Special prime families (twin, Sophie Germain, safe, cousin, sexy, Chen, and isolated primes) are central objects of analytic number theory, yet no efficiently computable probabilistic filter exists for identifying likely members among known primes at large scale. Classical sieves assign no probability weights to surviving candidates, and prior machine learning approaches are limited by the algorithmic randomness of the prime indicator sequence, yielding near-zero true positive rates. We present PrimeFamilyNet, a multi-head residual network conditioned on the backward prime gap and modular primorial residues of a known prime $p$, learning probabilistic filters for all seven families simultaneously and generalising across nine orders of magnitude from training ($10^7$--$10^9$) to evaluation at $10^{16}$. Isolated prime recall increased monotonically from $0.809$ at $5\times10^8$ to $0.984$ at $10^{16}$, a gain of $17.5$ percentage points and the only family among seven to improve with scale. Because recall is invariant to class prevalence, this reflects genuine decision boundary sharpening, not the rising isolated-prime fraction at extreme scales. A model trained only to $10^9$ reproduced the correct asymptotic direction without density supervision, corroborating Hardy--Littlewood $k$-tuple predictions. The causal model retained over $95\%$ recall for five families near $10^{10}$ while reducing the search space by $62$--$88\%$. For Chen primes, causal recall exceeded non-causal recall at every scale (margin $+0.245$ at $10^{16}$) because $g^+=2$ encodes only the prime case of the Chen condition. Focal Loss collapsed sparse algebraic family recall to $0.000$. Asymmetric Loss outperformed weighted BCE in-distribution but degraded more steeply out-of-distribution, showing that in-distribution recall alone is a misleading criterion for scale-generalisation tasks.
In this paper, we classify all finite groups $G$ which have the following property: for all subsets $A \subseteq G$, we have $|AA^{-1}| = |A^{-1}A|$. This question is motivated by the problem in additive combinatorics of More Sums Than Difference sets and answers several questions posed in arXiv:2509.00611 [math.NT].
The present paper is devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. As an application of our main result, we establish Deligne's conjecture for a class of CM-automorphic motives, which we construct in this paper. Our proof uses the results of our recent joint work with Raghuram in combination with the Ichino--Ikeda--Neal-Harris (IINH) formula for unitary groups -- which is now a theorem -- and an analysis of cup products of coherent cohomological automorphic forms on Shimura varieties to establish relations between certain automorphic periods and critical values of Rankin-Selberg and Asai $L$-functions of $\GL(n)\times\GL(m)$ over CM fields. By reinterpreting these critical values in terms of automorphic periods of holomorphic automorphic forms on unitary groups, we show that the automorphic periods of holomorphic forms can be factored as products of coherent cohomological forms, compatibly with a motivic factorization predicted by the Tate conjecture. All of these results are stated under a certain regularity condition and an hypothesis of rationality on archimedean zeta-integrals.
The Wright-Euler Mersenne Exponent Hypothesis proposes that Euler's quadratic polynomial C(n) = n^2 + n + 41, combined with nearest-integer rounding n_closest = round((-1 + sqrt(4p - 163))/2), identifies candidate exponents for Mersenne primes 2^p - 1. Applied to the 43 known Mersenne prime exponents with indices x = 10 through 52 (excluding p <= 31), the method produces seven exact matches (a 16.3% success rate, e.g., x = 38, p = 6972593 and x = 52, p = 136279841) and four close approximations (e.g., x = 34, p = 1257787, C(1121) = 1257803), with a mean absolute error of approximately 614 over the range x = 30 to 52. By comparison, an exponential regression model y = 11111.14 e^{0.1787x} captures the overall growth trend (R^2 approx 0.974) but yields no exact matches and a mean absolute error of 10,466,686. Graphical analysis, including scatter plots of C(n_closest) versus actual exponents and absolute deviations d = |n - n_closest|, demonstrates the hypothesis's precision when nearest-integer rounding is applied. From approximately 50 prime values of C(n) identified among 560 unique candidates, five cases with d < 0.1 are selected for targeted GIMPS testing, reducing the effective search space by approximately 74%.
Let $S$ be a numerical semigroup with Frobenius number $f$, genus $g$ and embedding dimension $e$. % In 1978 Wilf asked the question, whether $\frac{f+1-g}{f+1}\geq\frac1e$. As is well known, this holds in the cases $e=2$ and $e=3$. For $e\geq4$, we derive from results of Zhai [5] the following (substantially weaker) lower bound \[\frac{f+1-g}{f+1}>\left(\frac{2N+1}{(2N+2)(e-2)}\right)^e\text{ with }\lfloor N\rfloor=104978\,.\] To the best of our knowledge this is the first explicit lower bound for $\frac{f+1-g}{f+1}$ in terms of the embedding dimension.
We study several aspects of the study of Ind-Banach modules over Banach rings thereby synthesizing some aspects of homological algebra and functional analysis. This includes a study of nuclear modules and of modules which are flat with respect to the projective tensor product. We also study metrizable and Fréchet Ind-Banach modules. We give explicit descriptions of projective limits of Banach rings as ind-objects. We study exactness properties of projective tensor product with respect to kernels and countable products. As applications, we describe a theory of quasi-coherent modules in Banach algebraic geometry. We prove descent theorems for quasi-coherent modules in various analytic and arithmetic contexts.
Let $p$ be a prime. Let $n\in\mathbb N-\{0\}$. Let $\mathcal C$ be an $F^n$-crystal over a locally noetherian $\mathbb F_p$-scheme $S$. Let $(a,b)\in\mathbb N^2$. We show that the reduced locally closed subscheme of $S$ whose points are exactly those $x\in S$ such that $(a,b)$ is a break point of the Newton polygon of the fiber $\mathcal C_x$ of $\mathcal C$ at $x$ is pure in $S$, i.e., it is an affine $S$-scheme. This result refines and reobtains previous results of de Jong--Oort, Vasiu, and Yang. As an application, we show that for all $m\in \mathbb N$ the reduced locally closed subscheme of $S$ whose points are exactly those $x\in S$ for which the $p$-rank of $\mathcal C_x$ is $m$ is pure in $S$; the case $n=1$ was previously obtained by Deligne (unpublished) and the general case $n\ge 1$ refines and reobtains a result of Zink.
In this paper, we give identities involving cyclic sums of regularized multiple zeta values of depth less than $5$. As a corollary, we present two kinds of extensions of Hoffman's theorem for symmetric sums of multiple zeta values for this case.
Let $φ$ be a rank $r$ Drinfeld $\BF_q[T]$-module determined by $φ_T(X) = TX+g_1X^q+...+g_{r-1}X^{q^{r-1}}+X^{q^r}$, where $g_1,...,g_{r-1}$ are algebraically independent over $\BF_q(T)$. Let $N\in\BF_q[T]$ be a polynomial, and $k/\BF_q$ an algebraic extension. We show that the Galois group of $φ_N(X)$ over $k(T,g_1,...,g_{r-1})$ is isomorphic to $\GL_r(\BF_q[T]/N\BF_q[T])$, settling a conjecture of Abhyankar.
The main goal of this paper is to construct non-commutative Hilbert modular symbols. However, we also construct commutative Hilbert modular symbols. Both the commutative and the non-commutative Hilbert modular symbols are generalizations of Manin's classical and non-commutative modular symbols. We prove that many cases of (non-)commutative Hilbert modular symbols are periods in the sense on Kontsevich-Zagier. Hecke operators act naturally on them. Manin defines the non-commutative modilar symbol in terms of iterated path integrals. In order to define non-commutative Hilbert modular symbols, we use a generalization of iterated path integrals to higher dimensions, which we call iterated integrals on membranes. Manin examines similarities between non-commutative modular symbol and multiple zeta values both in terms of infinite series and in terms of iterated path integrals. Here we examine similarities in the formulas for non-commutative Hilbert modular symbol and multiple Dedekind zeta values, recently defined by the author, both in terms of infinite series and in terms of iterated integrals on membranes.
We continue the study of the spectral theory associated to integrable metrics, started in our previous paper arXiv:1301.1793 [math.SP]. We introduce the notion of 1-integrable metric on line-bundles on a compact Riemann surface. We extend the spectral theory of generalized Laplacians to line-bundles equipped with 1-integrable metrics. As an application, we recover the following identity: [ζ'_{Δ_{\bar{\mathcal{O}(m)}_\infty}}(0)=T_g\bigl((\p^1,ω_\infty); \bar{\mathcal{O}(m)}_\infty \bigr),] obtained using direct computations in arXiv:1301.1792 [math.NT].
We define the algebraic fundamental group functor of a reductive group scheme over an arbitrary (non-empty) base scheme and prove that this functor is exact.
In this note, we explore the notion of hyperbolicity of topologically finitely generated profinite groups. Some applications to diophantine geometry are suggested and we try to reformulate certain problems in diophantine geometry in terms of hyperbolic profinite groups. Then, we introduce many occasions in which Galois groups are free profinite and try to explore implications of this condition in the world of diophantine geometry. In particular, we prove that, Grothendieck's "section conjecture" plus Shafarevich's "freeness conjecture" imply that hyperbolic curves have infinitely many solutions over the maximal abelian extension of a global field. This makes Mordell's conjecture, which was proved by Faltings, more interesting.
This paper is devoted to deformation theory of graded Lie algebras over $\Z$ or $\Z_l$ with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artin local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations. In the second part, we use a version of Schlessinger criteria for functors on the Artinian category of nilpotent Lie algebras which is formulated by Pridham, and explore arithmetic deformations using this technique.