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 $$ \begin{align*} \frac{1}{Q}\sum_{Q
more »
« less
Abstract Inspired by Lehmer’s conjecture on the non-vanishing of the Ramanujan $$\tau $$ τ -function, one may ask whether an odd integer $$\alpha $$ α can be equal to $$\tau (n)$$ τ ( n ) or any coefficient of a newform f ( z ). Balakrishnan, Craig, Ono and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight $$k\ge 4$$ k ≥ 4 . We use these methods for weight 2 and 3 newforms and apply our results to L -functions of modular elliptic curves and certain K 3 surfaces with Picard number $$\ge 19$$ ≥ 19 . In particular, for the complete list of weight 3 newforms $$f_\lambda (z)=\sum a_\lambda (n)q^n$$ f λ ( z ) = ∑ a λ ( n ) q n that are $$\eta $$ η -products, and for $$N_\lambda $$ N λ the conductor of some elliptic curve $$E_\lambda $$ E λ , we show that if $$|a_\lambda (n)|<100$$ | a λ ( n ) | < 100 is odd with $$n>1$$ n > 1 and $$(n,2N_\lambda )=1$$ ( n , 2 N λ ) = 1 , then $$\begin{aligned} a_\lambda (n) \in&\{-5,9,\pm 11,25, \pm 41, \pm 43, -45,\pm 47,49, \pm 53,55, \pm 59, \pm 61,\\&\pm 67, -69,\pm 71,\pm 73,75, \pm 79,\pm 81, \pm 83, \pm 89,\pm 93 \pm 97, 99\}. \end{aligned}$$ a λ ( n ) ∈ { - 5 , 9 , ± 11 , 25 , ± 41 , ± 43 , - 45 , ± 47 , 49 , ± 53 , 55 , ± 59 , ± 61 , ± 67 , - 69 , ± 71 , ± 73 , 75 , ± 79 , ± 81 , ± 83 , ± 89 , ± 93 ± 97 , 99 } . Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving $$\begin{aligned} a_\lambda (n) \in \{-5,9,\pm 11,25,-45,49,55,-69,75,\pm 81,\pm 93, 99\}. \end{aligned}$$ a λ ( n ) ∈ { - 5 , 9 , ± 11 , 25 , - 45 , 49 , 55 , - 69 , 75 , ± 81 , ± 93 , 99 } .
more »
« less
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
