned}$$

(13.7)

Riemann’s idea then was to prove that this last integral contributes less than x to the formula, and hence obtain

$$\begin{aligned} \sum _{n < x} \varLambda (n) \sim x, \quad x \rightarrow \infty . \end{aligned}$$

(13.8)

Exercise 13.16 can now be used to prove

$$\begin{aligned} \#\{p \le x \} \sim \frac{x}{\log x} \end{aligned}$$

(13.9)

which is the celebrated Prime Number Theorem, conjectured by Gauss. Also, knowing the specific value of $c'$ would lead to error estimates for the Prime Number Theorem. So, the question that Riemann was faced with was to determine how far back the contour could be moved. In general, the logarithmic derivative of a meromorphic function has poles whenever the function has poles or zeros. In particular in order to know the poles of $\zeta '(s)/\zeta (s)$ we need to know where the function $\zeta (s)$ is zero. Riemann computed several zeros of the zeta function in the domain $\mathfrak {R}s >0$ and observed that they are all on the line $\mathfrak {R}s =1/2$, and conjectured that this would be the case for all zeros. If one assumes the Riemann Hypothesis, then it follows that

$$ \#\{p \le x\} = \mathrm {Li} \, x + O(x^{1/2+\varepsilon }) $$

for all $\varepsilon > 0$, with

$$ \mathrm {Li}\, x = \int _2^x \frac{dt}{\log t}. $$

At present, the Riemann Hypothesis appears out of reach.

Titchmarsh’s classic [52] is a much recommended, comprehensive introduction to the theory of the Riemann zeta function.