Editing Hurwitz quaternion (section)

==The lattice of Hurwitz quaternions==

The [[field norm|(arithmetic, or field) norm]] of a Hurwitz quaternion {{nowrap|''a'' + ''bi'' + ''cj'' + ''dk''}}, given by {{nowrap|''a''{{sup|2}} + ''b''{{sup|2}} + ''c''{{sup|2}} + ''d''{{sup|2}}}}, is always an integer. By a [[Lagrange's four-square theorem|theorem of Lagrange]] every nonnegative integer can be written as a sum of at most four [[square (algebra)|squares]]. Thus, every nonnegative integer is the norm of some Lipschitz (or Hurwitz) quaternion. More precisely, 
the number ''c''(''n'') of Hurwitz quaternions of given positive norm ''n'' is 24 times the sum of the odd [[divisor]]s of ''n''. The generating function of the numbers ''c''(''n'') is given by the level 2 weight 2 [[modular form]]
:<math>2E_2(2\tau)-E_2(\tau) = \sum_nc(n)q^n = 1 + 24q + 24q^2 + 96q^3 + 24q^4 + 144q^5 + \dots</math>   {{oeis|A004011}}
where 
:<math>q=e^{2\pi i \tau}</math>
and
:<math>E_2(\tau) = 1-24\sum_n\sigma_1(n)q^n</math>
is the weight 2 level 1 [[Eisenstein series]] (which is a [[quasimodular form]]) and ''σ''<sub>1</sub>(''n'') is the sum of the divisors of ''n''.