Papers

We give an elementary proof that every positive rational number $a/b$ with $b$ squarefree is a finite sum of distinct unit fractions $1/n$, where each $n$ is a product of two distinct primes (Erdős Problem #306). After a reduction to small targets, we take a single complete bipartite graph between the primes in $(y^2,2y^2]$, together with $2$ and the primes of $b$, and a tuned initial segment of the primes in $(y^8,y^9]$, and show that some subgraph has reciprocal sum congruent to $a/b$ modulo $1$; the small total mass then forces equality. Writing the number of such subgraphs as a finite Fourier sum, we sort the frequencies into three cases using a table indexed by the two sides of the graph. The small integer frequencies give a positive main term, and all other frequencies are negligible by a divisor-counting argument and a no-wrap-around form of the Chinese remainder theorem. The only inputs about primes are Chebyshev-type bounds. The circle-method framework comes from Tang's Lean development, which gave the first proof; our construction removes its anchor-synchronisation step. The proof has been formalised in Lean 4, apart from a cited inequality of Ramanujan. This work is a human-AI collaboration: AI tools contributed substantially to the construction, the experiments and the writing.

We study the additive and multiplicative arithmetic of algebraic monogenic semidomains of the form $S_\alpha$. We first relate their structure to the conjugates of $\alpha$, characterize when $S_\alpha$ is a ring, and determine when its additive monoid is free. We then investigate units and prove a Dirichlet-type theorem showing that $S_\alpha^\times$ is the product of a finite cyclic group of roots of unity and a finitely generated free abelian group; in particular, the unit group is free abelian when $\alpha$ is positive. We also give a criterion that produces non-atomic algebraic monogenic semidomains. Turning to finiteness properties, we prove that $\mathbb{Z}[\alpha]$ is a finite factorization domain for every algebraic number $\alpha$, establish criteria ensuring that $S_\alpha$ is a finite factorization semidomain, and construct infinitely many positive cubic generators yielding finite factorization semidomains that are not rings. Finally, we give a membership criterion for rational elements of $\mathbb{Z}[\beta]$ for arbitrary algebraic $\beta$, and prove that, for algebraic integers $\alpha$, unique factorization ascends from $\mathbb{Z}[\alpha]$ to $\mathbb{Z}[\alpha/n]$ for every $n\in\mathbb{N}$.

Let $F$ be a totally real field and $K/F$ a totally real quadratic extension. For a Hilbert cusp newform $f$ over $F$ with base change $f_K$ to $K,$ we determine the exact level of $f_K$ in terms of the level of $f,$ using only local representation theory: for each place $\mathfrak p$ of $F$ we compute how the conductor of the local component of $f$ changes under quadratic base change. The tame case (odd residue characteristic) follows from existing base change theory, but the case $p=2$ requires a finer $\varepsilon$-factor analysis that does not appear in the literature. We give a complete treatment of the dyadic imprimitive representations and, more delicately, the exceptional supercuspidal representations, for which no closed-form base change formula was previously known; in particular we determine, in terms of local root numbers, the base change of every exceptional supercuspidal representation of $GL(2,\mathbb Q_2)$ with trivial central character and conductor $3.$ This yields an explicit formula for the level of $f_K,$ specialized here to $F=\mathbb Q,$ along with a partial converse.

We prove sharp lower bounds for the Schrödinger maximal function on the torus T^d for all dimensions d at least 2, and as a corollary obtain sharp regularity conditions for pointwise convergence of the periodic Schrödinger equation. Combined with sufficiency results established by Compaan-Lucá-Staffilani, this yields a full resolution of Carleson's problem up to endpoint for the periodic Schrödinger equation in all dimensions at least 2, and disproves the conjectured regularity condition of Miao-Yuan-Zhao, in contrast to the corresponding question in R^d. We also prove sharp estimates on the dimensions of divergence sets for the equation for high dimensional tori. Our approach uses complex multiplication on abelian varieties.

Recent developments in theories of partitions, quasimodular forms, multiple zeta values and $q$-multiple zeta values have led to connections between partitions and primes. Central to this progress are prime-detecting quasimodular forms, which are $q$-series whose non-negative coefficients vanish if and only if the input is prime. In this paper, we use properties of these quasimodular forms to motivate a proof of the infinitude of primes, to construct a novel partition theoretic factorization algorithm, and to show that the distribution of these coefficients relate to existence of primes in intervals. We also prove analogs of these results for primes in arithmetic progressions and discuss potential extensions of our results, including to Linnik's problem.

For a positive integer $\alpha$, let $p_\alpha(n)$ denote the number of $\alpha$-colored partitions of $n$. The Rademacher-type expansion of Iskander, Jain, and Talvola is valid for every real $\alpha>0$, but when $\alpha>24$ it involves several polar terms and a two-parameter family of exponential sums $A_k^{(\alpha)}(n,m)$. For integral $\alpha$, we prove multiplicativity and prime-power reduction formulas for these sums, expressing their local factors as classical or quadratically twisted Kloosterman sums. Together with explicit truncation and precision bounds, these formulas yield an efficient algorithm for computing $p_\alpha(n)$ exactly. Our SageMath implementation computes the 1,113,767-digit integer $p_{100}(10^{10})$ in less than one hour. As an application, we use the algorithm to certify new Ramanujan-type congruences.

This is a continuation of the author's earlier paper Block-count constrained harmonic sums: spectral expansion and block-directed Euler–Maclaurin. We presented in that reference the modal expansion of block-Irwin sums $I(b,w,k)$ for $k\geq1$. The case $k=0$ was fully developed when $w$ is a single digit, but it was presented only as a very short result summary for $w$ a multi-digit block. Although the framework is already completely exposed in the earlier paper, the case $k=0$ requires adding a separate, dedicated discussion, which is provided here.

We study the divisibilities of Tamagawa numbers $ c(E) $ of elliptic curves $ E $ over global function fields $ K $ in terms of their torsion subgroups $ E(K)_{\operatorname{tors}} $. In particular, for a non-isotrivial elliptic curve $ E / k(t) $, where $ k $ is a finite field of characteristic greater than $ 3 $, we prove that $ |E(k(t))_{\operatorname{tors}}|^2 $ divides $ c(E) $, except possibly in four exceptional torsion families. More specifically, we give a complete characterisation of divisibilities for $ c(E) $ in each torsion family, and provide explicit examples to prove that they are best possible. Over a general global function field, we also prove that a rational point of prime order $ 5 \le N \le 101 $ on $ E / K $ forces $ N^2 $ to divide $ c(E) $, which motivates our result. Finally, we formulate a conjecture on the leading coefficient of the $ L $-function of $ E / K $, motivated by the integrality of Birch–Swinnerton-Dyer quotients.

We study the Galois group $G_f$ of a random polynomial $f$ in the family of polynomials of degree $2n$ satisfying the twisted reciprocal relation $f(x) = x^{2n}/b^n \cdot f(b/x)$. We use a Euclidean height adapted to this relation. Our main result is an asymptotic theorem of van der Waerden–Bhargava type: for fixed $b \neq 0$ and $n \geq 4$, the number of polynomials of height at most $H$ whose Galois group is not the full hyperoctahedral group $S_2 \wr S_n$ is an explicit constant times $H^n\log H$, with an error of order $H^n$. We determine the dependence of the leading constant on $b$; the leading-order group $G_1$ is of index $2$. This paper is a sequel to a recent paper by Anderson, Bertelli, and the author addressing reciprocal polynomials (i.e. the case $b = 1$).

Let $K/E$ be a finite unramified extension of $p$-adic local fields, and let $H$ be a one-dimensional formal $\mathcal{O}_E$-module of finite height over $\mathcal{O}_K$. We introduce the exponential period map of $H$ and use it to construct a complete regular local ring $R_{H,K}$ with imperfect residue field, an endomorphism $\varphi_E$, and a commuting action of $\Gamma=\mathrm{Gal}(K(H[p^\infty](\overline{K}))/K)$. We prove an equivalence of categories between finitely generated $\mathcal{O}_E$-modules with a continuous $\mathrm{Gal}_K$-action and étale $(\varphi_E,\Gamma)$-modules over $R_{H,K}$. This recovers the classical cyclotomic and Lubin-Tate equivalences in the corresponding cases. In general, $R_{H,K}$ can have Krull dimension greater than one, and $\varphi_E$ need not lift the $q_E$-power Frobenius modulo $\pi_E$. The proof uses $F$-dynamical systems over $\mathcal{O}_E$, which combine contraction modulo $\pi_E$ with Frobenius on the residue field. For every flat $F$-dynamical system, we establish an equivalence of categories between étale $\varphi$-modules and continuous representations of the absolute Galois group of the residue field on finitely generated $\mathcal{O}_E$-modules.

Fix a degree $n$, a nonzero integer constant term, and some further coefficients of a monic integer polynomial, leaving $s$ coefficients to vary in $[-H,H]$. We first prove that, when only the leading and constant coefficients are fixed, the number of polynomials whose Galois group is not $S_n$ is of order $H^{n-2}$ for every $n\ge4$. We then prove a half-dimensional extension: for $n\ge6$ the corresponding exceptional count is of order $H^{s-1}$ whenever $s>n/2+1$, provided the coefficient slice meets the higher-multiplicity loci with a specified dimension bound. This geometric condition holds for every prescribed value if the free coefficients form an initial or final block, and for a nonempty Zariski-open set of prescribed values for arbitrary coefficient positions. It also holds for every value of one additional prescribed interior coefficient. Thus, in degrees seven and eight, any one interior coefficient can be fixed in addition to the leading and constant coefficients. The proof combines a fixed-norm version of Bhargava's primitive-group sieve with sparse Hermite interpolation. A separate imprimitive estimate, using a minimal intermediate field, gives $O(H^{\beta(n)})$ outside rational partial-product hypersurfaces, where $\beta(n)=\ell+n/\ell-2\le n/2$ for composite $n$ with smallest prime divisor $\ell$, and has no logarithmic loss. The fixed-constant quartic and quintic endpoints require separate local arguments: exceptional Fourier directions for $A_4$ quartics and a quadratic Gauss sum for quintics. We isolate further even-degree endpoint results, explain the distinct cubic obstruction, and conclude with conditional improvements under the upper-bound form of Malle's conjecture.

Given power series $f_1,\dots,f_r\in z^2K[[z]]$ over a field $K$, we show that the semigroup $\langle f_1,\dots,f_r\rangle$ generated by the $f$'s under composition is free of rank $r$ whenever the multiplicative semigroup $\langle\mathrm{lc}(f_1),\dots,\mathrm{lc}(f_r)\rangle$ in $K^\times$ is free commutative of rank $r$; here $\mathrm{lc}(f)$ denotes the first nonzero coefficient of $f$. To do this, we associate an affine linear transformation to every $F\in\langle f_1,\dots,f_r\rangle$ and then apply a ping-pong lemma. In particular, combining this with work of Pappalardi, Sha, Shparlinski, and Stewart, it follows that most semigroups of polynomials over a number field $K$ are free, and we make this statement precise through a counting argument.

In this note, we study the infinite reciprocal sum $\sum_{k=n}^{\infty}1/B_k^4$ involving the fourth powers of balancing numbers $B_n$. We show that, for every $n\geq2$, \begin{equation*} \left\lfloor \left( \sum_{k=n}^{\infty}\frac{1}{B_k^4} \right)^{-1} \right\rfloor = B_n^4-B_{n-1}^4 -\left\lceil\frac{B_{2n-1}}{280}\right\rceil +\varepsilon_n, \end{equation*} where $\varepsilon_n=1$ if $n\equiv1\pmod{12}$ and $\varepsilon_n=0$ otherwise. This result extends the corresponding reciprocal-sum result for Fibonacci numbers due to Hwang, Park and Song to balancing numbers.

Let $E$ be an elliptic curve defined over rational numbers. Further assume $E$ has good ordinary reduction at $p=2$ and $E[4]$ is reducible as a $G_\mathbb{Q}$-representation. In this paper, we offer sufficient and necessary computational criteria for the algebraic Iwasawa $\mu$-invariant over the cyclotomic $\mathbb{Z}_2$-extension of rational number to be $0$, $1$ or $\geq 2$. We also show that elliptic curves with isomorphic mod-$4$ representations have algebraic $\mu$-invariant equal to $0$ if one of the curves does.

Let $E$ be an elliptic curve define over $\mathbb{Q}$. Further assume $E$ has good ordinary reduction at $p=2$. In this article, we prove Greenberg's conjecture on the value of the algebraic Iwasawa $\mu$-invariant when $E$ has a rational isogeny of degree 2, and obtain a sharp absolute upper bound of the $2$-adic $\mu$-invariants for all such curves. We also prove the vanishing of the algebraic $\mu$-invariant of the fine Selmer group introduced by Coates and Sujatha at $p=2$.

The notion of sum-freedom of binary functions was introduced recently by C. Carlet as a generalization of the APN functions used in cryptography; the $q$-ary version of the notion is a natural extension. For each integer $k$ with $0\le k\le n$, there is a $k$th order sum-free function on $\Bbb F_{2^n}$. It is also known that when $k/n$ is not close to 0 or 1, the multiplicative inverse function on $\Bbb F_{2^n}$ is not $k$th order sum-free. We generalize these two results to $q$-ary functions. APN functions have a coding theoretic characterization. We generalize the characterization to sum-free functions of arbitrary order over any finite field. It is well known that the Welch functions is 2nd order sum-free. We give an alternative proof for this result which leads to a more general algebraic question. We also investigate that the 3rd order sum-freedom of the Welch function and power functions of algebraic degree 3. We formulate a conjecture about the 3rd order sum-freedom of the Welch function which is supported by strong numerical evidence.

In this work, we enumerate the ternary perfect forms over all imaginary quadratic fields of absolute discriminant up to $91$ and study the number and combinatorial types of their associated polytopes. We prove a bound on the number of combinatorial types that can arise regardless of the discriminant. We analyze the data and make several observations concerning the number and complexity of the perfect forms that arise in the scope of our calculation.

The classical ADE classification implies that $E_8$ is the unique irreducible unimodular Euclidean lattice generated by vectors of norm $2$. In sharp contrast, for every positive integer $m$, we construct an irreducible odd unimodular lattice of rank $24m$ and minimum norm $3$ that is generated by its minimal vectors.

Let $p>3$ be a prime, and let $b,c$ be integers with $p\nmid b+2c$. We obtain an explicit congruence modulo $p^2$ for the sum of $\binom{2k}{k}T_{2k}(b,c^2)/(4^k(k+1)(b+2c)^{2k})$ from $k=0$ to $(p-1)/2$, where $T_n(b,c)$ denotes the generalized central trinomial coefficient. This gives a parametric formula for the sum considered in a question of Wang and Cui. The proof uses a constant term representation and a logarithmic generating function to reduce the sum to two coefficients of an algebraic generating function. A differential identity and a coefficientwise polynomial congruence then yield the desired evaluation. The formula also applies when $p$ divides $b$ or $b-2c$.

Let $E_{A,B}\colon y^2 +Axy + By = x^3$ be an elliptic curve over $\mathbb{Q}$. The $3$-torsion point $(0,0)$ induces a $3$-isogeny $\phi\colon E_{A,B}\to E_{A,B}'$. Assuming $E_{A,B}$ has good reduction at $3$, we construct an explicit $(m+t)\times m$ matrix $M_{A,B}'$ over $\mathbb{F}_3$, whose kernel encodes the dual isogeny Selmer group $\operatorname{Sel}_{\widehat{\phi}}(E_{A,B}')$ modulo the image of the torsion point $(0,0)$. Here, $m = \omega(B) - 1$, and $t$ encodes the global Selmer ratio $3^{t-2}$. We compute $M_{A,B}'$ in various regimes for billions of elliptic curves $E_{A,B}$. Based on our data, and motivated by prevalence of random linear algebraic models throughout number theory, we conjecture that, for fixed $m$ and $t$, the matrices $M_{A,B}'$ become uniformly distributed, and we formulate a corresponding conjecture for the distribution of $\operatorname{Sel}_{\phi}(E_{A,B})$. Our model predicts that, for fixed $t$, the average size of $\operatorname{Sel}_{\phi}(E_{A,B})$ is $1 + 3^t$.

Finite-aperture near-field sensing leads to a super-resolution geometry fundamentally different from the translation-invariant Fourier setting. In the Fresnel regime, wavefront curvature introduces a range-dependent quadratic aperture phase, so distinguishability is governed by incomplete quadratic exponential sums of the form \( \sum_{n=0}^{N_r-1} a_n e^{i(\omega_1 n+\omega_2 n^2)} \), rather than by angular separation alone. We develop a deterministic recovery theory for sparse measures with ranges on a finite grid and continuous angles, and introduce a support-uniform quadratic-phase aperture criterion replacing classical minimum separation. Under this criterion, total-variation minimization exactly recovers every sparse measure in the admissible support class, uniformly over all nonzero complex amplitudes. The proof develops nonasymptotic support-uniform bounds for finite quadratic sums and combines them with a gauged Hermite dual certificate controlling interpolation, local curvature, and off-support leakage. We further construct a finite-harmonic Bessel-Vandermonde lift with explicit truncation error. In the far-field limit, the quadratic phase disappears and the theory reduces to Fourier-type angular super-resolution.

We study the right derivative of the zeta-star correspondence between binary expansions and multiple zeta-star values of infinite depth. At a finite-index point, this derivative is a normalized difference of two finite-depth multiple zeta-star values. We use this identity to express its local behavior in terms of lower-truncated star sums and their boundary coefficients. The homogeneous contribution at summation variable three contracts with ratio $2/3$; the full coefficient also receives a positive forcing term. A weighted sum of the remaining coefficients contracts by at most $1/2$ at each binary step. On the open parameter interval, we prove that the derivative is right-continuous everywhere and continuous precisely at the non-dyadic points. Its left jump at a dyadic point is given by an explicit positive multiple series, with divergence exactly at $1/2-2^{-M}$, $M\ge2$. A refinement of the index gives an infinite upper right Dini derivative at every interior point. We also prove that the Hausdorff, packing, and modified upper box dimensions of the graph are all $\log_2(8/3)$. The Hausdorff lower bound uses the dimension of the Bernoulli convolution with parameter $2/3$, together with a separate measure-transfer argument. Finally, the pointwise Hölder exponent at a non-dyadic point is $\log_2(3/2)$ divided by its dyadic approximation exponent, and the corresponding Hausdorff spectrum is linear.

Andrews and Newman \cite{AN19, AN20} defined the minimal excludant or mex of a partition. The mex of a partition is the smallest positive integer which is absent from the partition. Baruah, Bhoria, Eyyunni and Maji \cite{BBEM} studied the sum of mex split according to parity, together with its $\ell$-th moments. Ballantine and Merca \cite{BM20} generalized mex to the least $s$-gap, which is the smallest natural number that does not occur minimum $s$ times in the partition. Baruah and Talukdar \cite{BT} conjectured a corresponding asymptotic equivalence between the sums of odd and even least $s$-gaps for every $s>1$, generalizing a theorem of Barman and Singh \cite{BS} for the classical mex. In this article, we derive exact formulas for the $\ell$-th moments of $\smex(\pi)$ for every fixed $\ell\ge 1$, in terms of partition functions. We give a complete proof of the conjecture of Baruah and Talukdar \cite{BT} for every natural number $s>1$. We also generalize an identity of Hopkins, Sellers and Stanton \cite{HSS} to the least $s$-gap setting.

The Duffin-Schaeffer Conjecture, proven in 2020, gives a zero-full dichotomy for the approximation of irrational numbers with reduced fractions for arbitrary approximation functions. We construct approximation functions such that the inhomogeneous analogue of the Duffin-Schaeffer Conjecture fails. The counterexamples are obtained for all non-zero rationals and certain Liouville numbers.

Shallit's law of leap years in Pierce expansions gives, for Lebesgue-almost every $x\in[0,1]$, the upper and lower limits of a normalized difference between $Nx$ and the number of leap years up to the year $N$ determined by the Pierce expansion digits of $x$. In this paper, we show that, for every $x\in[0,1]$, both limits are determined by the lower limit of a normalized logarithm of the product of the first $n$ digits of $x$. In particular, the upper and lower limits always have the same absolute value. As applications, we characterize the sets of points with prescribed upper and lower limits, and show that each of these sets, if non-empty, is dense in $[0,1]$ and its intersection with any non-empty open subset of $[0,1]$ has full Hausdorff dimension. We also compare these sets with the sets defined by the growth rate of the digits, and show that, for each finite parameter, their difference has full Hausdorff dimension.

Let $\theta\in\mathbb Q\setminus\{0\}$. We construct an approximating function $\psi:\mathbb N\to[0,\frac12)$ for which \[ \sum_{q=1}^{\infty}\frac{\varphi(q)}{q}\psi(q)=\infty, \] but the set of $x\in[0,1]$ for which \[ |qx-a-\theta|<\psi(q),\qquad \gcd(a,q)=1, \] holds for infinitely many $(a,q)\in\mathbb Z\times\mathbb N$ has Lebesgue measure zero. Thus the inhomogeneous analogue of the Duffin–Schaeffer conjecture fails for every nonzero rational shift. By a modification of the construction, we further show that the set of shifts for which the inhomogeneous Duffin–Schaeffer conjecture fails is residual in $\mathbb R$.

We study multiple zeta values in positive characteristic. We construct, using $\infty$-adic multiple zeta values, a coaction of the $\infty$-adic multiple zeta value algebra modulo $\zeta_A(q-1)$ on the $\infty$-adic multiple zeta value algebra itself, and prove that it agrees with Mishiba's coaction constructed via special values of Carlitz multiple polylogarithms. We also determine the coaction on the $\star$-inverse values and show that the antipode on the $\infty$-adic multiple zeta value algebra modulo $\zeta_A(q-1)$ is given by the $\star$-inverse operation. In particular, these results prove Mishiba's conjecture concerning the coaction and the antipode for $\infty$-adic multiple zeta values.

For $a$ in $\mathbb{R}$, a subset $V$ contained in $\mathbb{R}^{n}$ is said to be $a$-convex if $x, y \in V \Longrightarrow a x + (1-a) y \in V$. According to Pinch [Math. Proc. Cambridge Philos. Soc., 1985], the $a$-convex hull of $V$ is the intersection of all of the $a$-convex subsets of $\mathbb{R}^{n}$ that contain $V$, and Pinch also defines $D(a)$ as the $a$-convex hull of $\{ 0, 1 \}$ in $\mathbb{R}^{1}$. Pinch conjectured that if $a$ is a totally real algebraic integer and $D(a)$ has no limit points, then every algebraic conjugate of $a$ other than $a$ is in $(0, 1)$. We succeed in proving this conjecture, which seems to have remained open.

In 2018, Soloveychik, Xiang and Tarokh considered a pseudo-random symmetric circulant matrix constructed from binary Golomb sequences of length $n=2^m-1$, and claimed a proof that its empirical spectral distribution converges almost surely to the semicircle law as $n$ grows to infinity by the method of moments. In this comment note we show that their argument contains several technical flaws and inappropriate applications of technical lemmas. Instead we demonstrate that the eigenvalues of the matrix are simply a normalized twisted Kloosterman sum varying over the multiplicative character, to which we apply Katz's result directly to establish the asymptotically semicircle spectral distribution deterministically.

We prove that fiberwise-connected finite locally free commutative group schemes over a ring $R$ are equivalent to an explicit full subcategory of the derived category of modules over the Cartier ring of $R$.