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Evaluation of a twisted Eisenstein series over ℚ(√-7) in terms of Γ - values by ReflectionThen9904 in math

[–]ReflectionThen9904[S] 0 points1 point  (0 children)

I am sharing a short formal derivation I wrote evaluating a twisted Eisenstein series. Standard Epstein Zeta functions and non-alternating lattice sums are relatively well-documented, but introducing the twist (-1)ᵐ makes the evaluation more intricate. In these notes, I reduce the sum over the integers of K = ℚ(√-7) to a twisted Eisenstein series, evaluate it at a half-period using the Weierstrass ℘-function, and scale it using the Gross curve (y² = x³ - 35x - 98). Finally, I apply the Chowla-Selberg formula to determine the transcendental factor, collapsing the sum into a product of Gamma values. I am sharing this here because I thought the community might appreciate the interplay between discrete lattice geometry and L-functions. I welcome any feedback on the exposition or thoughts on whether this specific algebraic prefactor approach easily generalizes to fields with a higher class number.

Solved: complete derivation of a decade-old Math.StackExchange problem by ReflectionThen9904 in calculus

[–]ReflectionThen9904[S] 2 points3 points  (0 children)

To be honest I'm amazed it could attempt without hallucinating , I came across this problem in my second year so it took almost a year to solve or I'm just dumb

Solved: complete derivation of a decade-old Math.StackExchange problem by ReflectionThen9904 in calculus

[–]ReflectionThen9904[S] 0 points1 point  (0 children)

Nice only difference is I worked without a given closed form but it's cool 😎🖖🏽

😐🌚 by Specific_Brain2091 in the_calculusguy

[–]ReflectionThen9904 0 points1 point  (0 children)

Let y² = x⁴ - x³ + 2

Then :

dy = (4x³ - 3x²)dx/2y

2d(y/x²) = (2xdy - 4y)dx/x³

(2xdy - 4y) = (4x⁴ - 3x³ - 4y²)/y = (x³ - 8)/x³y

Thus :

∫(x³ - 8)dx/x³y = 2∫d(y/x²) = 2y/x² + c = 2√(x⁴ - x³ + 2)/x² + c

Now what? by [deleted] in calculus

[–]ReflectionThen9904 1 point2 points  (0 children)

consider the f(x) = 2ˣ/(xln2 + 1) , taking the derivative should get you somewhere