Fungrim home page

Fungrim entry: 097efc

τFj(τ)w(τ)ordz=τE2k ⁣(z)=120k2ζ ⁣(12k)   where w(τ)={12,τ=i1,otherwise\sum_{\tau \in \mathcal{F}} j(\tau) w(\tau) \mathop{\operatorname{ord}}\limits_{z=\tau} E_{2 k}\!\left(z\right) = 120 k - \frac{2}{\zeta\!\left(1 - 2 k\right)}\; \text{ where } w(\tau) = \begin{cases} \frac{1}{2}, & \tau = i\\1, & \text{otherwise}\\ \end{cases}
Assumptions:kZ2k \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}} j(\tau) w(\tau) \mathop{\operatorname{ord}}\limits_{z=\tau} E_{2 k}\!\left(z\right) = 120 k - \frac{2}{\zeta\!\left(1 - 2 k\right)}\; \text{ where } w(\tau) = \begin{cases} \frac{1}{2}, & \tau = i\\1, & \text{otherwise}\\ \end{cases}

k \in \mathbb{Z}_{\ge 2}
Definitions:
Fungrim symbol Notation Short description
Sumnf(n)\sum_{n} f(n) Sum
ModularJj(τ)j(\tau) Modular j-invariant
ComplexZeroMultiplicityordz=cf(z)\mathop{\operatorname{ord}}\limits_{z=c} f(z) Multiplicity (order) of complex zero
EisensteinEEk ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series
ModularGroupFundamentalDomainF\mathcal{F} Fundamental domain for action of the modular group
RiemannZetaζ(s)\zeta(s) Riemann zeta function
ConstIii Imaginary unit
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("097efc"),
    Formula(Where(Equal(Sum(Mul(Mul(ModularJ(tau), w(tau)), ComplexZeroMultiplicity(EisensteinE(Mul(2, k), z), For(z, tau))), ForElement(tau, ModularGroupFundamentalDomain)), Sub(Mul(120, k), Div(2, RiemannZeta(Sub(1, Mul(2, k)))))), Equal(w(tau), Cases(Tuple(Div(1, 2), Equal(tau, ConstI)), 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC