Papers

Let $p_n$ be the $n$th prime, and let $r,s\ge2$ be fixed multiplicatively independent integers. We count the integers up to $x$ whose prime factors all have the form $p_{r^a s^b}$ with integers $a,b\ge0$. Our asymptotic formula for this count has relative error $o(1)$ and is explicit down to the multiplicative constant. For $(r,s)=(2,3)$ these integers are the prime codes of the ordinals below $\omega^{\omega^2}$, so the formula settles that case of the counting problem of Vernaeve, Vindas and Weiermann.

We prove that the twisted endoscopic character identities between triality $\mathrm{PGSO}_8$ and the endoscopic group $\mathrm{G}_2$ imply the Hiraga-Ichino-Ikeda conjecture on formal degrees for $\mathrm{G}_2$. In the course of proving the main result, we also establish some fundamental results on representation theory of $\mathrm{PGSO}_8$ with triality. A key feature of our approach is an application of the adjoint group of type $E_6$.

We prove that there is an absolute constant $c>0$ such that for every finite $(\alpha,M)$-Ahlfors–David regular set $A\subseteq[N]$ with $0\leq\alpha<1$ and $M\geq2$, we have the sumset estimate \[|A+A|\geq |A|^{1+c(1-\alpha)/\log M}.\] We also prove a corresponding statement for Ahlfors–David regular sets in $[0,1]$. The proof combines a multiscale entropy decomposition with inverse results from additive combinatorics, showing that Ahlfors–David regularity forces a definite entropy gain at each scale. The dependence $1/\log M$ in the exponent is of optimal order.

We solve the sup-norm problem for minimal weight vectors in an arbitrary cuspidal representation $\pi$ of $\mathrm{GL}(n,\mathbb{Z})\backslash\mathrm{GL}(n,\mathbb{R})$ with a uniform power saving bound in terms of the archimedean data of $\pi$, including its spectral parameters and the dimension of its minimal $K$-type. As a key ingredient, we establish new uniform decay bounds for generalized spherical functions.

Let $q$ be a prime power and $\mathbb{F}_{q^2}$ be the finite fields of $q^2$ elements. The Hermitian function field $H=\mathbb{F}_{q^2}(x,y)$ defined by $y^q+y=x^{q+1}$ is a well-known maximal function field with the largest possible genus. Let $A(P_\infty)$ be the decomposition group of the infinity place $P_\infty$ of $H$ which is the common pole of $x$ and $y$. For every subgroup $G\le A(P_\infty)$, we construct explicit generators of Galois subfield $H^G$ of $H$ with respect to $G$ and determine an absolutely irreducible equation defining the smooth affine plane model for such a Galois subfield. For $p$-subgroups, the generators can be chosen so that the defining equation has an additive polynomial on the left-hand side and an $\mathbb{F}_p$-quadratic polynomial on the right-hand side. For $q=27$, we can construct a genus-two subfield $D\subset H$ that is not isomorphic to $H^J$ for any subgroup $J\le \text{Aut}(H)$ from the explicit equations of Galois subfields of the Hermitian function field. To the best of our knowledge, this is the first example of a maximal function field covered but not Galois-covered by the same Hermitian function field.

We study the $\theta$-congruent number problem for $\cos\theta=\pm3/5$ and $\pm4/5$ using the generalized theta series construction of Sirolli–Tornaría. We describe its specialization to newforms of weight $2$ over $\mathbb Q$ with nontrivial square-free odd part of the level, and explain the reduction of quadratic twists to odd fundamental discriminants. The same construction gives an effective procedure for every $\theta$-congruent number problem with nonzero rational cosine. For the four angles, we construct explicit forms of weight $3/2$ whose Fourier coefficients determine the central $L$-values of the associated elliptic curves. This gives Tunnell-type criteria for every positive square-free integer: a nonzero coefficient implies non-$\theta$-congruence unconditionally, and the converse holds assuming the Birch–Swinnerton-Dyer conjecture. We also prove unconditional non-$\theta$-congruence for primes in explicit arithmetic progressions.

We prove the André–Oort conjecture for hypersurfaces $V \subset Y(1)^n \cong \mathbb{A}^n_\mathbb{C}$ defined by an equation $a_1 x_1^m + \ldots + a_n x_n^m = b$, where $a_1, \ldots, a_n, b \in \overline{\mathbb{Q}}$ and $m \in \mathbb{Z}_{>0}$. Unlike previous proofs, our result is both effective and uniform in the height of the coefficients $a_1, \ldots, a_n, b$. This is the first effective proof of a uniform André–Oort statement for a class of subvarieties with arbitrary dimension and non-empty special locus. We also prove an analogous result for hypersurfaces $V \subset Y(1)^n \times \mathbb{G}_m^l$.

Taming the exponential increase of the Hilbert space dimension with system size in the simulation of gapped quantum lattice models is of the highest relevance for understanding and designing exotic quantum materials, where nonlocal interactions are of particular interest. High-order linked-cluster expansions provide access the solution of the eigenvalue problem for the infinite system, yet rely on the computation of high-dimensional oscillatory lattice sums with a graph structure, only approachable with Monte Carlo methods so far. This work resolves this issue, rendering all required graph lattice sums, referred to as graph zeta functions for kernels involving power-laws, computable. The resulting method reduces the evaluation time for state-of-the art series expansions from tenthousands of core-hours to minutes. After factorizing the lattice sum over blocks, each block is evaluated by the cheapest available strategy depending on its treewidth $\mathrm{tw}$. Basic blocks admit analytic forms in terms of generalized zeta functions. Series-parallel blocks with $\mathrm{tw}\le 2$ can be computed at linear cost in the number of graph nodes and in the size of the momentum grid using a semi-analytical algebra based on Epstein zeta functions and rapidly decaying Fourier series. Finally, for $\mathrm{tw}>2$, the method is combined with tensor-network bucket elimination yielding polynomial scaling of numerical work and memory in momentum grid size with exponents only growing with $\mathrm{tw}$ rather than with the number of vertices. Through use of FFT, the full momentum grid is recovered at the cost of a single momentum evaluation. We provide a detailed analysis of the precision and runtime of our method against analytic and numerical benchmarks. We further reproduce published Monte Carlo data for the transverse-field Ising model on different 1D, 2D, and 3D lattices, obtaining full agreement.

Browning and Sawin conjectured that random hypersurfaces with sign coefficients are smooth with probability tending to one as the degree grows. We prove this conjecture and obtain a quantitative bound. For each $n\geq1$, a degree $d$ form in $n+1$ variables, with independent uniform coefficients in $\{-1,1\}$, defines a singular complex hypersurface with probability $O_n(d^{-1/2})$. The positive-dimensional singular loci occur with exponentially small probability. For $n\geq3$, the same exponential bound holds for failure of absolute irreducibility. These results have been formalized in Lean by AxiomProver assuming existing literature.

In this paper, we study multiple Eisenstein series (MES) in positive characteristic. By computing and analyzing the $t$-expansions of MES, we determine the precise "weights" of their coefficients. This framework enables us to establish a graded algebra structure for MES, extending the direct sum result for Thakur's multiple zeta values proved in [Cha14]. Our result may also be viewed as a function field analogue of the corresponding direct sum result of Bachmann and Kanno [BK26].

Let $(G,X)$ be a Shimura datum of abelian type satisfying the hypotheses of the main theorem, and suppose that the associated Shimura varieties have hyperspecial good reduction at $p$. Their special fibres are stratified by the $\sigma$-conjugacy classes $b\in B(G_{\mathbb{Q}_p},\mu_h^{-1})$. For an individual Newton stratum, this paper constructs a stabilized formula for the alternating traces of Frobenius–Hecke correspondences on its compactly supported $\ell$-adic cohomology, with coefficients in an $\ell$-adic local system. The formula, obtained by modifying the Langlands–Kottwitz method, expresses these Lefschetz numbers as elliptic stable geometric distributions on endoscopic groups of $G$, placing the cohomological traces in a form suitable for comparison with automorphic spectral data. We also give a group-theoretic criterion determining which elliptic endoscopic groups can contribute to a given Newton stratum. For certain unitary Shimura varieties, we exhibit an intermediate Newton stratum with no non-trivial endoscopic contribution, although there are non-trivial endoscopic contributions in the cohomology of the ambient Shimura variety.

Euler-Kronecker constants are analogues of the Euler-Mascheroni constant for number fields. Assuming the Generalized Riemann Hypothesis for Hecke $L$-functions, we obtain extreme values of Euler-Kronecker constants of abelian cubic fields. We also prove a similar result for quadratic fields.

Complementing an earlier paper, which focused on $3$-sumfree sequences, we here consider only $2$-sumfree sequences: starting with positive integers $f$ and $g>f$, the infinite, increasing 2-sumfree sequence $S_{f,g}$ is constructed as follows. After any initial segment, the next entry is the smallest positive integer exceeding all previous ones that differs from all sums of distinct pairs in the sequence. It follows from a theorem in the previous paper that for every $f\geq 1$ and all $f+1\leq g<2f$ the sequence $S_{f,g}$ exhibits ultimately periodic behaviour. In this paper we state precise conjectures that, if true, would imply that every $2$-sumfree sequence is ultimately periodic. Here periodicity of an increasing sequence is understood to mean periodicity of the sequence of first differences, or, equivalently, of its characteristic sequence. We supply much computational evidence to support the conjectures.

We develop explicit prime-recovery formulas from the values $\Phi_n(2)$ of cyclotomic polynomials. Binary divisibility patterns detect repeated prime factors and identify the least prime divisor of a squarefree index, while small corrections to $\log_2\Phi_n(2)$ allow successive recovery of the distinct prime factors. Specializing the index gives identities for prime products and the least prime above a given integer. The same mechanism extends from finite factorizations to an infinite prime sequence: a normalized limit of cyclotomic values along the odd primorials defines a real constant $\Omega=0.25061403238015047218\ldots$, from which every odd prime can be recovered by a recursive rounding rule.

Parczyk and Spiegel initiated the study of an anti-Ramsey multiplicity variant of Schur's theorem and proved that the maximum fraction of Schur triples that can be rainbow in a $3$-coloring of $\{ 1, \dots ,n \}$ is bounded asymptotically between $0.4$ and $0.66364$. Furthermore, they conjectured that their lower bound is optimal. We disprove this conjecture and prove new bounds. In particular, we show that the maximum fraction of rainbow Schur triples that can be rainbow in a $3$-coloring of $\{ 1, \dots ,n \}$ lies between $9/22$ and $8/15$ asymptotically. Moreover, we study the problem in the general $k$-color setting and establish new non-trivial bounds.

In this paper, we establish a new $\Omega$-result for the values $|L(\sigma,\chi_P)|$ of quadratic Dirichlet $L$-functions at and near the central point, where \[ \sigma=\frac12+\frac{\kappa}{\ln g}, \qquad \kappa\geq 0, \] and $P$ ranges over monic irreducible polynomials associated with hyperelliptic curves of genus $g$ over a fixed finite field $\mathbb{F}_q$. We consider the asymptotic setting in which $q$ is fixed and $g\to\infty$. More precisely, for every $\epsilon \in (0,1/2)$ andevery fixed $\kappa\geq 0$, we prove that \[ \max_{P \in \mathcal{P}_{2g+1}} |L(\sigma,\chi_P)| \gg \exp \left(\left(e^{-\kappa} \ln q \,\sqrt{\frac{\sqrt{q}+1}{\sqrt{q}-1}(1/2-\epsilon)} +o(1)\right)\sqrt{\frac{g\ln_2 g}{\ln g}} \right), \] where $\mathcal{P}_{2g+1}$ is the set of all monic irreducible polynomials of degree $2g+1$ in $\mathbb{F}_q[t]$. In particular, when $\kappa=0$, this gives the corresponding result at the central point $\sigma=1/2$. Our result extends the recent work of Darbar and Maiti (2024) and yields an improved lower bound for the extreme values in this family. In addition, for $1/2<\sigma<1$ and sufficiently large $n$, we study the extreme values of $L(\sigma,\chi_P)$, where $P\in\mathcal{P}_n$, and obtain an improved lower bound compared with the result of Lumley (2021).

Let $m$ be a non-negative integer and $f(x)$ a positive, non-decreasing function satisfying certain conditions. We give an explicit lower bound for the number of integers $n\leq x$ such that $\#([n, n+f(n)] \cap \mathbb{P})=m$ for sufficiently large $x$, where $\mathbb{P}$ denotes the set of prime numbers. This work extends the results of Mastrostefano and of Freiberg, and also makes the lower bound of Masrtrostefano explicit. In addition, we show that if the interval length is sufficiently large, then there exist infinitely many bounded intervals of the same length that contain exactly a prescribed number of primes.

In this paper, for $N=8,12,16,20,32$, we determine the explicit structure of the graded algebra of modular forms for $\varGamma_0(N)$ of both integral and half-integral weights, with all quadratic Dirichlet characters modulo $N$. We give explicit eta quotient generators and determine the relations among them. Using these structures, for a given weight, a level among those considered above, and a quadratic character, we obtain a basis for the corresponding space of modular forms consisting of eta quotients. Using these eta quotient bases, we obtain necessary and sufficient conditions for the periodicity of the signs of the Fourier coefficients of modular forms in three specific spaces. Finally, we determine all eta quotients in these spaces whose Fourier coefficient signs are periodic.

The Kaneko-Zagier conjecture predicts a correspondence between finite and symmetric multiple zeta values. Under this correspondence, $\zeta(3)$ corresponds to an element $Z(3)$ defined by Bernoulli numbers. We prove a conjecture of Tasaka relating $Z(3)$ to the quotient of two solutions of a recurrence. A two-index $q$-recurrence connects this quotient to a finite harmonic $q$-series. Using a method of the author, Takeyama, and Tasaka, we obtain the algebraic and analytic limits $3Z(3)/4$ and $3\zeta(3)/4$ at roots of unity.

We study the nonvanishing of central values of ray class $L$-functions over a fixed imaginary quadratic field $K$. We prove that, as $\mathrm{N}(\mathfrak f)$ tends to infinity, at least a proportion $1/3-\mathsf{o}_K(1)$ of the primitive ray class characters $\chi$ modulo $\mathfrak f$ satisfy $L(1/2,\chi)\neq 0$. Our proof adapts the classical mollifier method to the ray class setting and establishes asymptotic formulas for the first and second mollified moments.

The Gál–Koksma lemma is a standard tool for converting quadratic-mean estimates on consecutive blocks into almost-everywhere bounds for partial sums, without assumptions of independence, mixing, or orthogonality. A natural open problem is to determine exactly which universal growth normalizations are forced by this hypothesis alone. The corresponding universal normalization problem under the abstract consecutive-block second-moment hypothesis is resolved by characterizing exactly which non-decreasing normalizations are valid uniformly over the entire admissible class. The resulting necessary-and-sufficient summability criterion is sharp even for bounded exactly centred systems with constant majorants and exact linear block variance, and determines the critical logarithmic and iterated-logarithmic thresholds.

Let $f_n$ be the $n$-th Fibonacci number with $f_1=f_2=1$. Recently exact formulas for the integer parts of the tails of inverse reciprocal Fibonacci numbers have been obtained in several cases. However, the cubic case ($s=3$) is much more complicated because of highly oscillating error terms. Thus it is difficult to construct a precise continuous approximation and algebraic estimates simultaneously. In this paper, we give a complete and unified algebraic method to solve this difficulty. More precisely we construct an explicit closed form of sequence $g_n$, preserving the principal part $f_n^3-f_{n-1}^3$, such that $\ds \lim_{n\rightarrow\infty}\left\{ \left( \sum^\infty_{k=n}\frac{1}{f_k^3} \right)^{-1}-g_n \right\}=0. $ By decomposing the error terms and investigating the algebraic identities, we establish the lower and upper bounds $ \ds g_n<\left( \sum^\infty_{k=n}\frac{1}{f_k^3} \right)^{-1}<g_n+2/f_n $ for sufficiently large $n$. As an application of the explicit form of the sequence $g_n$ and these estimates, we completely determine the exact value of the floor function for $s=3$.

We study the geometry of Shimura varieties associated with a Hermitian form of signature $(p,q)$ over an imaginary quadratic field $E$, where $2\leq p\leq q$. We prove that when $p\geq w_E$ and $(p,q)\neq(2,2),(2,3)$, where $w_E:=\#\mathscr{O}_E^\times$, there exists a toroidal compactification with at worst canonical singularities. As an application, combining this singularity analysis with Arthur's multiplicity formula, we prove that only finitely many pairs $(p,q)$ satisfying the above conditions and $p+q\equiv1\pmod{w_E}$ give rise to unitary Shimura varieties that are not of general type. Our method also improves the singularity bound of Gritsenko–Hulek–Sankaran (Invent.\ Math., 2007) for $\mathrm{O}^+(2,n)$ to the range $n\geq6$ and shows that this bound is sharp.

We prove that the Enots Wolley sequence contains every positive integer with at least two distinct prime divisors. Suppose, toward a contradiction, that some eligible integer is omitted, and consider its finite set of prime divisors. The local rules then severely restrict how terms involving these primes can occur: after a finite initial segment, terms divisible by some but not all of them can outnumber terms divisible by all of them by at most a fixed constant. A prime-exchange construction gives the opposite conclusion at large scales. From almost every term divisible by all of the chosen primes, it produces enough smaller earlier terms divisible by only some of them; a weighted double count makes this excess quantitative and yields a contradiction. It follows that any omission would force every sufficiently late term to have a prime divisor in one fixed finite set. Prime recurrence and a disjoint-cover argument rule out such a finite obstruction, proving surjectivity. The only analytic number-theoretic inputs are the prime number theorem and Mertens' estimate for reciprocal primes.

For an algebraically closed field $k$ of characteristic $p>0$, we study unramified $\mathbf{Z}_p$-towers $\{X_n\}_{n\ge 0}$ of smooth, projective and connected curves over $k$. Our main result provides an Iwasawa-theoretic description of the Newton polygons of their Jacobians, establishing that the slopes of the new part at level $n$ are eventually affine functions of $p^{-n}$ with rational coefficients.

We extend the discrete complex complement quotient (DCCQ) framework from binary Bernoulli counts to multinomial count compositions. For m+1 categories, m is the number of independent probability degrees of freedom. Integer count vectors modulo common scaling determine rational points of the m-dimensional probability simplex. Building on standard simplex and log-ratio coordinate geometry, for m >= 2 we define the full multinomial DCCQ coordinate map and show that it is a real-analytic diffeomorphism The previously established binary baseline m=1 gives a critical-line coordinate, while the ternary case m=2 gives the full open critical strip; higher multinomial models retain m-2 additional real contrasts. We also give a one-versus-rest specialization, an exact integer-lattice realization of the ternary coordinate, and a hyperbolic representation of its log-ratio. No zero-location theorem or proof of the Riemann Hypothesis is claimed.

We prove that broad families of finite Dirichlet sums exhibit deterministic concentration of normalized vertical zero density on a single line. The main result gives a general criterion under which the normalized Jessen potentials converge locally uniformly to a piecewise-linear convex function with a single corner. The associated normalized measures describing the distribution of real parts of zeros in vertical mean density then converge weakly to a Dirac mass. Thus, for each fixed truncation, zeros may occupy a nontrivial range of real parts, while asymptotically their normalized vertical density concentrates on one line. The proof combines the finite Bohr lift with a translation-uniform anti-concentration estimate for isolated prime coordinates. We apply the criterion to partial sums of the Riemann zeta function, fixed Dirichlet $L$-functions, and primitive holomorphic Hecke eigenforms of fixed level, trivial Dirichlet character, and without complex multiplication. The concentration line is $\operatorname{Re} s=1/2$ in the normalized setting and becomes $\operatorname{Re} s=k/2$ for the classical Fourier coefficients of a form of weight $k$.

In this paper, we investigate integral solutions satisfying the system of two forms in many variables of different degrees. Let $d_1$ and $d_2$ be natural numbers with $d_2>d_1\geq 2$. Let $F_{i}(\boldsymbol{x}) \ (i=1,2)$ be forms in $n$ variables of degrees $d_i\ (i=1,2)$, respectively. Define $$N(\boldsymbol{F};P):=\#\{\boldsymbol{x}\in [-P,P]^n\cap \mathbb{Z}^n:\ F_{i}(\boldsymbol{x})=0\ (i=1,2)\}.$$ When each dimension of singular loci of $F_1=0$ and $F_2=0$ is small, we obtain a number $n_0:=n_0(\boldsymbol{F})$ such that whenever $n> n_0$ one has the expected asymptotic formula \begin{equation*} N(\boldsymbol{F};P)=c_{\boldsymbol{F}}\cdot P^{n-d_1-d_2}+O(P^{n-d_1-d_2-\delta}),\ \text{for some }\delta>0, \end{equation*} where the constant $c_{\boldsymbol{F}}$ is the product of local densities. We note that this asymptotic formula agrees with the Manin-Peyre conjecture. Compared to the previous work, we lower the admissible threshold $n_0(\boldsymbol{F})$ in most cases, with the exception of case $d_2-d_1=1$. In particular, if $F_1$ and $F_2$ are non-singular forms, then we obtain $$n_0(\boldsymbol{F})=3(d_2-1)2^{d_2-1}+(d_1-1)2^{d_1},$$ provided that $d_2\geq 5d_1$ with $d_1\geq2$. This yields a substantial improvement over the previous bound $n_0(\boldsymbol{F})=(d_1+2)(d_2-1)2^{d_2-1}+d_12^{d_1-1}$. To achieve this, we develop a new differencing argument together with the van der Corput differencing argument, delivering an efficient upper-bound estimate for mean values of exponential sums associated with two forms in many variables of different degrees, when the difference between degrees is sufficiently large. Furthermore, the method described in this paper is flexible enough to apply to forms in many variables of differing degrees in general.

In this paper, we study supersingularity of modular Jacobians $J_0(N)$ and abelian varieties $A_f$ of $\mathrm{GL}_{2}$-type from both the vertical perspective (where the prime $p$ is fixed, and the level $N$ varies) and the horizontal perspective (where the newform $f$ is fixed, and the prime $p$ varies). Vertically, we prove that $J_{0}(N)$ is supersingular modulo a given prime $p$ for only finitely many levels $N$. Horizontally, we show that for sufficiently large $p$, the reduction of $A_f$ modulo $p$ is supersingular if and only if it isogenous to a power of a supersingular elliptic curve.

For a fixed prime $p$, let $\mu_p$ denote the $p$-adic Plancherel measure. Then the first main goal of this paper is to prove effective (and moreover explicit) $\mu_p$-equidistribution of the $p$-th Hecke eigenvalues over the Atkin–Lehner subspaces $S_k^\sigma(N) \subseteq S_k(N)$ and $S_k^{\operatorname{new}, \sigma}(N) \subseteq S_k^{\operatorname{new}}(N)$. We then highlight five applications of this explicit equidistribution result. For the first application, we generalize Kim's vertical analog of the Atkin–Serre conjecture to the Atkin–Lehner setting. For the second application, we obtain explicit bounds on the number of newforms $f \in S_k^{\operatorname{new}, \sigma}(N)$ for which $p$ is extremal over $S_k^{\operatorname{new}, \sigma}(N)$. For the third application, we prove explicit asymptotics for the number of $\mathbb{F}_{p^r}$-points on the modular Jacobian $J_0(N),$ as well as on its factors $J_0^{\operatorname{new}}(N),$ $J_0^\sigma(N),$ and $J_0^{\operatorname{new}, \sigma}(N)$. We also make explicit an asymptotic result of Serre concerning point counts of the modular curves $X_0(N)$. For the fourth application, we generalize lower bounds due to Murty and Sinha on the sizes of large $\mathbb{Q}$-simple factors of $J_0(N)$ to analogous bounds for $J_0^\sigma(N)$. Finally, for the fifth application (the details of which are given in a separate paper), we use our explicit equidistribution result to prove that only finitely many modular Jacobians are supersingular modulo any fixed prime.