Fungrim entry: 13cac5

\sum_{\tau \in \mathcal{F}} w(\tau) \mathop{\operatorname{ord}}\limits_{z=\tau} E_{2 k}\!\left(z\right) = \frac{2 k}{12}\; \text{ where } w(\tau) = \begin{cases} \frac{1}{2}, & \tau = i\\\frac{1}{3}, & \tau = {e}^{2 \pi i / 3}\\1, & \text{otherwise}\\ \end{cases}

Assumptions:

k \in \mathbb{Z}_{\ge 2}

References:

K. Ono and M. A. Papanikolas (2004). p-Adic Properties of Values of the Modular j-Function. In: Hashimoto K., Miyake K., Nakamura H. (eds) Galois Theory and Modular Forms. Developments in Mathematics, vol 11. Springer, Boston, MA, https://doi.org/10.1007/978-1-4613-0249-0_19
S. Garthwaite, L. Long, H. Swisher, S. Treneer. Zeros of classical Eisenstein series and recent developments, Fields Communications Volume 60, WIN - Women In Numbers, Proceedings of the WIN Workshop, (2011), 251-263. http://math.oregonstate.edu/~swisherh/C1P.pdf

TeX:

\sum_{\tau \in \mathcal{F}} w(\tau) \mathop{\operatorname{ord}}\limits_{z=\tau} E_{2 k}\!\left(z\right) = \frac{2 k}{12}\; \text{ where } w(\tau) = \begin{cases} \frac{1}{2}, & \tau = i\\\frac{1}{3}, & \tau = {e}^{2 \pi i / 3}\\1, & \text{otherwise}\\ \end{cases}

k \in \mathbb{Z}_{\ge 2}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`ComplexZeroMultiplicity`	$\mathop{\operatorname{ord}}\limits_{z=c} f(z)$	Multiplicity (order) of complex zero
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("13cac5"),
    Formula(Where(Equal(Sum(Mul(w(tau), ComplexZeroMultiplicity(EisensteinE(Mul(2, k), z), For(z, tau))), ForElement(tau, ModularGroupFundamentalDomain)), Div(Mul(2, k), 12)), Equal(w(tau), Cases(Tuple(Div(1, 2), Equal(tau, ConstI)), Tuple(Div(1, 3), Equal(tau, Exp(Div(Mul(Mul(2, Pi), ConstI), 3)))), Tuple(1, Otherwise))))),
    Variables(k),
    Assumptions(Element(k, ZZGreaterEqual(2))),
    References("K. Ono and M. A. Papanikolas (2004). p-Adic Properties of Values of the Modular j-Function. In: Hashimoto K., Miyake K., Nakamura H. (eds) Galois Theory and Modular Forms. Developments in Mathematics, vol 11. Springer, Boston, MA, https://doi.org/10.1007/978-1-4613-0249-0_19", "S. Garthwaite, L. Long, H. Swisher, S. Treneer. Zeros of classical Eisenstein series and recent developments, Fields Communications Volume 60, WIN - Women In Numbers, Proceedings of the WIN Workshop, (2011), 251-263. http://math.oregonstate.edu/~swisherh/C1P.pdf"))

Topics using this entry

Eisenstein series