Papers

Let $\kappa_{\mathbb{K}}$ be the residue at $1$ of the Dedekind zeta function of a number field $\mathbb{K}$. We prove explicit upper and lower bounds for $\kappa_{\mathbb{K}}$ under the Generalized Riemann Hypothesis (GRH). Previous explicit bounds of comparable generality are weaker and require a weak form of Artin's holomorphy conjecture: our bounds improve the constants while assuming only GRH for the relevant Dedekind zeta functions. The new lower bound enables all imaginary quadratic fields with class number $\leq 2000$ to be detected (assuming GRH). Furthermore, we demonstrate that the upper bound can be significantly reduced when considering extensions of any fixed, nontrivial field.

The second Jones problem (2011) consists in determining whether there exists a convergent positive series for which the ratio of consecutive terms has a limit and the set of subsums is a Cantorval; in the survey by Gł{\k a}b and Prus-Wiśniowski (2026) this problem is included in the list of open problems. In this paper, a family of such series is constructed explicitly as combinations of blocks of the form $\frac{2^k-1}{2^t} + \frac{2^{k-1}}{2^t} + \frac{2^{k-2}}{2^t} + \cdots + \frac{2}{2^t}$, where $t$ is the index of the last term of the block in the series and $k\ge2$ is the length of the block. For each series of the family it is proved that its set of subsums contains an interval and is not a finite union of closed intervals, and hence is a Cantorval. The proof relies on mixed-radix numerical systems with a redundant alphabet and uses an idea of Guthrie and Nymann (1988). If the lengths of the blocks tend to infinity, then the limit of the ratio of consecutive terms of such series is $\tfrac{1}{2}$, which, by a result of Jones, is the only possible value of the limit for which the set of subsums of a series can be a Cantorval. Thus, an affirmative answer to the second Jones problem is obtained.

In his paper, ”Sur la distribution des fractions irréductibles de dénominateur $n$ ou de dénominateur au plus égal à $x$”, Hubert Delange considers the family integrals defined by \begin{equation*} \int_0^1\left(\frac{N(n,t)}{\phi(n)}-t\right)^{2q}dt, \end{equation*} where $n>1$ and $q>0$ are positive integers, $\phi$ is the Euler totient function, and for any real number $t>0$, $N(n,t)$ counts the number reduced fractions whith denominator $n$ that belong to the interval $]0,t]$. He was able to evaluate the integral for $q=1$ but stated that ”Malheureusement, nous n'avons pas su jusqu'à présent faire cette évaluation pour $q>1$”. In addition to its intrinsic interest, this calculation would help understand the asymptotic distribution of the $\phi(n)$ totatives of $n$ relative to $\phi(n)$ equidistributed numbers in $[0,n]$. In this paper, we first provide some general properties of the integral for any $q>1$ and reduce the problem to its calculation for odd square-free integers with at least two distinct prime factors. We then analyse into more detail the case $q=2$ for which we provide a formula that reduces the problem to the evaluation of the fourfold Möbius sum, of integer valued modular sums related to generalized Dedekind sums.

We determine the dynamical degrees of bimeromorphic automorphisms of hyperkähler manifolds of all known deformation types with irreducible characteristic polynomial.

We resolve the long-standing problem of Rudin concerning uniformly bounded orthonormal bases of homogeneous polynomials on complex unit spheres. Specifically, we show that for every $K \geq 2$ and $N \geq 0$ there exists a basis of degree $N$ homogeneous polynomials in $\mathbb{C}^K$ that is orthonormal with respect to the surface measure of the unit sphere of $\mathbb{C}^K$ and uniformly bounded by a constant depending only on $K$ on the unit ball of $\mathbb{C}^K$. Bourgain resolved the case $K = 2$ in 1985 and the case $K = 3$ in 2016. No cases $K \geq 4$ were previously known.

We study the size of sets $\mathcal{B}\subseteq\mathbb{Z}_{\ge0}$ for which $\mathbb{P}+\mathcal{B}$ contains every sufficiently large integer. We obtain $B(X)\gg\log X\log\log X$, where $B(X)=\#(\mathcal{B}\cap[0,X])$ and the implied constant is absolute. This gives a quantitative form of Conjecture 2 of Ruzsa and excludes additive complements of size $O(\log X)$, as asked in one part of Erdős Problem 32. More generally, we construct a block of $\lfloor c\log X\log\log X/(B(X)+1)\rfloor$ consecutive integers outside $\mathbb{P}+\mathcal{B}$ in $[X/2,X]$, whenever this number is positive.

We use the theory of canonical lifts to prove the integral geometric Tate conjecture for ordinary Calabi-Yau threefolds over finite fields, and also for divisor classes on certain ordinary Calabi-Yau varieties.

Euler's constant $\gamma$, also known as the Euler-Mascheroni constant, remains one of the most elusive constants in mathematics. Despite extensive study, its irrationality is still unproven, and Apéry-like approximations of it are remarkably scarce. We present a novel identity that expresses $\gamma$ in terms of a Meijer G-function and its first few derivatives. This identity is then generalized and used to systematically generate infinitely many Apéry-like approximations of $\gamma$ using a Conservative Matrix Field construction based on the Meijer G-function.

The $(0,x,1,2)$-hiccup sequence starts at $x$ and increases by 1 when the current index has already appeared as a value, and by 2 otherwise. We prove directly that for $x\geq3$ the gaps between its missing values are $1,2,3,\ldots$, each repeated $x-2$ times, and we derive closed formulas, an explicit error bound, and a generating function in terms of Ramanujan's theta functions. The counting functions of the complements satisfy nested recurrences. The sequences of missing values are the partial complements of polygonal numbers. In particular, the $(x-2)$-fold composition of the enumeration of the positive integers that are not $x$-gonal is the hiccup sequence minus 1. We extend the nested recurrence and a tree interpretation to every miss increment $z\geq2$ and every initial value $x\geq2$. Further interpretations involve polyomino vertices, monochromatic pairs, and, by a theorem of Bickle, Ramsey core numbers.

Lazard theory is becoming increasingly important in $p$-adic representation theory, especially when studying mod $p$ representations. For $G$ a connected reductive group defined over a finite extension $K/\mathbb Q_p$, a question going back to Lazard is when a pro-$p$ Iwahori subgroup of $G(K)$ admits a saturated $p$-valuation. In this article, we establish a deep connection to Moy-Prasad theory, and show that the Moy-Prasad filtration defines a saturated $p$-valuation on $G_{x,\frac{1}{p-1}+}$ for any point $x$ in the semisimple Bruhat-Tits building of $G$, given a mild assumption on the maximal torus of $G$ and that $p>2$. We also compute Lazard Lie algebras defined over $\mathbb Z_p$ and the polynomial ring $\mathbb F_p[\varepsilon]$ associated to the saturated group $G_{x,\frac{1}{p-1}+}$, which allows us to reinterpret the Moy-Prasad isomorphism as an isomorphism of Lie algebras over $\mathbb F_p[\varepsilon]$.

We find the Dirichlet series and convergence bounds for the counting function of sum systems (collections of sets of non-negative integers whose Minkowski sum is an arithmetic progression). This motivates the definition and study of divisor function polynomials, which are generating functions for the non-trivial divisor function and for the balanced divisor function that counts, up to a sign, non-trivial ordered factorisations into squarefree factors. These divisor function polynomials turn out to be intimately connected between each other and with Bernoulli numbers and ordered Bell polynomials.

We compare three heights attached to a curve of genus two: the weighted height of its Igusa–Clebsch invariants, the GIT height of a binary sextic defining it, and the Néron–Tate height on its Jacobian. For a morphism from an abelian variety to a weighted projective space we define a weighted Néron–Tate height and show that the weighted height differs from it by a linear form and a bounded function; the linear form vanishes when inversion is induced by an automorphism of the weighted projective space. For binary sextics we prove the explicit inequality $h_{\inv}\le h_{\text{GIT}} + \frac{1}{2} \log 302$, with nonnegative local differences at the finite places that vanish at places of good reduction. We then construct a family of semistable binary sextics over $E\times E$, where $E:y^2=x^3-2$, whose relative Jacobian carries a section of infinite order, and we show that the Jacobian over the base point is absolutely simple. Along every curve in the parameter space through the base point, the Néron–Tate height of the section is a positive multiple of the weighted moduli height up to an error of square-root size, and up to a bounded error when the curve is rational. As applications, we obtain explicit families of genus-two Jacobians of positive rank over $\mathbf{Q}$, an infinite congruence lattice of fibres of positive height, and canonical-limit formulas along elliptic curves in $E\times E$.

Let $r_1,\ldots,r_t$ be positive integers, and let $\mathcal A$ be the set of integers of the form $2^{k_1^{r_1}}+\cdots+2^{k_t^{r_t}}$, where $k_1,\ldots,k_t$ are positive integers. For any positive integer $n$, let $r(n)$ be the number of representations $n=p+a$ with $p$ an odd prime and $a\in \mathcal A$. We prove that if $r_1^{-1}+\cdots+r_t^{-1}=1$, then, for every fixed positive integer $m$, the set of integers $n$ with $r(n)\ge m$ has positive lower asymptotic density. In particular, this confirms a conjecture of Chen.

We apply the generalized monoid of words and logarithmic-prefix functionals of \cite{C1} to the regularization of the divergent product \[ P_{\omega}=\prod_{n=1}^{\infty} n^{(-1)^{\omega(n)}}, \] where \(\omega(n)\) is the number of distinct prime divisors of \(n\). The associated Dirichlet series \(D_{\omega}(s)=\sum_{n\ge1}(-1)^{\omega(n)}n^{-s}\) has a natural boundary at \(\Re(s)=0\), as established by Meretov (1987) and recovered here as a special case of the Estermann–Dahlquist theory of Euler products, which makes zeta regularization impossible: the derivative \(D_{\omega}'(0)\) does not exist. This product therefore lies completely beyond the reach of classical regularization methods. We construct the canonical net encoding the sign sequence, define the logarithmic-prefix functional with exponent \(\alpha=2\), and extract a finite value through the weighted evaluation functional of the monoid framework. The evaluation functional is shown to converge absolutely, and this convergence is unconditional: it follows from the classical zero-free region of \(\zeta(s)\) (de la Vallée Poussin), which yields the effective bound \(T_{\omega}(x)=O(x\,e^{-c\sqrt{\log x}})\) for some \(c>0\). Numerical computation up to \(N=10^7\) gives the regularized value \[ P_{\omega,\mathrm{ren}}\approx 0.7731, \] a new constant of arithmetic origin that cannot be obtained by any classical summation method. We also compute the second derivative of the Dirichlet series at \(s=1\), \[ D_{\omega}''(1)=8\,C_2\ln 2, \] where \(C_2\) is the Hardy–Littlewood twin prime constant, which provides an unconditional analytical link to the arithmetic structure of the sequence \((-1)^{\omega(n)}\). A self-contained proof of the natural boundary, based on the Estermann–Dahlquist theorem, is given in Appendix~A.

We place the Appell shift theorem for asymptotic expansions inside a Borel–Laplace framework and apply it to two classical functions of analytic number theory: the Hurwitz zeta function \(\zeta(m,q)\) and the Lerch transcendent \(\Phi(z,m,a)\). The resulting shifted expansions have coefficients given by Appell polynomials evaluated at the shift, with remainders controlled uniformly on compact sets. The generating sequences are identified explicitly in terms of the Taylor coefficients of the Borel kernels. In a second direction, we extend the Appell–Mellin framework of Navas, Ruiz and Varona to Dirichlet \(L\)-functions. For a primitive non-principal character \(\chi\) modulo \(q\), we construct an explicit Appell–Mellin generating function \(A_\chi(t)\) whose associated Appell sequence satisfies \(R_n^\chi(x) = L(-n,\chi,x)\), and we prove the functional equation \(A_\chi(t) = \chi(-1)A_\chi(-t)\). We further establish five structural properties of the polynomials \(R_n^\chi\): identification with generalized Bernoulli polynomials, a difference equation, an explicit formula, a Fourier expansion, and the trivial zeros of \(L(s,\chi)\). To the best of our knowledge, these results do not appear in the literature.

For every smooth projective generalised Markoff surface $X_m$, every hyperplane section $H$, and every integer $n\ge2$, we establish Campana strong approximation with the Brauer–Manin obstruction for $\bigl(X_m,(1-1/n)H\bigr)$.

We consider the question which functions $f$ may appear as limiting functions of the $N$-point correlation. We obtain sharp point-wise lower and upper bounds. The explicit examples constructed to show the sharpness of the bounds are based on van der Corput sequences.

We give a short autonomous recurrence that generates every odd prime, in increasing order, from the single initial value $8$. Its integer state determines both its output and its next update: the lowest set bit gives the next prime, while the highest set bit determines the sieve period. The same state also determines the following prime gap. A related recurrence represents the prime-counting function as the $2$-adic valuation of a single integer. Independently, a direct packing of a finite divisor sieve yields fixed arithmetic constructions for the $n$-th prime, the prime-counting function, and the twin-prime counting function. We express these constructions as straight-line programs using addition, truncated subtraction, multiplication, integer division, exponentiation, and remainder. Counting operations with reuse of computed values gives programs of lengths $166$, $132$, and $167$, respectively; one autonomous recurrence update uses $99$ operations. Explicit arithmetic subroutines remove all auxiliary operations, including bitwise AND. The programs unfold into fixed arithmetic terms without variable-length sums, products, or searches.

In this chapter, we discuss the interaction between low-dimensional dynamics, symbolic dynamics, and geometric topology, focusing on the algebraic characterization and asymptotic behaviour of stretch factors or equivalently pseudo-Anosov dilatations. We first review the algebraic foundations linking Perron–Frobenius theory, multi-variable Euclidean algorithms, and Lind's characterization of topological entropy for shifts of finite type, emphasizing recent constraints on the Perron–Frobenius degree of expansion constants. We then relate these dynamical invariants to the combinatorial world, examining train tracks for surfaces and automorphisms of free groups. Finally, we briefly review the relevance of fibered 3-manifolds, detailing how the Thurston norm, Fried's concavity theorem over fibered cones, and McMullen's Teichmüller polynomial come into play.

We consider Hecke modules of orthogonal modular forms attached to definite ternary lattices of half-discriminant $p^2$ with $p$ prime and $4p^2$ with $p$ an odd prime. Where possible, we prove explicit Jacquet-Langlands style correspondences with spaces of classical modular forms (in the spirit of Birch and Hein-Tornaria-Voight), and give conjectures for the remaining genera. This is done by applying sequences of Watson-style transformations that reduce the discriminant while preserving the adelic orthogonal group.

We prove that any subset $A \subseteq \{1, \dots, N\}$ lacking non-trivial polynomial progressions $(x, x + y, x + y^2)$ satisfies $\lvert A \rvert \le CN \exp\bigl(-c(\log \log N)^\beta\bigr)$ for effective absolute constants $C, c, \beta > 0$. The argument develops a density increment framework localized to anisotropic quadratic Bohr sets. To handle mixed linear and quadratic frequencies, we establish a multilinear transference principle coupled with a relative quadratic inverse theorem based on Balog–Szemerédi–Gowers additive energy rigidity. Boundary stability across increments is maintained via semi-algebraic model regularization using cylindrical algebraic decomposition and Cauchy–Crofton estimates, while dimension growth is controlled by dissociated basis reduction and multidimensional Erdős–Turán–Koksma equidistribution.

Heyer's left adjoint to derived parabolic induction gives rise to higher mod $p$ Satake transforms \[ \mathcal S_V^i: \mathcal H_G(V) \longrightarrow \mathcal H_M\bigl(H^i(K_U,V)\bigr), \] whose extreme degrees recover Herzig's invariant and coinvariant transforms. For $G=\mathbf G(F)$, where $\mathbf G$ is a connected reductive group defined over a finite extension $F/\mathbb{Q}_p$, we describe these maps explicitly using Mackey decompositions and corestriction in group cohomology. For $G=\mathrm{GL}_n(F)$, we evaluate them on the standard Hecke generators, showing that their support lies in a single Weyl orbit and that their values on the individual support cosets are concrete corestriction operators. For $G=\mathrm{GL}_2(F)$, we determine the image in every cohomological degree for each non-Steinberg Serre weight. The endpoint images are the classical polynomial Satake images, while every interior image is of the form \[ k[x,y^{\pm1}]/(x^r) \] for some positive integer $r$. Thus, every interior image is a finite nilpotent thickening of the Laurent polynomial algebra $k[y^{\pm1}]$. We give explicit bounds for its thickness, prove that they are sharp, and show that in the unramified case every interior image is either reduced or a square-zero thickening.

We investigate restrictions on the dimensions and coordinate fields of real Weyl-Heisenberg fiducial vectors. Our main result is that such vectors can exist only in dimensions $d\equiv 3\pmod{4}$. This result is motivated by evidence showing that real Weyl-Heisenberg fiducial vectors can exist only in dimensions $d = 4m^2 + 3, m \in \mathbb{Z}_{\geq 0}$. We further restrict the possible dimensions when the coordinate field is a totally real number field, an even or odd degree extension of $\mathbb{Q}$, a real quadratic field, or a maximal real subfield of a cyclotomic field. Lastly, we establish non-existence over odd-degree number fields under either positivity of all non-trivial cyclic-shift overlaps or a conjectural containment of a distinguished quadratic field in the SIC field.

After removing suitable thin subsets, we count rational points of bounded height on some hypersurfaces of small bidegree in $\mathbb{P}^3\times\mathbb{P}^3$ and in $\mathbb{P}^3\times\mathbb{P}^2$. One of the ingredients is a new estimate for the number of rational points of bounded height on diagonal quadratic surfaces which improves as the coefficients grow.

We investigate generalized Stieltjes constants attached to Hurwitz-type Dirichlet series of the form $$ L(s, f, x)=\sum_{n \geq 0} \frac{f(n)}{(n+x)^s} $$ where $f$ is a periodic arithmetic function and $x\in\mathbb C\backslash\{0,-1,-2,\dots\}.$ We prove that these series admit meromorphic continuation to the whole complex plane, with at most a simple pole at $s=1$. We further derive functional equations satisfied by $L(s,f,x).$ Our main result gives an explicit formulas of the Stieltjes constants $\gamma_n(f, x)$ appearing in the Laurent expansion at $s=1$. In particular, we obtain explicit evaluations in the case of rational $x$. As an application, we derive explicit closed-form expressions for the first three generalized Stieltjes constants associated with nonprincipal real primitive odd Dirichlet characters arising from quadratic fields. These expressions are given in terms of the class number of the corresponding quadratic number field. These results extend and unify recent work of Blagouchine and others on generalized Stieltjes constants and Hurwitz-type $L$-functions.

Following the celebrated breakthrough in Bénard-He-Zhang, we prove Khintchine dichotomy for push-forwards of self-similar measures on $\mathbb{R}^d$ under uniform $C^{1+\alpha}$ maps. For this, we also study almost random walks on $\mathrm{SL}(d+1,\mathbb{R})/\mathrm{SL}(d+1,\mathbb{Z})$ and prove their effective equidistribution.

We prove that, for every fixed $d\geq 2$ and $\varepsilon>0$, the $\mathbb{F}_q$-points of a geometrically irreducible reflexive plane curve of degree $d$ over $\mathbb{F}_q$ can be covered by at most $\left(\frac{1}{d}+\varepsilon\right)q$ many $\mathbb{F}_q$-lines, provided that $q$ is sufficiently large. The same conclusion holds for $\mathbb{F}_q$-irreducible curves that are not geometrically irreducible, since they have only $O(d^2)$ $\mathbb{F}_q$-points. In particular, for sufficiently large $q$, the bound applies to every $\mathbb{F}_q$-irreducible plane curve of degree $d$ when $\operatorname{char}(\mathbb{F}_q)>d$. Our bound is asymptotically sharp. The proof combines Entin's Chebotarev density theorem for varieties over finite fields with the Pippenger–Spencer theorem on covers of nearly regular hypergraphs.

Let $p$ be a prime number. In 1986, David Lampert asked if one expands a $\mathbf{Q}_p$-algebraic number as a $p$-adic Hahn series, then what is the possible order type of its support? In 2001, Kedlaya made a vague conjecture that the possible ordinals can only be finite ordinals, $\omega$ and $\omega^\omega$, where $\omega$ is the first infinite ordinal. In this article, we solve Lampert's problem by confirming Kedlaya's intuition. Moreover, we completely characterize the $\mathbf{Q}_p$-algebraic numbers with order type of support at most $\omega$. As an application, we determine the bi-algebraic set of the $p$-adic shadow map.

We study full-level Hilbert Hecke eigenform product identities over totally real fields of degree at least two, allowing arbitrary compatible narrow class characters. Combining our results with previous work, we classify products of two Eisenstein eigenforms of positive parallel integral weights and products of an Eisenstein eigenform of weight at least two with a cuspidal eigenform of positive parallel integral weight. The unique identity in each of these two classes is given by $E_{4}=60 E_{2}^{2}$ and $h_{8}=120 E_{2} \cdot h_{6}$, respectively, both over $\mathbb{Q}(\sqrt{5})$. The proofs use constant-term comparisons, indecomposable Fourier exponents, integrality of reciprocal Eisenstein constant terms, and a Rankin-Selberg formula for Petersson inner products. In the remaining case of a weight-one Eisenstein factor and a cuspidal factor, we also establish $h_5=12E_1(1,\varepsilon)\cdot h_4$ over $\mathbb Q(\sqrt3)$, where $\varepsilon$ is the nontrivial narrow class character. All eigenforms in these identities have unit-ideal coefficient one.

For a non-singular quadratic form $F$ in four variables, we study integer solutions to $F=0$ with an arbitrary weight attached to one of the variables. This allows us to investigate genuinely sparse subsets of the corresponding variety. Our main input is to obtain cancellation in sums of character sums over the moduli arising in the application of the DFI's delta method to the quadratic hypersurface. This cancellation relies crucially on the equidistribution of quadratic roots and the Weil bound for Kloosterman sums.