Page:EB1911 - Volume 19.djvu/892

so that 𝑘²＋𝑘′²＝1, we shall have

${\theta _{11}(v) \over \theta _{01}(v)}={\sqrt {\kappa }}\cdot {\mbox{sn}}\ u,\ \ {\theta _{10}(v) \over \theta _{01}(v)}={\sqrt {\kappa \over \kappa '}}\cdot {\mbox{cn}}\ u,\;{\theta _{00}(v) \over \theta _{01}(v)}={1 \over {\sqrt {\kappa '}}}\cdot {\mbox{dn}}\ u$

and, supposing for simplicity that 𝑖𝜔 is a real negative quantity,

𝜋𝜃₀₀²＝2𝖪,    𝜔𝜋𝜃₀₀²＝2𝑖𝖪′,    𝜔＝𝑖𝖪′/𝖪,

the notation being that which is now usual for the elliptic functions. It is found that

${\begin{matrix}{\kappa {\mbox{K}} \over \pi }{\mbox{sn}}\;2{\mbox{K}}u&=&2\sum _{1}^{\infty }{q^{s-{1 \over 2}} \over 1-q^{2s-1}}\sin \;(2s-1)\;\pi u,\\{\kappa {\mbox{K}} \over \pi }{\mbox{cn}}\;2{\mbox{K}}u&=&2\sum _{1}^{\infty }{q^{s-{1 \over 2}} \over 1-q^{2s-1}}\cos \;(2s-1)\;\pi u,\\{{\mbox{K}} \over \pi }{\mbox{dn}}\;2{\mbox{K}}u&=&{1 \over 2}+2\sum _{1}^{\infty }{q^{s} \over 1+q^{2s}}\cos \;2s\pi u.\end{matrix}}$

From the last formula, by putting 𝑢＝0, we obtain

$1+4\sum _{1}^{\infty }{q^{s} \over 1+q^{2s}}={2{\mbox{K}} \over \pi }=\theta _{00}{}^{2}(1+2q+2q^{4}+2q^{9}+...)^{2}$,

and hence, by expanding both sides in ascending powers of 𝑞, and equating the coefficients of 𝑞^𝑛, we arrive at a formula for the number of ways of expressing 𝑛 as the sum of two squares. If 𝛿 is any odd divisor of 𝑛, including 1 and 𝑛 itself if 𝑛 is odd, we find as the coefficient of 𝑞^𝑛 in the expansion of the left-hand side 4𝚺(－1)^{⁠1/2⁠(𝛿－1)}; on the right-hand side the coefficient enumerates all the solutions 𝑛＝(±𝑥)²＋(±𝑦)², taking account of the different signs (except for 0²) and of the order in which the terms are written (except when 𝑥²＝𝑦²). Thus if 𝑛 is an odd prime of the form 4𝑘＋1, 𝚺(－1)^{⁠1/2⁠(𝛿－1)}＝2, and the coefficient of 𝑞^𝑛 is 8, which is right, because the one possible composition 𝑛＝𝑎²＋𝑏² may be written 𝑛＝(±𝑎)²＋(±𝑏)²＝(±𝑏)²＋(±𝑎)², giving eight representations.

By methods of a similar character formulae can be found for the number of representations of a number as the sum of 4, 6, 8 squares respectively. The four-square theorem has been stated in § 41; the eight-square theorem is that the number of representations of a number as the sum of eight squares is sixteen times the sum of the cubes of its factors, if the given number is odd, while for an even number it is sixteen times the excess of the cubes of the even factors above the cubes of the odd factors. The five-square and seven-square theorems have not been derived from 𝑞-series, but from the general theory of quadratic forms.

68. Still more remarkable results are deducible from the theory of the transformation of the theta functions. The elementary formulae are

${\begin{matrix}\theta _{11}(u,\;\omega +1)=e^{\pi i/4}\theta _{11}(u,\;\omega ),&\theta _{10}(u,\;\omega +1)=e^{\pi i/4}\theta _{10}(u,\;\omega )\\\theta _{01}(u,\;\omega +1)=\theta _{00}(u,\;\omega ),&\theta _{00}(u,\;\omega +1)=\theta _{01}(u,\;\omega )\end{matrix}}$

${\begin{matrix}e^{-\pi iu^{2}/\omega }\theta _{11}\left({u \over \omega },-{1 \over \omega }\right)&=&-i{\sqrt {-i\omega }}\theta _{11}(u,\;\omega ),\\e^{-\pi iu^{2}/\omega }\theta _{10}\left({u \over \omega },-{1 \over \omega }\right)&=&{\sqrt {-i\omega }}\theta _{10}(u,\;\omega ),\\e^{-\pi iu^{2}/\omega }\theta _{10}\left({u \over \omega },-{1 \over \omega }\right)&=&-i{\sqrt {-i\omega }}\theta _{01}(u,\;\omega ),\\e^{-\pi iu^{2}/\omega }\theta _{00}\left({u \over \omega },-{1 \over \omega }\right)&=&-i{\sqrt {-i\omega }}\theta _{00}(u,\;\omega ),\end{matrix}}$

where √－𝑖𝜔 is to be taken in such a way that its real part is positive. Taking the definition of 𝜅 given in § 67, and considering 𝜅 as a function of 𝜔, we find

𝜅(𝜔＋1)＝𝑖𝜃₁₀²/𝜃₀₁²＝𝑖𝜅(𝜔)/𝜅′(𝜔)

$\kappa \left(-{1 \over \omega }\right)=\theta _{01}{}^{2}/\theta _{00}{}^{2}=\kappa '(\omega )$.

For convenience let 𝜅²(𝜔)＝𝜎: then the substitutions (𝜔,𝜔＋1) and (𝜔,－𝜔^－1) convert 𝜎 into 𝜎/(𝜎－1) and (1－𝜎) respectively. Now if 𝛼, 𝛽, 𝛾, 𝛿 are any real integers such that 𝛼𝛿－𝛽𝛾＝1, the substitution [𝜔,(𝛼𝜔＋𝛽)/(𝛾𝜔＋𝛿)] can be compounded of (𝜔, 𝜔＋1) and (𝜔,－𝜔^－1); the effect on 𝜎 will be the same as if we apply a corresponding substitution compounded of [𝜎, 𝜎/(𝜎－1)] and [𝜎, 1－𝜎]. But these are periodic and of order 3, 2 respectively; therefore we cannot get more than six values of 𝜎, namely

𝜎, 1－𝜎, ⁠𝜎/𝜎－1⁠, ⁠1/1－𝜎⁠, ⁠𝜎－1/𝜎⁠, ⁠1/𝜎⁠,

and any symmetrical function of these will have the same value at any two equivalent places in the modular dissection (§ 33). Their sum is constant, but the sum of their squares may be put into the form

⁠2(𝜎²－𝜎＋1)³/𝜎²(𝜎－1)²⁠－3;

hence (𝜎²－𝜎＋1)³÷𝜎²(𝜎－1)² has the same value at equivalent places. F. Klein writes

𝖩＝⁠4(𝜎²－𝜎＋1)³/27𝜎²(𝜎－1)²⁠;

this is a transcendental function of 𝜔, which is a special case of a Fuchsian or automorphic function. It is an analytical function of 𝑞², and may be expanded in the form

𝖩＝⁠1/1728⁠{𝑞^－2＋744＋𝑐₁𝑞²＋𝑐₂𝑞⁴＋ . . . }

where 𝑐₁, 𝑐₂, &c., are rational integers.

69. Suppose, now, that 𝑎, 𝑏, 𝑐, 𝑑 are rational integers, such that dv(𝑎, 𝑏, 𝑐, 𝑑)＝1 and 𝑎𝑑－𝑏𝑐＝𝑛, a positive integer. Let (𝑎𝜔＋𝑏)/(𝑐𝜔＋𝑑)＝𝜔′; then the equation 𝖩(𝜔′)＝𝖩(𝜔) is satisfied if and only if 𝜔′∼ 𝜔, that is, if there are integers 𝛼, 𝛽, 𝛾, 𝛿 such that 𝛼𝛿－𝛽𝛾＝1, and

(𝑎𝜔＋𝑏)(𝛾𝜔＋𝛿)－(𝑐𝜔＋𝑑)(𝛼𝜔＋𝛽)＝0.

If we write 𝜓(𝑛)＝𝑛𝚷(1＋𝑝^－1), where the product extends to all prime factors (𝑝) of 𝑛, it is found that the values of 𝜔 fall into 𝜓(𝑛) equivalent sets, so that when 𝜔 is given there are not more than 𝜓(𝑛) different values of 𝖩(𝜔′). Putting 𝖩(𝜔′)＝𝖩′, 𝖩(𝜔)＝𝖩 we have a modular equation

𝑓₁(𝖩′, 𝖩)＝0

symmetrical in 𝖩, 𝖩′, with integral coefficients and of degree 𝜓(𝑛). Similarly when dv(𝑎, 𝑏, 𝑐, 𝑑)＝𝜏 we have an equation 𝑓_𝜏(𝖩′, 𝖩)＝0 of order 𝜓(𝑛/𝜏²); hence the complete modular equation for transformations of the 𝑛th order is

𝖥(𝖩′,𝖩)＝𝚷𝑓_𝜏(𝖩′, 𝖩)＝0,

the degree of which is 𝚽(𝑛), the sum of the divisors of 𝑛.

Now if in 𝖥(𝖩′, 𝖩) we put 𝖩′＝𝖩, the result is a polynomial in 𝖩 alone, which we may call 𝖦(𝖩). To every linear factor of 𝖦 corresponds a class of quadratic forms of determinant (𝜅²－4𝑛) where 𝜅²＜4𝑛 and 𝜅 is an integer or zero: conversely from every such form we can derive a linear factor (𝖩－𝛼) of 𝖦. Moreover, if with each form we associate its weight (§ 41) we find that with the notation of § 39 the degree of 𝖦 is precisely 𝚺𝖧(4𝑛－𝜅²)－𝜖_𝑛, where 𝜖_𝑛＝1 when 𝑛 is a square, and is zero in other cases. But this degree may be found in another way as follows. A complete representative set of transformations of order 𝑛 is given by 𝜔′＝(𝑎𝜔＋𝑏)/𝑑, with 𝑎𝑑＝𝑛, 0⩽𝑏＜𝑑; hence

$G(J)=\Pi \left\{J(\omega )-J\left({a\omega +b \over d}\right)\right\}$

and by substituting for 𝖩(𝜔) and $J\left({a\omega +b \over d}\right)$ their values in terms of 𝑞, we find that the lowest term in the factor expressed above is either 𝑞^－2/1728 or 𝑞^{－2𝑎/𝑑}/1728, or a constant, according as 𝑎＜𝑑, 𝑎＞𝑑 or 𝑎＝𝑑. Hence if 𝜈 is the order of 𝖦(𝖩), so that its expansion in 𝑞 begins with a term in 𝑞^－2𝜈 we must have

$\nu =\sum _{d>{\sqrt {n}}}(1\cdot d)+\sum _{d>{\sqrt {n}}}\left({a \over d}\cdot d\right)=\sum _{d>{\sqrt {n}}}d+\sum _{a>{\sqrt {n}}}a$

extending to all divisors of 𝑛 which exceed √𝑛. Comparing this with the other value, we have

$\sum _{\kappa }{\mbox{H}}(4n-\kappa ^{2})=2\sum _{d>{\sqrt {n}}}d+\epsilon _{n}=\Phi (n)+\Psi (n)$,

as stated in § 39.

70. Each of the singular moduli which are the roots of 𝖦(𝖩)＝0 corresponds to exactly one primitive class of definite quadratic forms, and conversely.

Corresponding to every given negative determinant －𝚫 there is an irreducible equation 𝜓(𝑗)＝0, where 𝑗＝1728𝖩, the coefficients of which are rational integers, and the degree of which is ℎ(－𝚫). The coefficient of the highest power of 𝑗 is unity, so that 𝑗 is an arithmetical integer, and its conjugate values belong one to each primitive class of determinant －𝚫. By adjoining the square roots of the prime factors of 𝚫 the function 𝜓(𝑗) may be resolved into the product of as many factors as there are genera of primitive classes, and the degree of each factor is equal to the number of classes in each genus. In particular, if {1, 1, ⁠1/4⁠(𝚫＋1)} is the only reduced form for the determinant －𝚫, the value of 𝑗 is a real negative rational cube. At the same time its approximate value is $\exp \left[-2\pi i\cdot {1+i{\sqrt {\Delta }} \over 2}\right]+744=744-e^{\pi {\sqrt {\Delta }}}$, so that, approximately, 𝑒^𝜋√𝚫＝𝑚³＋744 where 𝑚 is a rational integer. For instance 𝑒^𝜋√43＝884736743.9997775 . . . ＝ 960³＋744 very nearly, and for the class (1, 1, 11) the exact value of 𝑗 is －960³. Four and only four other similar determinants are known to exist, namely －11, －19, －67, －163, although thousands have been classified. According to Hermite the decimal part of 𝑒^𝜋√163 begins with twelve nines; in this case Weber has shown that the exact value of 𝑗 is －2¹⁸⋅3³⋅5³⋅23³⋅29³.

71. The function 𝑗(𝜔) is the most fundamental of a set of quantities called class-invariants. Let (𝑎, 𝑏, 𝑐) be the representative of any class of definite quadratic forms, and let 𝜔 be the root of 𝑎𝑥²＋𝑏𝑥＋𝑐＝0 which has a positive imaginary part; then 𝖥 (𝜔) is said to be a class-invariant for (𝑎, 𝑏, 𝑐) if $F\left({\alpha \omega +\beta \over \gamma \omega +\delta }\right)=F(\omega )$ for all real integers 𝛼, 𝛽, 𝛾, 𝛿 such that 𝛼𝛿－𝛽𝛾＝1. This is true for 𝑗(𝜔) whatever 𝜔 may be, and it is for this reason that 𝑗 is so fundamental. But, as will be seen from the above examples, the value of 𝑗 soon becomes so large that its calculation is impracticable. Moreover, there is the difficulty of constructing the modular equation 𝑓₁(𝖩, 𝖩′)＝0 (§ 69), which