## Christian Táfula

### Publications

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.

### Old projects

• "On $$\Re(\frac{L'}{L}(1,\chi))$$ and zero-free regions near $$s=1$$".
[old version (2020), included in 3.]

Let $$q\geq 2$$ be an integer, $$\chi~(\mathrm{mod}~q)$$ a primitive Dirichlet character, and $$f:\mathbb{Z}_{\geq 2} \to \mathbb{R}$$ a function satisfying $$2\leq f(q) \ll \log q$$. We show that, if $$L(s,\chi)$$ has no zeros in the region $\left\{\sigma + it \in \mathbb{C} ~\bigg|~ \sigma > 1- \frac{1}{f(q)},\ |t| < \frac{1}{\sqrt{f(q)}} \right\},$ then $$\Re(\frac{L'}{L}(1,\chi)) \ll \sqrt{f(q)\log q}$$ uniformly for primitive $$\chi~(\mathrm{mod}~q)$$. As an example of an application, we show that the uniform $$abc$$-conjecture implies a strong version of "no Siegel zeros" for odd real characters of $$q^{o(1)}$$-smooth moduli, by using our result in T. [arXiv:1911.07215] together with a theorem of Chang [DOI:10.1007/s11854-014-0012-y] on zero-free regions.

• "Classification of the conjugacy classes of $$\widetilde{\mathrm{SL}}_2(\mathbb{R})$$"
In this paper, we classify the conjugacy classes of $$\widetilde{\mathrm{SL}}_2(\mathbb{R})$$, the universal covering group of $$\mathrm{PSL}_2(\mathbb{R})$$. For any non-central element $$\alpha \in \widetilde{\mathrm{SL}}_2(\mathbb{R})$$, we show that its conjugacy class may be determined by three invariants:
1. Trace: the trace (valued in the set of positive real numbers $$\mathbb{R}_{+}$$) of its image $$\overline{\alpha}$$ in $$\mathrm{PSL}_2(\mathbb{R})$$;
2. Direction type: the sign behavior of the induced self-homeomorphism of $$\mathbb{R}$$ determined by the lifting $$\widetilde{\mathrm{SL}}_2(\mathbb{R}) \curvearrowright \mathbb{R}$$ of the action $$\mathrm{PSL}_2(\mathbb{R}) \curvearrowright \mathbb{S}^{1}$$;
• "Effective Schnirelmann's method for $$O$$-regular sequences".
In this paper we introduce the notion of pre-basis, which is a sequence such that some of its $$h$$-fold sumsets has positive lower asymptotic density, the least such $$h$$ being its pre-order. Schnirelmann's classical theory of sumsets is reinterpreted as characterizing additive bases in terms of pre-bases, and estimates for the order in terms of the pre-order are derived from the deep theorems of Mann and Kneser.
Under certain regularity assumptions on the representation functions, we then derive estimates to the pre-order. This is achieved by studying sequences $$\mathscr{A}=\{a_0 < a_1 < a_2 < \ldots\}\subseteq \mathbb{N}$$ for which $$A(2x) = O(A(x))$$ and $$a_{2n} = O(a_n)$$, and ends up providing a small shortcut to the proofs of the Schnirelmann–Goldbach theorem and Linnik's elementary solution of Waring's problem.