The level-one Eisenstein series E_k and the weight-two series E_2

For even $k\ge 4$ put

$G_k(\tau )={{\sum }_{(m,n)\in {\mathbb{Z}}^2}}^{\mathrm{\prime }}(m\tau +n{)}^{-k},\tau \in \mathfrak{H},$

the prime excluding $(m,n)=(0,0)$; this is an absolutely summable family whose sum is holomorphic in $\tau$ (Absolute convergence and holomorphy of the lattice Eisenstein sums). Define the normalised Eisenstein series

$E_k:=\frac{G_k}{2\zeta (k)},\zeta (k)={\sum }_{n\ge 1}{n}^{-k},$

the Riemann zeta value (The Riemann zeta function on the half-plane $\mathrm{Re}s>1$). For $k>1$ the series for $\zeta (k)$ converges (For rational $p>0$, $\sum 1\mathrm{/}{k}^p$ converges iff $p>1$), so $2\zeta (k)>0$ and the normalisation is well defined.

The constant term and the vanishing of the higher terms. Splitting the absolutely convergent sum into $m=0$ and $m\mathrm{\ne }0$, and bounding the latter with the Lipschitz formula (The Lipschitz formula for the reciprocal-power sums),

$G_k(\tau )=2\zeta (k)+2{\sum }_{m\ge 1}\frac{(-2\pi i{)}^k}{(k-1)!}{\sum }_{r\ge 1}{r}^{k-1}{q}^{mr},q=e^{2\pi i\tau },$

so $E_k(\tau )=1+O(q)$ as $\mathrm{Im}\tau \to \mathrm{\infty }$: the correction is a power series in $q$ divisible by $q$, convergent for $\mathrm{\mid }q\mathrm{\mid }<1$. The same computation yields the Fourier expansion of Eisenstein series are modular forms; their Fourier coefficients.

The weight-two series. Separately define

$E_2(\tau ):=1-24{\sum }_{n\ge 1}{\sigma }_1(n)q^n,q=e^{2\pi i\tau },$

with ${\sigma }_1(n)$ the sum of the positive divisors of $n$ (The divisor power sums ${\sigma }_k$). Since ${\sigma }_1(n)\le n\cdot n=n^2$ and ${\sum }_{n\ge 1}{n}^2\mathrm{\mid }q{\mathrm{\mid }}^n$ converges for $\mathrm{\mid }q\mathrm{\mid }<1$ by the ratio test (Ratio test: $\mathrm{lim\hspace{0.17em}sup}\mathrm{\mid }{a}_{k+1}\mathrm{/}{a}_k\mathrm{\mid }<1$ gives absolute convergence and hence convergence, and $\mathrm{lim\hspace{0.17em}inf}\mathrm{\mid }{a}_{k+1}\mathrm{/}{a}_k\mathrm{\mid }>1$ gives divergence), the $M$-test (Weierstrass M-test for complex-valued function series, A complex power series converges absolutely and uniformly on every closed subdisc strictly inside its disc of convergence) makes $E_2$ a holomorphic function of $\tau \in \mathfrak{H}$. It is a quasimodular comparison object used in the discriminant product proof and in the counterexample item of the companion page; it is not itself a modular form, since its transformation law carries a nonzero correction term. The half-plane conventions are The modular group and its action on the upper half-plane.