On the Form of Odd Perfect Gaussian Integers

Let Z[i] = {a + bi : a, b ∈ Z} be the ring of Gaussian integers. All Gaussian integers will be represented by Greek letters and rational integers by ordinary Latin letters. Primes will be denoted by π and p respectively. Units will be denoted by ε = ±1,±i and 1 respectively. In 1961 Robert Spira [3] defined the sum-of-divisors function on Z[i] as follows. Let η = εΠπi i be a Gaussian integer. This representation is unique in that we will choose our unit ε such that each πi is in the first quadrant (Re(πi) > 0 and Im(πi) ≥ 0). Spira defined the sum-of-divisors function σ as