Jacobi's theorem (number theory)

The set of Jacobi (according to C. Jacobi ) is a statement from the additive theory of numbers on the number of representations of a natural number as the sum of four squares.

Jacobi's theorem is used, among other things, in geometric number theory z. B. when determining the number of grid points in a -dimensional sphere. ${\displaystyle n}$

sentence

For every natural number let through ${\displaystyle n}$ ${\displaystyle r_{4}(n)}$

${\displaystyle r_{4}(n)=\#\{(n_{1},n_{2},n_{3},n_{4}):n_{1},n_{2},n_{3},n_{4}\in \mathbb {Z} ,\ n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+n_{4}^{2}=n\}}$

Are defined. Then

${\displaystyle r_{4}(n)={\begin{cases}8\sigma (n),&4\nmid n\\8\sigma (n)-32\sigma \left({\frac {n}{4}}\right),&4\mid n\end{cases}}\ ,}$

where is the divisor function (i.e. the sum of all the divisors of n including n itself). ${\displaystyle \sigma (n)}$

This can also be expressed:

${\displaystyle r_{4}(n)={\begin{cases}8\sum \limits _{m|n}m&{\text{für }}n{\text{ ungerade}}\\[12pt]24\sum \limits _{\begin{smallmatrix}m|n\\m{\text{ ungerade}}\end{smallmatrix}}m&{\text{für }}n{\text{ gerade}}.\end{cases}}}$

or:

${\displaystyle r_{4}(n)=8\sum _{d\mid n,4\nmid d}d}$

(Sum of the divisors of n that are not divisible by 4)

Jacobi found this theorem with the help of the theta functions about identity that he introduced in his theory of elliptic functions :

${\displaystyle {\left(\sum _{n=-\infty }^{\infty }q^{n^{2}}\right)}^{4}=\sum _{a,b,c,d\in \mathbb {Z} }q^{a^{2}+b^{2}+c^{2}+d^{2}}=1+8\sum _{m=1}^{\infty }\left(\sum _{f\mid m,4\nmid f}f\right)q^{m}}$

with , . The theta functions on the left and the Eisenstein row on the right are both modular forms (for the congruence subgroup and weight k = 2). ${\displaystyle q=e^{2\pi iz}}$ ${\displaystyle z\in \mathbb {H} =\{z\in \mathbb {C} \mid \Im (z)>0\}}$ ${\displaystyle \Gamma _{1}(4)}$

example

For results from Jacobi's theorem ${\displaystyle n=2}$

${\displaystyle r_{4}(2)=8\sigma (2)=8\cdot (1+2)=24.}$

It is with the help of the multinomial coefficient to calculate the number of permutations of the tuple or : For there are permutations, for are there and there are permutations, for a total possible tuples. ${\displaystyle 2=0^{2}+0^{2}+1^{2}+1^{2}=0^{2}+0^{2}+1^{2}+(-1)^{2}=0^{2}+0^{2}+(-1)^{2}+(-1)^{2}.}$ ${\displaystyle (0,0,1,1),(0,0,1,-1)}$ ${\displaystyle (0,0,-1,-1)}$ ${\displaystyle (0,0,1,1)}$ ${\displaystyle {\frac {4!}{2!\cdot 2!}}=6}$ ${\displaystyle (0,0,1,-1)}$ ${\displaystyle {\frac {4!}{2!\cdot 1!\cdot 1!}}=12}$ ${\displaystyle (0,0,-1,-1)}$ ${\displaystyle {\frac {4!}{2!\cdot 2!}}=6}$ ${\displaystyle 6+12+6=24}$

See also

• Four squares set
• Sum of squares function

Web links

• Eric W. Weisstein : Sum of Squares Function . In: MathWorld (English).

References and comments

1. ↑ ^{a b} E. Krätzel: Number theory . VEB Deutscher Verlag der Wissenschaften, Berlin 1981, ISBN 978-3-8171-1287-6 , 6.2, 6.6
2. ↑ H. Siemon: Introduction to Number Theory . Publishing house Dr. Kovac, Hamburg 2002, ISBN 978-3-8300-0674-9 , 5.5
3. ↑ For example Ila Varma: Sums of Squares, Modular Forms and Hecke Characters . ( Memento of the original from September 9, 2016 in the Internet Archive ; PDF) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. Master thesis, University of Leiden 2010, p. 38