Under two assumptions, we determine the distribution of the difference between two functions each counting the numbers less than or equal to $x$ that are in a given arithmetic progression modulo $q$ and the product of two primes. The two assumptions are (i) the extended Riemann hypothesis for Dirichlet $L$-functions modulo $q$, and (ii) that the imaginary parts of the nontrivial zeros of these $L$-functions are linearly independent over the rationals. Our results are analogues of similar results proved for primes in arithmetic progressions by Rubinstein and Sarnak.

Given fixed $0 = q_0 < q_1 < q_2 < \cdots < q_k = 1$, a constellation in $[n]$ is a scaled translated realization of the $q_i$ with all elements in $[n]$, i.e.,

$$p, p + q_1d, p + q_2d, \dots , p + q_{k−1}d, p + d.$$

We consider the problem of minimizing the number of monochromatic constellations in a two-coloring of $[n]$. We show how, given a coloring based on a block pattern, to find the number of monochromatic solutions to a lower-order term, and also how experimentally we might find an optimal block pattern. We also show for the case $k = 2$ that there is always a block pattern that beats random coloring.

We describe CW complexes for complex projective algebraic surfaces in the context of practical computation of topological invariants.

Let $d(n)$ denote the corank of $I + A$ over the field with two elements, where $A$ is the adjacency matrix of the discrete torus $C_n × C_n$, and $I$ is the identity matrix. We shall prove that $d(2n) = 2d(n)$ and $d(2^r + 1) = d(2^r − 1) + 4$. For the proof of the latter result, we use an elliptic curve. Our motivation for this study is the “lights out” puzzle.

Herein, with the aid of substantial symbolic computation, we solve previously open problems in the theory of $n$-dimensional box integrals $B_n(s) := \langle|\vec{r}|^s\rangle, \vec{r} ∈ [0, 1]^n$. In particular, we resolve an elusive integral called $\mathcal{K}_5$ that previously acted as a “blockade” against closed-form evaluation in $n = 5$ dimensions. In consequence, we now know that $B_n$(integer) can be given a closed form for $n = 1, 2, 3, 4, 5$. We also find the general residue at the pole at $s = −n$, this leading to new relations and definite integrals; for example, we are able to give the first nontrivial closed forms for six-dimensional box integrals and to show hyperclosure of $B_6$(even). The Clausen function and its generalizations play a central role in these higher-dimensional evaluations. Our results provide stringent test scenarios for symbolic-algebra simplification methods.

Bertini classified the birational involutions of the complex projective plane, but his geometric approach does not allow one easily to exhibit these maps explicitly. In this article, we present an effective approach to this problem by associating to each quadratic foliation a birational involution that is, in the generic case, a Geiser involution; this subject has already been covered by Geiser, Milinowski, Williams, and others. We end by conducting some experiments, from which we obtain trivolutions from some foliations of degree 3.

La classification des involutions birationnelles du plan projectif complexe remonte essentiellement à Bertini; cette classification de nature géométrique permet difficilement la construction explicite de telles involutions. Nous proposons une approche effective permettant d’associer à tout feuilletage quadratique une involution qui, dans le cas générique, est de Geiser; ce sujet a déjà été étudié par Geiser, Milinowski, Williams et alt. Nous présentons ensuite quelques expériences qui produisent des trivolutions à partir de feuilletages cubiques très spéciaux.

We propose a modification of the predictions of the Cohen– Lenstra heuristic for class groups of number fields in the case that roots of unity are present in the base field. As evidence for this modified formula we provide a large set of computational data that show close agreement with it. Furthermore, our predicted formula agrees with results on class groups of function fields in positive characteristic for which the base field contains appropriate roots of unity.

In this paper, we compute the minimum discriminants of imprimitive degree-10 fields for different combinations of Galois group and signature. We use class field theory when there is a quintic subfield, and a Martinet search in the more difficult case in which there is only a quadratic subfield.

We study the convergence properties of the series $Ψ_s(α) := \sum_{n≥1} \frac{||n^2α||} {n^{s+1}||nα||}$ with respect to the values of the real numbers $α$ and $s$, where $||x||$ is the distance of $x$ to $\mathbb{Z}$. For example, when $s ∈ (0, 1]$, the convergence of $Ψ_s(α)$ strongly depends on the Diophantine nature of $α$, mainly its irrationality exponent. We also conjecture that $Ψ_s(α)$ is minimal at $\sqrt{5}$ for $s ∈ (0, 1]$, and we present evidence in favor of that conjecture. For $s = 1$, we formulate a more precise conjecture about the value of the abscissa $u_k$ where the $F_k$-partial sum of $Ψ_1(α)$ is minimal, $F_k$ being the $k$th Fibonacci number. A similar study is made for the partial sums of the series $\tilde{Ψ}_1(α) := \sum_{n≥1}(−1)^n \frac{||n^2α||} {n^2||nα||}$, which we conjecture to be minimal at $\sqrt{2}/2$.

Let $K$ be a number field. The $\operatorname{Gal}(\bar{K}/K)$-action on the torsion of an elliptic curve $E/K$ gives rise to an adelic representation $ρ_E: \operatorname{Gal}(\bar{K}/K) \to \mathrm{GL}_2(\hat{\mathbb{Z}})$. From an analysis of maximal closed subgroups of $\mathrm{GL}_2(\hat{\mathbb{Z}})$ we derive useful necessary and sufficient conditions for $ρ_E$ to be surjective. Using these conditions, we compute an example of a number field $K$ and an elliptic curve $E/K$ that admits a surjective adelic Galois representation.