Rationality of the zeta function of the subgroups of abelian pp-groups

Given a finite abelian $p$-group $F$, we prove an efficient recursive formula for $\sigma_a(F)=\sum_{\substack{H\leq F}}|H|^a$ where $H$ ranges over the subgroups of $F$. We infer from this formula that the $p$-component of the corresponding zeta-function on groups of $p$-rank bounded by some constant $r$ is rational with a simple denominator. We also provide two explicit examples in rank $r=3$ and $r=4$ as well as a closed formula for $\sigma_a(F)$

A Moebius sum

We provide numerical bounds for $\Sigma(X)=\sum_{\substack{d_1,d_2\le X}}\frac{\mu(d_1)\mu(d_2)}{[d_1,d_2]}$. We show in particular that $0\le \Sigma(X)\le 17/25$ for every $X\ge2$

A probabilistic language-free proof of an explicit Croot-Laba-Sisask Lemma

This note proposes a probabilistic language-free proof of the famous Croot-Laba-Sisask Lemma. In between, we do the same for the Khintchine and Marcinkiewicz-Zygmund inequalities and explicitate the implied constants

Explicit count of integral ideals of an imaginary quadratic field

We provide explicit bounds for the number of integral ideals of norms at most $X$ is $\mathbb{Q}[\sqrt{d}]$ when $d <0$ is a fundamendal discriminant with an error term of size $O(X^{1/3})$. In particular, we prove that, when $\chi$ is the non-principal character modulo $3$ and $X\ge1$, we have $\sum_{n\le X}(1\star\chi)(n) = \frac{\pi}{3\sqrt{3}}X +O^*( 1.94\,X^{1/3})$, and that, when $\chi$ is the non-principal character modulo $4$ and $X\ge1$, we have $\sum_{n\le X}(1\star\chi)(n) = \frac{\pi }{4}X+ O^*( 1.4\,X^{1/3})$

Should Diamond OA be viewed as a threat to librarians?

International audienceOpen access has become a growth opportunity for publishers with the " Gold Open Access APC " model and the hybrid model. During the period 2012-2015, these proposals generated a lot of lively discussions. has made some recommendations to French mathematicians for their publications asking them not to choose the author-pays option for open access, especially for hybrid journals, and not to include publication fees (APC, author processing charges) in funding requests. The rejection of the author-pays model by French mathematicians gives a boost to the development of other economic models of scientific edition, especially the Diamond OA (Gold OA without APC). This new model requires new skills from librarians. In this paper we present the Diamond OA publishing model, with a stress on the French situation in mathematics, and then explore the problem of how this model could impact libraries and librarians. Everybody says that Open Access is a real opportunity for libraries but we would like to be a little provocative: is Diamond Open Access a threat to librarians