Papers

For a prime $p$, a pair $(s,m)\in\mathbb{N}^2$ with $m$ relatively prime to $p$, a homomorphism $\chi:C_m\rightarrow\operatorname{Aut}(C_{p^s})$, and an algebraically closed field $k$ of characteristic $p$, we consider the semidirect product $G=C_{p^s}\rtimes_{\chi} C_m$, the $p$-Sylow subgroup $H\cong C_{p^s}$ of $G$, and a $k[G]$-module $V$. Let $R$ be a complete discrete valuation ring of residue field $k$ and mixed characteristic $(0,p)$ that contains a primitive $p^s$-th root of unity. If $\chi$ is injective, we present two necessary and sufficient criteria for lifting $V$ to an $R[G]$-module $\widetilde{V}$ which is a free $R$-module: (i) when no extra requirement is made on $\widetilde{V}$ and (ii) when we require $\widetilde{V}^H=\{0\}$. The criteria correct several results in the literature and we use them to prove that, if $\chi$ is injective and $G$ acts faithfully on a connected smooth projective curve $X$ over $k$, then, under mild hypotheses satisfied if $X\rightarrow X/G$ is a Harbater–Katz–Gabber cover, the $k[G]$-module $H^0(X,\Omega_X)$ has a lift $\widetilde{V}$ to $R$ with $\widetilde{V}^H=\{0\}$. Also, with $B:=R[\frac{1}{p}]$, we prove the following obstruction when $p$ is odd, $m$ is even, $\chi$ is injective, and $X/C_{p^s}\cong\mathbb{P}^1_k$: if no such lift $\widetilde{V}$ exists with the $B[H]$-module $\widetilde{V}\otimes_R B$ defined over $\mathbb{Q}$, then the action of $G$ on $X$ does not lift to $R$.

For a prime $p\geq 5$, let $\mathscr S_g$ be the moduli space over $\overline{\mathbb F}_p$ of $g$-dimensional principally polarized supersingular abelian varieties. We show that each of the following loci contains an open dense subscheme on which the principally polarized abelian varieties have automorphism group $\{\pm1\}$: (i) certain supersingular Ekedahl–Oort strata when $g$ is even, (ii) the loci in $\mathscr S_g$ with non-supersingular Ekedahl–Oort invariants of positive Coxeter type when $g\geq 3$, and (iii) the locus in $\mathscr S_g$ with $a$-number at least $2$ when $g\geq 4$. Consequently, for $g\geq 4$, the complement in $\mathscr S_g$ of the open locus where the automorphism group is $\{\pm1\}$ has codimension at least $2$. These results confirm Oort's conjecture for $p\geq 5$. We reduce them to statements about affine Deligne–Lusztig varieties for $\operatorname{GSp}_{2g}$ and prove analogues of (ii) for $\operatorname{GL}_{2g}$ and $\operatorname{GSO}_{4m}$.

Motivated by Schoof-Pila-type point-counting algorithms, we study the fields of definition and factorization patterns of Galois orbits of $\ell$-torsion points on the Jacobians of genus-3 hyperelliptic curves over finite fields. We show that the degree of the field of definition of the $\ell$-torsion points can be bounded by $O(\ell^4)$, improving the previously expected $O(\ell^6)$ bound (which reflects the size of the $\ell$-torsion subgroup, of order $\ell^6$ for a genus 3 Jacobian). Moreover, we establish a precise correspondence between the rank of the $\ell$-torsion subgroup and the Galois orbits of $\ell$-torsion divisors. Our approach relies on a detailed analysis of the Jordan decomposition of the Frobenius action on $J$ and its nilpotent part.

Let $R$ be a commutative ring and $f(a_1, \ldots, a_n) = \sum_{i=1}^k r_{i1}^{a_1} \cdots r_{in}^{a_n} m_i$ be a linear-exponential map over an $R$-module $M$. Dong and Shafrir (2026) showed that, when $\ell M = 0$ for some $\ell \in \mathbb{N}_{>0}$, the zero set of $f$ is the intersection of effectively computable $p$-normal sets, where $p$ ranges over the prime divisors of $\ell$. This generalizes an earlier theorem of Derksen and Masser (2012) on the solution set of $S$-unit equations over fields of positive characteristic. The purpose of this paper is twofold. First, we give a shorter proof of Dong and Shafrir's result, using the theorem of Derksen-Masser as a blackbox. Our proof also yields a decomposition of the zero set as a positive Boolean combination of affine transformations of zero sets of linear-exponential equations over fields. Second, we prove a multi-dimensional generalization of the Skolem-Mahler-Lech theorem over rings of finite characteristic. Specifically, we show that the zero set of every $n$-dimensional linear recurrence sequence over an $R$-module $M$ satisfying $\ell M = 0$ is the intersection of effectively computable $p$-normal sets (in $\mathbb{N}^n$), where $p$ ranges over the prime divisors of $\ell$. For example, this gives a decision procedure for whether two classical linear recurrence sequences have a common value over a ring of characteristic $p^a$ or $p^a q^b$, where $p$ and $q$ are primes.

We prove a Berry-Tabor theorem for the repulsive three-particle trigonometric Sutherland Hamiltonian. For fixed total momentum, its relative spectrum is an affine translate of the positive-definite $A_2$-form. For a Diophantine coupling, the desymmetrized spectrum has Poissonian pair correlation. The proof reduces the problem to a finite-congruence and Weyl-sector version of Marklof's theorem on inhomogeneous quadratic forms.

In this article we focus on the solutions of a congruence equation: $\binom{2n}{n}\equiv 2^n \bmod n$. Using the main result of this article and the SageMath software, we improve largely the number of known solutions. Furthermore, we prove that some famous numbers like even perfect numbers and Wieferich primes are connected to solutions of this equation.

Let $q=p^h>7$ be odd, put $m=(q+1)/2$, and suppose that $i=(m-2)/2$ satisfies $\gcd(i,m)=\gcd(i+2,m)=1$. For the $\mathbf{F}_{q^2}$-maximal function field $\mathcal{F}_i=\mathbf{F}_{q^2}(x,y)$ defined by $y^m=x^i(x^2+1)$, Peter Beelen, Maria Montanucci, Jonathan Niemann, and Luciane Quoos showed that the geometric automorphism group contains a subgroup of order $4(q+1)$ and conjectured that its order is exactly $4(q+1)$. We prove this equality by establishing the reverse inequality.

We obtain a new, conceptually simpler, unconditional proof that more than $67.25\%$ of the non-trivial zeros of the Riemann zeta function are simple and on the critical line, and that at least $83.62\%$ of the non-trivial zeros are distinct. Our approach also yields two new unconditional estimates on simple zeros and zeros on the critical line. More precisely, we prove that the proportion of zeros that are simple or lie on the critical line (or both) is at least $88.76\%$, and that the average of the proportions of simple zeros and of zeros on the critical line is at least $83.62\%$. A proof of the bounds for simple zeros on the critical line and for distinct zeros was very recently produced by an internal research version of Claude developed by Anthropic and subsequently verified by two mathematicians at Anthropic, Levent Alpöge and Ralph Furman, whereas our two additional estimates are neither stated nor proved in the Claude paper. The argument produced by Claude is technically intricate, and its main mechanism is not immediately transparent. It combines several ingredients from linear algebra, including a finite-dimensional matrix representation of Weil's Hermitian form and a rank–trace inequality for Hermitian matrices, with a second moment calculation over the zeros using the explicit formula. Our new approach proceeds by replacing the entire finite-dimensional matrix framework by a single Hilbert space inequality, which allows for a direct application of Montgomery's theorem on the pair correlation of zeros of the zeta function, in the unconditional form obtained by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh.

The first paper worked out one family of four lattices in full – $|\det| = 12, 24, 36$ and $72$. This paper generalizes it. We enumerate the 44 symmetrizable rank-three hyperbolic Cartan matrices and their 56 depth-one edits, 100 root bases in all, and compute the obstruction space $S_{5/2}(\rho_L)$ of the 98 within our weight-$5/2$ budget: 10 vacuous, 34 unobstructed, 54 obstructed. The vacuous ten are the lattices Bruinier, Ehlen and Freitag call simple, read in signature $(2,3)$. Of their fifteen, our ten realize five; of the rest, five need more than three generators and cannot be the discriminant form of a rank-three lattice at all, three fail $|\det L| = 2k^2$, and two are simply not root bases. Within the rank-three hyperbolic world the simple lattices are exactly the Feingold-Frenkel neighbours of index $k \le 4$. The discriminant group $L'/L$ carries a finite quadratic form, and the finite group of its isometries acts on the weight-$3/2$ cusp forms for $\rho_L$, the bottom antisymmetric rung. We survey the lattices on which that action is absolutely irreducible of dimension at least two, so that the Petersson pairing there is pinned down up to a single scalar. The condition alone cuts the census to three: $L_4$ of the first paper, and two new ones at $|\det| = 40$ and $88$. The quaternion discriminants that occur are 6, 10 and 22, the three for which the Shimura curve $X^D$ has genus zero. Rigidity uses no quaternion input, so we record it as an observation, not a characterisation. Both invariants of the first paper – the shadow norm $\|\Xi\|^2$ and the Petersson scalar $t$ – were single points there. Three rigid lattices instead of one are where their special faces come off: $\|\Xi\|^2$ generalizes against $L(f,1)$ where the first paper read $L(f,2)$, and $t$ against an elliptic curve's imaginary period where it read $\Gamma(1/3)$. Both numerically, to 26-58 digits.

Let $R$ be a discrete valuation ring. We prove that every proper flat finitely presented commutative $R$-group scheme $P$ fits into an exact sequence \[ 0\longrightarrow E\longrightarrow P\longrightarrow B\longrightarrow0, \] where $E$ is finite flat and $B$ is an abelian scheme. For a quasiabelian model—that is, a proper flat finitely presented $R$-group scheme with abelian generic fibre—we prove a finer structure theorem: its normalization is an abelian scheme, and the model is obtained from it by a pushout involving finite flat group schemes. An analogous abelian quotient exists for proper flat commutative group stacks with finite flat inertia. These structure theorems give explicit quotient presentations for duals. Over an arbitrary base, we show that the dual of a proper flat finitely presented commutative group algebraic space is algebraic, proper, flat, and finitely presented, and biduality holds. When $2$ is invertible, the corresponding result for group stacks proves Brochard's duality conjecture. As an application over a discrete valuation ring, Polishchuk's kernel-algebra Fourier–Mukai argument gives equivalences between the unbounded derived categories of quasi-coherent sheaves on every proper flat finitely presented commutative group scheme and its dual, and between their bounded derived categories of coherent sheaves, with no tameness or residue-characteristic hypothesis. For a quasiabelian model, the dual category is a finite-flat equivariant category on its abelian normalization.

Let $k$ be a number field and let $G$ be a connected linear algebraic group over $k$. We compare three approaches to strong approximation for the classifying stack $BG$ with respect to its full Brauer group: the homogeneous-space method of \cite{DhillonClassifying}, Kottwitz's local–global sequence, and, for reductive groups, Borovoi's localization theorem. Under the natural identification \[ Br(BG)/Br(k)\simeq Pic(G), \] we identify the Kottwitz and Borovoi obstruction maps with Brauer evaluation. The three approaches therefore give the same exact description of the localization image as the projected full Brauer–Manin set. We then study strong approximation with respect to the ordinary unramified Brauer group. We prove \[ Br^{un}(BG)/ Br(k)\simeq Sha^1_{\mathrm{cyc}}(k,\widehat G), \qquad \widehat G=X^*(G_{\bar{k}}), \] and give an exact local criterion for unramified Brauer approximation off a finite set of places. Examples show that this approximation can fail even when a finite place is omitted, and exhibit a torus for which the unramified Brauer group cuts out the global image as a proper subset of the adelic space.

An explicit formula for the prime-counting function $\pi(x)$, usually attributed to Riemann and von Mangoldt, is prominently stated as the equation $\pi(x)=R(x)-\sum_\rho R(x^\rho)$, where the sum runs over all zeros $\rho$ of the Riemann $\zeta$-function, the non-trivial ones being ordered by increasing absolute value of their imaginary parts and counted with multiplicity. This particularly entails the claim that the partial sums over the non-trivial zeros, $\Sigma R_T(x):=\sum_{0<|\Im m(\rho)|\le T} R(x^\rho)$ converge as ${T\to\infty}$. Writing $\Theta:=\sup\{\Re e(\rho):\ \zeta(\rho)=0,\ 0<\Re e(\rho)<1\},$ for what has recently been called “Riemann's constant”, we prove that, for every fixed $x>1$ and every $\theta<\Theta$, the sums $\Sigma R_T(x)$ are not $O(T^\theta)$. As a consequence, $\limsup_{{T\to\infty}}|\Sigma R_T(x)|=\infty$ and $\sum_\rho R(x^\rho)$ diverges. We conclude the paper by showing that an adapted, but simpler strategy also gives the analogous result – and hence divergence – for the contribution of the trivial zeros to $\sum_\rho R(x^\rho)$. In both cases the exponent $\Theta$ is proved to be sharp.

We prove the Lonely Runner Conjecture for fourteen and fifteen runners. Our proof combines a stronger bound on the speed product in a primitive counterexample with exhaustive computations modulo primes. To obtain the bound, we work with a projected lattice basis and use cases of the conjecture with fewer runners to bound partial sums of its squared Gram-Schmidt lengths. For fifteen runners, the product bound derived from the work of Malikiosis, Santos, and Schymura gives a logarithmic threshold of about $810$; ours reduces this to about $415$, making the computation feasible. For fourteen runners, the new bound reduces the required number of primes from $111$ to $61$. The computations start with a complete two-branch covering search, followed by binary lifting. Since $14$ and $15$ are composite, the polynomial argument used in earlier work does not apply directly. We finish the fourteen-runner case by a direct search. For fifteen runners, we use the factorisation $15=3\cdot5$ and a shifting argument when all but a few speeds share a common divisor. The code and certificates are publicly archived.

Recently, the first and third authors initiated a study of arithmetical relationships between Ramanujan's theta functions $\varphi(-q)$ and $\psi(q)$. In this work, we prove eight families of internal congruences modulo arbitrary powers of $3$ and $5$ for infinite series related to the two theta functions. We also show that there exist isomorphisms between the congruence families for $\varphi(-q)$ and $\psi(q)$, which can be realized by the study of congruence multiplicities previously studied by Garvan, Sellers, and the second author. We believe that such equivalences are exclusive, at least on the congruence subgroups $\Gamma_0(6)$ and $\Gamma_0(10)$. In the end, we show how a simple manipulation of function field extensions allows us to predict whether additional isomorphisms to our congruences occur.

Let f be a labeling of the edges of a finite graph G by positive integers, and let the weight of a path be the sum of the labels of its edges. The labeling is a geodesic Leech labeling if the weights of the geodesics are exactly 1, 2, ..., t_gp(G), each occurring once, where t_gp(G) is the geodesic path number of G. Let W_n be the wheel on n vertices, a hub joined to an (n-1)-cycle. Our main result is an upper bound: if n >= 5 and W_n is geodesic Leech, then n <= 40. The proof quantifies, via a finite Fourier kernel, the Sidon-type structure of the spoke labels, in which only the cyclically adjacent pairs are allowed as defects, and closes the last three cases with a six-variable Parseval argument. In the other direction, explicit labelings of W_7, ..., W_13, found by a computer search, answer in the negative a problem of Lakshmanan S. and Manattu, who had found labelings of W_5 and W_6 and expected every W_n with n >= 7 to be a non-geodesic Leech graph. Writing E for the set of n >= 5 for which W_n is geodesic Leech, we obtain {5, 6, ..., 13} is contained in E, which is contained in {5, 6, ..., 40}.

For an integer $d\ge 3$, put $\Delta_d=\min{2^{d-1},d(d-1)}$. Let $a/q$ be reduced, let $P(X)=\frac{a}{q}X^d+\alpha_{d-1}X^{d-1}+\cdots+\alpha_0$, and let $\mathcal{I}$ be an interval of $H\le q$ consecutive integers. We prove $\left|\sum_{n\in\mathcal{I}}e(P(n))\right|\ll_{d,\varepsilon}q^{1/d}H^\varepsilon+H^{1-1/\Delta_d+\varepsilon}$. Consequently, for every prime $p>d$, every degree-$d$ polynomial $P\in\mathbb{F}p[X]$, and every interval $\mathcal{I}$ of $H$ consecutive integers with $p^{1/d}<H<p^{1/(d-1)}$, writing $H^d/p=H^u$, one has $\left|\sum{n\in\mathcal{I}}e_p(P(n))\right|\ll_{d,\varepsilon}H^{1-\min{u/d,1/\Delta_d}+\varepsilon}$. This strictly improves throughout the full natural short-interval window the best generic estimate obtained by combining classical Weyl differencing with the optimal Vinogradov mean value theorem.

For a finite extension of global fields $K/k$, the norm map $N_{K/k} : K^\times \to k^\times$ extends to a map on idèle groups. The Hasse norm principle holds if every element of $k^\times$ that is a norm everywhere locally is also a norm globally. In this paper, we study the statistics of the Hasse norm principle in a setting that is out of reach in the number field context, namely that of $A_4$-quartic extensions. We show that failures of the Hasse norm principle are generally rare for $A_4$-quartic extensions of global function fields $\mathbb{F}_q(t)$. We achieve this by introducing a decorated Hurwitz space parametrising the failures of the Hasse norm principle and then using the Chebotarev density theorem to estimate their frequency.

We study the parity-perturbed Hofstadter $Q$-recursion $$ \widetilde Q(1)=\widetilde Q(2)=1,\qquad \widetilde Q(n)=\widetilde Q(n-\widetilde Q(n-1)) +\widetilde Q(n-\widetilde Q(n-2))+(-1)^n, $$ and the associated two-variable Dirichlet series $$ Z_{\widetilde Q}(s,t)=\sum_{n\ge1}n^{-s}\widetilde Q(n)^{-t}. $$ The estimate $\widetilde Q(n)=n/2+O(n/\sqrt{\log n})$ gives the exact domain of absolute convergence $\operatorname{Re}(s+t)>1$. With $w=s+t$, we separate the universal term $2^t\zeta(w)$ and derive exact transport, frequency-position, and dyadic renormalization identities. The main result concerns $t=-1$. For $E(n)=2\widetilde Q(n)-n$ and $A(X)=\sum_{n\le X}E(n)$, the binary-arch clock yields $$ A(X)=X\log_2X+X\Omega\!\left(\log_2\frac{3X}{32}\right) +O\!\left(\frac{X}{\sqrt{\log X}}\right), $$ where $\Omega$ is an explicit continuous periodic function. This continues the normalized correction to $\operatorname{Re}w>0$ and yields a boundary resonance lattice: a double resonance at $w=0$ and simple resonances at $2\pi i m/\log2$. After subtracting the full-slice order-$X$ skeleton, we analyze the negative-even arch channel. Its companion-forest layers have a weak Gaussian limit, and a canonical subsequence realizes the optimal $n/\sqrt{\log n}$ pointwise scale with an explicit signed constant. The negative-arch mass satisfies $$ A_r=\frac{512}{9\sqrt{2\pi}}\frac{16^r}{\sqrt r} \left(1-\frac{13}{16r}+O(r^{-2})\right). $$ We do not claim a full-slice continuation across $\operatorname{Re}w=0$.

In this paper, we apply the resonator method to exhibit large values at the central point of the families of cubic and quartic Dirichlet $L$-functions of prime moduli under the generalized Riemann hypothesis.

For $n$-dimensional simultaneous best Diophantine approximations in an arbitrary norm induced by an inner product, for $n\geq2$ we prove $q_{k+2^n}\geq q_k+\min\{q_{k+2^{n-1}},2q_{k+1}\}$. This yields $g_n(\alpha):=\liminf_{m\to\infty}(q_m)^{1/m}\geq\varphi^{1/2^{n-1}}, \text{ for } \ \varphi = \dfrac{1 + \sqrt{5}}{2}.$ Consequently, $\displaystyle G(n):=\inf_{\alpha\in\mathbb R^n\setminus\mathbb Q^n}g_n(\alpha)\geq\varphi^{1/2^{n-1}}$ and $\underline{\mathcal D}_n(\alpha)\leq\left\lfloor 2^{n-1}\frac{\log2}{\log\varphi}\right\rfloor+1$, where $\underline{\mathcal D}_n(\alpha)$ is a quantity related to multidimensional generalizations of the three-distance theorem. In particular, $g_2(\alpha)\geq\sqrt\varphi, \, g_3(\alpha)\geq\sqrt[4]{\varphi}, \, \underline{\mathcal D}_2(\alpha)\leq3, \, \underline{\mathcal D}_3(\alpha)\leq6.$

We endow spaces of decorated planar rooted trees with new dendriform and tridendrifrom algebra structures and provide their combinatorial description. We then show that the planar counterparts of Arborified Zeta Values are algebra morphisms for these shuffle and quasi-shuffle products of planar rooted trees. We also prove an arborified version of Hoffman's regularisation relation for Arborified Zeta Values. We conjecture that those give every rational relation between Arborified Zeta Values and show that this conjecture implies the regularised double shuffle conjecture for Multiple Zeta Values.

We study the distribution of square-full numbers under the assumptions of Riemann hypothesis and that the zeros of $\zeta(s)$ are simple. Thus the error term is improved to a considerable extent with an extra main term.

Let $P=\{0,a,b\}$, where $0<a<b$ and $\gcd(a,b)=1$. For a finite set $A\subset\mathbb Z$, let $M_P^+(A)$ count the copies $x,x+ad,x+bd\in A$ with $d>0$, and let $M_P(A)$ count the copies with any $d\ne0$. We prove that every such three-point pattern other than the arithmetic progression $\{0,1,2\}$ satisfies \[ M_P^+(A)\le \frac{99}{400}|A|^2+O(|A|), \qquad M_P(A)\le \frac{13}{28}|A|^2+O_P(|A|). \] For the particular pattern $P=\{0,1,3\}$ – the subject of a question raised by Ganguly and recorded as Problem 24 in Green's list of open problems – we sharpen the bound allowing both signs of the dilation to \[ M_{\{0,1,3\}}(A)\le \frac{47}{122}|A|^2+O(|A|). \]

Let $K$ be a number field containing a primitive $p$-th root of unity $\zeta_p$. Let $f(x)\in K[x]$ be a monic integral polynomial, and let $f_0$ denote its radical. Let $C/K$ be the superelliptic curve defined by $y^p=f(x)$, and let $J$ be its Jacobian variety. The variety $J$ admits multiplication by $\zeta_p$ over $K$; in particular, the endomorphism induced by $\Pi:=1-\zeta_p$ gives an isogeny of $J$ over $K$. Let $\operatorname{Sel}_\Pi(J)$ denote the Selmer group associated to $\Pi$. Under suitable hypotheses, we obtain bounds for $\operatorname{Sel}_\Pi(J)$ in terms of the $p$-torsion subgroup of the class group of $L:=K[x]/(f_0)$. Several examples illustrating the results are also discussed.

We construct an $\mathfrak{sl}_2$-action by derivations on the Lie algebra $\mathfrak B$ of swap invariant alternil bimoulds with the uri bracket of Kühn and Schneps. This Lie algebra plays the role for formal multiple Eisenstein series which Racinet's double shuffle Lie algebra $\mathfrak{dm}_0$ plays for multiple zeta values. The kernel $\mathfrak m$ of the lowering operator is a Lie subalgebra, $\mathfrak B$ is the direct sum of its iterates under the raising operator, and this gives Rankin-Cohen type operators on $\mathfrak m$. We define a Lie subalgebra $\mathfrak d$ of $\mathfrak B$ and show that it is isomorphic to $\mathfrak{dm}_0$ extended by an additional element in weight one.

Consider the polynomial $f=x^N+y^N-z^N$ where $x,\,y$ and $z$ are positive integers and $N \ge 3$ is an integer. By Fermat's Last Theorem, $f$ is never zero so that its reciprocal, $1/f$, has no singularities. We therefore study the finite sum of the reciprocal: $S(m,N)=\sum_{x=1}^m\sum_{y=1}^m\sum_{z=1}^{m}\frac{1}{f}$. The terms $1/f$ can be positive, negative and their magnitude is less than unity. A key observation is that $S(m,N)$ can be split into two convenient parts: a dominant contribution $D(m,N)$ that has a simple analytical expression and a remainder $R(m,N)$ which is more complicated but negligible compared to $D(m,N)$. Therefore, $S(m,N)$ is almost identical to $D(m,N)$. The analytical expression for $D(m,N)$ is $(2\,m-1)\,H_m^{(N)}$ where $H_m^{(N)}=\sum_{x=1}^m\frac{1}{x^N}$ approaches quickly the Riemann zeta function $\zeta(N)$ as $m$ increases. Therefore, the original sum $S(m,N)$ has a simple expression: it is basically linear in $m$ with slope equal to $2\,\zeta(N)$. Its linear behavior is not an asymptotic result; plots of $S(m,N)$ vs. $m$ for different $N$ show a straight line starting at $m=1$. $S(m,N)$ deviates slightly from a straight line over a small interval $8\le m\le 12$ for the case $N=3$. This slight deviation is due to Fermat near misses where $x^3+y^3-z^3=\pm 1$ (for $z\ne x$ and $z\ne y$); these create a jump in the remainder $R(m,3)$ at $m=9$. We make a numerical and analytical study of the remainder $R(m,N)$. From the numerical analysis, $R(m,N)$ converges for $N\ge 4$ but it was harder to tell whether $N=3$ converged. An analytical study based on a comparison of $R(m,N)$ to its Cauchy principal value integral, shows that $R(m,3)$ likely diverges logarithmically. It also shows that $R(m,N)$ converges for $N\ge 4$ in agreement with the numerical analysis. We discuss in the conclusion some interesting questions for future investigation.

A subset $A\subseteq \mathbb N_0$ is $k$-AP covering if there exists a constant $n_0$ such that for every integer $x>n_0$, there exists $d\in\mathbb N_0$ such that $x-d, x-2d,\dots,x-(k-1)d$ are all in $A$. Disproving a conjecture of Kiss, Sándor, and Yang, we prove that for every integer $k\geq 6$, there exists a constant $\varepsilon=\varepsilon_k>0$ and a $k$-AP covering set $A$ such that $|A\cap \{0,1,\dots,n\}| < n^{\frac{k-2}{k-1}-\varepsilon}$ for all sufficiently large $n$. We also relate this problem to the Arithmetic Kakeya Conjecture by Katz and Tao.

We begin the study of the beyond endoscopic trace formula for $\mathrm{GL}_2$ over $\mathbb{Q}$ attached to any symmetric power representation $\sigma_k$ of the dual group $\mathrm{G}L_2(\mathbb{C})$. For an adelic function that incorporates the $L$-functions $L(s_B,\pi,\sigma_k)$ through the basic functions at all the finite places, we integrate the cuspidal kernel against a corresponding Eisenstein series $E(g,s)$ and realize the trace formula as a residue at $s=1$, replacing Arthur's truncation operation by a continuously deformed trace formula. We argue Poisson summation on the trace variable of the Hitchin-Steinberg base and show that the dominant term of the elliptic part admits meromorphic continuation to $\mathfrak{R}(s_B) \geq 0$ with a pole of order $k$ at $s_B =1$.

The Fibonacci and Lucas sequences are old friends in recreational number theory. Here we ask a simple question: which of their terms $F_n$ and $L_n$ are Smith numbers? A Smith number is a composite integer whose decimal digit sum equals the sum of the decimal digits of its prime factors, counted with multiplicity. The question is easy to state, but for large $n$ it quickly becomes a factorization problem. Using available complete factorization data, we carried out computational searches in both sequences and obtained new Smith terms in each of them. For the Lucas sequence, the first author's search found ten Smith terms with indices below 1000: 3, 95, 105, 114, 183, 437, 609, 682, 827, 902. These results were published in the On-Line Encyclopedia of Integer Sequences as OEIS A395686. Four further indices, 1090, 1153, 1215, 1378, were subsequently added to the sequence by Sean A. Irvine. For the Fibonacci sequence, starting from the cases already recorded in OEIS A382922, our search produced seven further Smith numbers: $F_{1440}$, $F_{1554}$, $F_{1596}$, $F_{1863}$, $F_{2256}$, $F_{2277}$, $F_{2559}$. For the even-index cases, the identity $F_{2n}=F_nL_n$ allows available Fibonacci and Lucas factorization data to be combined to recover the complete factorizations needed for the Smith test. Surprisingly, the two searches also meet at index 827: both $F_{827}$ and $L_{827}$ are Smith numbers.

We determine the density of locally soluble fibres in certain families of varieties over a toric base. For this we develop a new form of the hyperbola method for toric bases which allows for counting functions with a growth behaviour of the form $B^a(\log B)^{-\Delta}$.