Christian Táfula


  1. "On Landau–Siegel zeros and heights of singular moduli"
    Acta Arithmetica 201(1), 1–28 (2021)
    <arXiv | DOI> [figures (python/bash scripts)]
    Let \(\chi_D\) be the Dirichlet character associated to \(\mathbb{Q}(\sqrt{D})\) where \(D < 0\) is a fundamental discriminant. Improving Granville–Stark [DOI:10.1007/s002229900036], we show that \[ \frac{L'}{L}(1,\chi_D) = \frac{1}{6}\, \mathrm{height}(j(\tau_D)) - \frac{1}{2}\log|D| + C + o_{D\to -\infty}(1), \] where \(\tau_D = \frac 12(-\delta+\sqrt{D})\) for \(D \equiv \delta ~(\mathrm{mod}~4)\) and \(j(\cdot)\) is the \(j\)-invariant function with \(C = -1.057770\ldots\). Assuming the ``uniform'' \(abc\)-conjecture for number fields, we deduce that \(L(\beta,\chi_D)\ne 0\) with \(\beta ≥ 1 - \frac{\sqrt{5}\varphi + o(1)}{\log|D|}\) where \(\varphi = \frac{1+\sqrt{5}}{2}\), which we improve for smooth \(D\).

  2. "An elementary heuristic for Hardy–Littlewood extended Goldbach's conjecture"
    São Paulo Journal of Mathematical Sciences 14(1), 391–405 (2020)
    <arXiv | DOI> [figures (python/bash scripts)]
    The goal of this paper is to describe an elementary combinatorial heuristic that predicts Hardy & Littlewood's extended Goldbach's conjecture. We examine common features of other heuristics in additive prime number theory, such as Cramér's model and density-type arguments, both of which our heuristic draws from. Apart from the prime number theorem, our argument is entirely elementary, in the sense of not involving complex analysis. The idea is to model sums of two primes by a hypergeometric probability distribution, and then draw heuristic conclusions from its concentration behavior, which follows from Hoeffding-type bounds.

  3. "An extension of the Erdős–Tetali theorem"
    Random Structures & Algorithms 55(1), 173–214 (2019)
    <arXiv | DOI>
    Given a sequence \(\mathscr{A}=\{a_0 < a_1 < a_2\ldots\}\subseteq \mathbb{N}\), let \(r_{\mathscr{A},h}(n)\) denote the number of ways \(n\) can be written as the sum of \(h\) elements of \(\mathscr{A}\). Fixing \(h\geq 2\), we show that if \(f\) is a suitable real function (namely: locally integrable, \(O\)-regularly varying and of positive increase) satisfying \[ x^{1/h}\log(x)^{1/h} \ll f(x) \ll \frac{x^{1/(h-1)}}{\log(x)^{\varepsilon}} \text{ for some } \varepsilon > 0, \] then there must exist \(\mathscr{A}\subseteq\mathbb{N}\) with \(|\mathscr{A}\cap [0,x]|=\Theta(f(x))\) for which \(r_{\mathscr{A},h+\ell}(n) = \Theta(f(n)^{h+\ell}/n)\) for all \(\ell \geq 0\). Furthermore, for \(h=2\) this condition can be weakened to \(x^{1/2}\log(x)^{1/2} \ll f(x) \ll x\). The proof is somewhat technical and the methods rely on ideas from regular variation theory, which are presented in an appendix with a view towards the general theory of additive bases. We also mention an application of these ideas to Schnirelmann's method.

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