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During the academic year 1980-1981 I was teaching at the Technion-the Israeli Institute of Technology-in Haifa. The audience was small, but con­ sisted of particularly gifted and eager listeners; unfortunately, their back­ ground varied widely. What could one offer such an audience, so as to do justice to all of them? I decided to discuss representations of natural integers as sums of squares, starting on the most elementary level, but with the inten­ tion of pushing ahead as far as possible in some of the different directions that offered themselves (quadratic forms, theory of genera, generalizations and modern developments, etc.), according to the interests of the audience. A few weeks after the start of the academic year I received a letter from Professor Gian-Carlo Rota, with the suggestion that I submit a manuscript for the Encyclopedia of Mathematical Sciences under his editorship. I answered that I did not have a ready manuscript to offer, but that I could use my notes on representations of integers by sums of squares as the basis for one. Indeed, about that time I had already started thinking about the possibility of such a book and had, in fact, quite precise ideas about the kind of book I wanted it to be.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
What do the relations
$$ {5^2} = {3^2} + {4^2} $$
(i)
$$ 6 = {1^2} + {1^2} + {1^2} + {1^2} + {1^2} + {1^2} = {2^2} + {1^2} + {1^2} $$
(ii)
$$ 7 \ne {a^2} + {b^2} + {c^2} $$
(iii)
have in common? Obviously, their right hand members are all sums of squares. One way to describe those relations is as follows:
i
The square 52 can be represented, in essentially one way only, as the sum of two squares,
 
ii
The integer 6 can be represented in (at least) two essentially distinct ways as a sum of squares.
 
iii
The integer 7 cannot be represented as a sum of three squares.
 
Emil Grosswald

Chapter 1. Preliminaries

Abstract
In this book, when we speak of a quadratic form,* we mean a rational, integral quadratic form, unless the contrary is stated explicitly. Given a quadratic form Q, let N Q be the set of values of Q where x ∈ ℤ k (i.e., x i ℤ for i = 1, 2, ..., k); clearly, N Q ⊂ ℤ. If \( Q = \sum\nolimits_{i = 1}^k {x_i^2} \), we denote N Q by N k . The main problems that we shall study can now be formulated as follows:
(a)
Given a quadratic form Q, determine N Q .
 
(b)
Given Q and n ∈ N Q , determine the number of representations of n by Q, i.e., the number of vectors x ∈ ℤk for which Q(x) = n.
 
Emil Grosswald

Chapter 2. Sums of Two Squares

Abstract
Logically, one should start with k = 1, i.e., the representation of a natural integer by a single (rational integral) square. In this case, however, both representation problems (i.e., problems (a), and (b) of Chapter 1) are trivial. For completeness only, we state here the result, without formal proof or further comments.
Emil Grosswald

Chapter 3. Triangular Numbers and the Representation of Integers as Sums of Four Squares

Abstract
After the discussion of representations as sums of two squares, it appears reasonable to consider next the case of representations as sums of three squares. However, as already indicated, the case of k odd is more difficult than that of k even and, what is more important, requires different methods.
Emil Grosswald

Chapter 4. Representations as Sums of Three Squares

Abstract
As already remarked in Chapter 3, it is easy to verify the “only if” part of Legendre’s Theorem 3.1, which we recall here in its complete form.
Emil Grosswald

Chapter 5. Legendre’s Theorem

Abstract
In this chapter we consider a diagonal form more general than a simple sum of squares. We shall be concerned with the ternary quadratic forms Q(x,y,z) = ax2 + by2 + cz2. If a, b, c are positive integers, then strictly speaking this form is also a sum of squares, because it can be written as
$$ \underbrace{{{x^2} + {x^2} + \cdots + {x^2}}}_{{a\;{\text{time}}}} + \underbrace{{{y^2} + {y^2} + \cdots + {y^2}}}_{{b\;{\text{time}}}} + \underbrace{{{z^2} + {z^2} + \cdots + {z^2}}}_{{c\;{\text{time}}}} $$
but the number of squares varies with the values of the coefficients a, b, c. In fact, however, Q is an arbitrary diagonal form, with a, b, c integers, but not necessarily positive.
Emil Grosswald

Chapter 6. Representations of Integers as Sums of Nonvanishing Squares

Abstract
It is clear that, for k ≥ 4, every nonnegative integer is representable as a sum of k squares. Indeed, one can always write \( n = \sum\nolimits_{{i = 1}}^4 {x_i^2} + {0^2} + \cdot \cdot \cdot + {0^2} \), with an arbitrary number of zeros. On the other hand, the number of representations r k (n) increases very rapidly with n (see Chapter 12), and besides the representations with k — 4 zeros, one usually finds others with fewer zeros or none at all. For example, if k = 5, then 5 = 22 + 12 + 02 + 02 + 02 = 12 + 12 + 12 + 12 + 12.
Emil Grosswald

Chapter 7. The Problem of the Uniqueness of Essentially Distinct Representations

Abstract
In 1948 D. H. Lehmer raised an interesting question [156] and answered it almost completely. The remaining gap has only been filled partially during the more than 30 years that have since passed [20].
Emil Grosswald

Chapter 8. Theta Functions

Abstract
We recall from Chapter 1 that the determination of r k (n) can be reduced to that of the coefficient a n (k) in the Taylor series expansion of the function \( {\left( {\sum\nolimits_{{m = - \infty }}^{\infty } {{x^{{{m^2}}}}} } \right)^k} = {\left( {1 + 2\sum\nolimits_{{m = 1}}^{\infty } {{x^{{{m^2}}}}} } \right)^k} \), because this series, denoted traditionally by |θ3(x)} k , equals \( \sum\nolimits_{{ - \infty < {m_i} < \infty }} {{x^{{m_1^2 + m_2^2 + \cdot \cdot \cdot + m_k^2}}}} = 1 + \sum\nolimits_{{n = 1}}^{\infty } {{r_k}} (n){x^n} \). It follows that if \( {\left\{ {{\theta_3}(x)} \right\}^k} = \sum\nolimits_{{n = 0}}^{\infty } {a_n^{{(k)}}{x^n}} \), then a 0 (k) = 1 for all k, and for n ≥ 1, r k (n) = a n (k) .
Emil Grosswald

Chapter 9. Representations of Integers as Sums of an Even Number of Squares

Abstract
In the previous chapter, we have presented those fundamental properties of the theta functions which will be used in the present one to obtain the number r k (n) of representations of a natural number n as a sum of an even number k of squares.
Emil Grosswald

Chapter 10. Various Results on Representations as Sums of Squares

Abstract
In [54] Dickson lists about 70 mathematicians who have made contributions to the problem of representations of integers by sums of five or more squares and to that of relations between the numbers of those representations. It is not possible to quote all these results, but in order to convey the flavor of some of them, we shall mention a few without proofs.
Emil Grosswald

Chapter 11. Preliminaries to the Circle Method and the Method of Modular Functions

Abstract
We have presented in Chapters 2, 3, and 4 the contributions made by mathematicians, from antiquity until well into the 19th century, to the problem of representations of natural integers by sums of k = 2, 4, and 3 squares. Theorems were obtained that presented the number r k (n) of these representations as sums of divisor functions (for k = 2 and 4) or sums over Jacobi symbols (for k = 3). Next, in Chapters 8 and 9, we studied a method developed mainly by Jacobi, based on the use of theta functions, by which one obtains similar results, involving divisor functions, for an even number of squares. The method is entirely successful for k ≤ 8, while for k ≥ 10 the formulae contain, besides sums of divisor functions, also more complicated additional terms. Even for k ≤ 8, the cases k = 5 and k = 7 were bypassed. Nevertheless, as mentioned earlier (see Chapter 10), Eisenstein presented formulae depending (as in the case k = 3) on Jacobi symbols for r5(n) and formulae for r5(n) of a different type were proposed and proved by Stieltjes (1856–1894) (see [259], [260]), and by Hurwitz [118]. Eisenstein also stated formulae for r7(n), expressed, like those for r5(n) with the help of the Legendre-Jacobi symbol. Proofs of Eisenstein’s formulae were given by Smith [253, 254], who deduced them from his own, of a different appearance, and by Minkowski (1864–1909) (see [178]).
Emil Grosswald

Chapter 12. The Circle Method

Abstract
We mentioned in Chapter 1 that the number r s (n) of solutions of the Diophantine equation
$$ \sum\limits_{{k = 1}}^s {x_i^2} = n $$
(12.1)
is the coefficient of x n in the Taylor expansion of the function \( 1 + \sum\nolimits_{{n = 1}}^{\infty } {{r_s}(n){x^n}} \). Here, as in Chapter 8, we write θ(x) for θ3(1;x) and we shall suppress the first entry, which will always be z = 1. From (12.1); it follows, by Cauchy’s theorem, that
$$ {r_s}(n) = \frac{1}{{2\pi i}}\int\limits_c {{x^{{ - n - 1}}}{\theta^s}(x)dx,} $$
(12.2)
where, we recall,
$$ \theta (x) = \sum\limits_{{ - \infty }}^{\infty } {{x^{{{n^2}}}}} = 1 + 2\sum\limits_{{n = 1}}^{\infty } {{x^{{{n^2}}}}} = \sum\limits_{{k = 0}}^{\infty } {{a_k}{x^k},\quad say,} $$
(12.3)
and 𝓒 is a sufficiently small circle around the origin.
Emil Grosswald

Chapter 13. Alternative Methods for Evaluating r s (n)

Abstract
Theorem 12.1 is formulated in [72] as follows:
Theorem 1.
For integers h, k with (h, k) = 1, let \( \lambda \left( {h/k} \right) = {\left( {2k} \right)^{{ - 1}}}\sum\nolimits_{{q = 1}}^{{2k}} {{e^{{2\pi ih{q^{2}}/2k}}}} \) and set \( {A_{k}} = \sum\nolimits_{{1 \le h \le 2k,\left( {h,k} \right) = 1}} {{\lambda ^{s}}{e^{{ - 2\pi ihn/2k}}}} \). Then, if \( S\left( n \right) = \sum\nolimits_{{k = 1}}^{\infty } {{A_{k}}} \) and s = 5, 6, 7, or 8, one has for some constant c = c(s), independent of n, that
$${r_{s}}\left( n \right) = c\left( s \right){n^{{\left( {s/2} \right) - 1}}}S\left( n \right) $$
.
 
Emil Grosswald

Chapter 14. Recent Work

Abstract
In the preceding chapters we have discussed the representation of natural integers as sums of squares of integers, and only occasionally (e.g., in Chapters 4 and 5) did particular cases of representations by more general quadratic forms occur.
Emil Grosswald

Backmatter

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