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Fungrim entry: 097efc

τFj ⁣(τ)w ⁣(τ)ordτE2k ⁣(τ)=120k2ζ ⁣(12k)   where w ⁣(τ)={12,τ=i1,otherwise\sum_{\tau \in \mathcal{F}} j\!\left(\tau\right) w\!\left(\tau\right) \mathop{\operatorname{ord}}\limits_{\tau} E_{2 k}\!\left(\tau\right) = 120 k - \frac{2}{\zeta\!\left(1 - 2 k\right)}\; \text{ where } w\!\left(\tau\right) = \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\!\left(\tau\right) w\!\left(\tau\right) \mathop{\operatorname{ord}}\limits_{\tau} E_{2 k}\!\left(\tau\right) = 120 k - \frac{2}{\zeta\!\left(1 - 2 k\right)}\; \text{ where } w\!\left(\tau\right) = \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\!\left(n\right) Sum
ModularJj ⁣(τ)j\!\left(\tau\right) Modular j-invariant
ComplexZeroMultiplicityordzf ⁣(z)\mathop{\operatorname{ord}}\limits_{z} f\!\left(z\right) 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\!\left(s\right) 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), tau), tau, tau)), tau, Element(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"))

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

2019-08-21 11:44:15.926409 UTC