A new formula for Chebotarev densities

We give a new formula for the Chebotarev densities of Frobenius elements in Galois groups. This formula is given in terms of smallest prime factors $p_{\mathrm{min}}(n)$ of integers $n\geq2$. More precisely, let $C$ be a conjugacy class of the Galois group of some finite Galois extension $K$ of $\mathbb{Q}$. Then we prove that $$-\lim_{X\rightarrow\infty}\sum_{\substack{2\leq n\leq X\\[1pt]\left[\frac{K/\mathbb{Q}}{p_{\mathrm{min}}(n)}\right]=C}}\frac{\mu(n)}{n}=\frac{\#C}{\#G}.$$ This theorem is a generalization of a result of Alladi from 1977 that asserts that largest prime divisors $p_{\mathrm{max}}(n)$ are equidistributed in arithmetic progressions modulo an integer $k$, which occurs when $K$ is a cyclotomic field $\mathbb{Q}(\zeta_k)$

Higher Width Moonshine

\textit{Weak moonshine} for a finite group $G$ is the phenomenon where an infinite dimensional graded $G$-module $$V_G=\bigoplus_{n\gg-\infty}V_G(n)$$ has the property that its trace functions, known as McKay-Thompson series, are modular functions. Recent work by DeHority, Gonzalez, Vafa, and Van Peski established that weak moonshine holds for every finite group. Since weak moonshine only relies on character tables, which are not isomorphism class invariants, non-isomorphic groups can have the same McKay-Thompson series. We address this problem by extending weak moonshine to arbitrary width $s\in\mathbb{Z}^+$. For each $1\leq r\leq s$ and each irreducible character $\chi_i$, we employ Frobenius' $r$-character extension $\chi_i^{(r)} \colon G^{(r)}\rightarrow\mathbb{C}$ to define \textit{width $r$ McKay-Thompson series} for $V_G^{(r)}:=V_G\times\cdots\times V_G$ ($r$ copies) for each $r$-tuple in $G^{(r)}:=G\times\cdots\times G$ ($r$ copies). These series are modular functions which then reflect differences between $r$-character values. Furthermore, we establish orthogonality relations for the Frobenius $r$-characters, which dictate the compatibility of the extension of weak moonshine for $V_G$ to width $s$ weak moonshine.Comment: Versions 2 and 3 address comments from the referee

Effective Bounds for the Andrews spt-function

In this paper, we establish an asymptotic formula with an effective bound on the error term for the Andrews smallest parts function $\mathrm{spt}(n)$. We use this formula to prove recent conjectures of Chen concerning inequalities which involve the partition function $p(n)$ and $\mathrm{spt}(n)$. Further, we strengthen one of the conjectures, and prove that for every $\epsilon>0$ there is an effectively computable constant $N(\epsilon) > 0$ such that for all $n\geq N(\epsilon)$, we have \begin{equation*} \frac{\sqrt{6}}{\pi}\sqrt{n}\,p(n)<\mathrm{spt}(n)<\left(\frac{\sqrt{6}}{\pi}+\epsilon\right) \sqrt{n}\,p(n). \end{equation*} Due to the conditional convergence of the Rademacher-type formula for $\mathrm{spt}(n)$, we must employ methods which are completely different from those used by Lehmer to give effective error bounds for $p(n)$. Instead, our approach relies on the fact that $p(n)$ and $\mathrm{spt}(n)$ can be expressed as traces of singular moduli.Comment: Changed the title. Added more details and simplified some arguments in Section

Polycyclic aromatic hydrocarbons and esophageal squamous cell carcinoma

Esophageal cancer (EC) is the 8th most common cancer and the 6th most frequent cause of cancer mortality worldwide. Esophageal squamous cell carcinoma (ESCC) is the most common type of EC. Exposure to polycyclic aromatic hydrocarbons (PAHs) has been suggested as a risk factor for developing ESCC. In this paper we will review different aspects of the relationship between PAH exposure and ESCC. PAHs are a group of compounds that are formed by incomplete combustion of organic matter. Studies in humans have shown an association between PAH exposure and development of ESCC in many populations. The results of a recent case-control study in a high risk population in northeastern Iran showed a dramatic dose-response relationship between PAH content in non-tumor esophageal tissue (the target tissue for esophageal carcinogenesis) and ESCC case status, consistent with a causal role for PAH exposure in the pathogenesis of ESCC. Identifying the main sources of exposure to PAHs may be the first and most important step in designing appropriate PAH-reduction interventions for controlling ESCC, especially in high risk areas. Coal smoke and drinking mate have been suggested as important modifiable sources of PAH exposure in China and Brazil, respectively. But the primary source of exposure to PAHs in other high risk areas for ESCC, such as northeastern Iran, has not yet been identified. Thus, environmental studies to determining important sources of PAH exposure should be considered as a high priority in future research projects in these areas

Multiquadratic fields generated by characters of AnA_n

For a finite group $G$, let $K(G)$ denote the field generated over $\mathbb{Q}$ by its character values. For $n>24$, G. R. Robinson and J. G. Thompson proved that $$K(A_n)=\mathbb{Q}\left (\{ \sqrt{p^*} \ : \ p\leq n \ {\text{ an odd prime with } p\neq n-2}\}\right),$$ where $p^*:=(-1)^{\frac{p-1}{2}}p$. Confirming a speculation of Thompson, we show that arbitrary suitable multiquadratic fields are similarly generated by the values of $A_n$-characters restricted to elements whose orders are only divisible by ramified primes. To be more precise, we say that a $\pi$-number is a positive integer whose prime factors belong to a set of odd primes $\pi:= \{p_1, p_2,\dots, p_t\}$. Let $K_{\pi}(A_n)$ be the field generated by the values of $A_n$-characters for even permutations whose orders are $\pi$-numbers. If $t\geq 2$, then we determine a constant $N_{\pi}$ with the property that for all $n> N_{\pi}$, we have $$K_{\pi}(A_n)=\mathbb{Q}\left(\sqrt{p_1^*}, \sqrt{p_2^*},\dots, \sqrt{p_t^*}\right).$