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Abstract The Euler–Mascheroni constant $$\gamma =0.5772\ldots \!$$ is the $$K={\mathbb Q}$$ example of an Euler–Kronecker constant $$\gamma _K$$ of a number field $K.$ In this note, we consider the size of the $$\gamma _q=\gamma _{K_q}$$ for cyclotomic fields $$K_q:={\mathbb Q}(\zeta _q).$$ Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that $$ \begin{align*} \frac{1}{Q}\sum_{Qmore » « less
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Using predictions in mirror symmetry, Căldăraru, He, and Huang recently formulated a “Moonshine Conjecture at Landau-Ginzburg points” [arXiv:2107.12405, 2021] for Klein’s modular j j -function at j = 0 j=0 and j = 1728. j=1728. The conjecture asserts that the j j -function, when specialized at specific flat coordinates on the moduli spaces of versal deformations of the corresponding CM elliptic curves, yields simple rational functions. We prove this conjecture, and show that these rational functions arise from classical 2 F 1 _2F_1 -hypergeometric inversion formulae for the j j -function.more » « less
