Papers

Andrews and Dastidar (2026) introduced $SOME(n)$ and $DSOME(n)$ functions related to partitions of a positive integer $n$, where $SOME(n)$ is the sum of all the odd parts in the partitions of $n$ minus the sum of all the even parts and $DSOME(n)$ is the sum of all the odd parts in the partitions of $n$ into distinct parts minusthe sum of all the even parts in the same partitions. The purpose of this paper is to establish the conjecture$SOME(\lambda)\equiv0\pmod{5^\alpha}$, $\alpha\ge 1$ and $\lambda\ge0$ are integers such that $24\lambda\equiv1\pmod{5^\alpha}$ due to Andrews and Dastidar, and the conjecture $DSOME(50n+21)\equiv0\pmod{8}$ due to Baruah and Gogoi (2026). In the process, we establish some new infinite families of congruences modulo 2, 4, and 8 for $DSOME(n)$.

George E. Andrews (2016) introduced the Kimberling index, $K(\pi)$, of a partition $\pi$ of a positive integer $n$, which is defined as $$K(\pi) = (\text{largest part of } \pi) - (\text{least part of } \pi) - (\text{number of parts of } \pi).$$ Based on Kimberling index, Andrews defined five partition functions, $K_>(n),$ $K_<(n),$ $K_\leq(n),$ $K_=(n),$ and $K_\geq(n)$, called Kimberling partition functions, which count the numbers of partitions of a positive integer $n$ for which the Kimberling index $K(\pi) $ is $>0$, $<0$, $\leq0$, $=0$ and $\geq 0$, respectively. He also gave the generating functions for $K_\le(n),$ $K_<(n),$ and $K_>(n)$ and established some relations connecting Kimberling partitions and other partition functions. Since then, the Kimberling partition functions and their generating functions remained unexplored. In this paper, we derive generating functions for $K_=(n),$ and $K_\geq(n)$, and establish some congruence relations of the five Kimberling partition functions by using the method of $q$-series identities.

We study the truncated Jacobi-symbol determinants $$\{c,d\}_n=\det\!\left[\left(\frac{j^2+cjk+dk^2}{n}\right)\right]_{2\le j,k\le n-2}$$ proposed by Zhi-Wei Sun in \cite{ZWSun} and prove Conjectures 5.1(i), 5.2, 5.3, 5.4, 5.5, 5.6(i), 5.7, 5.8, a case of 5.6(ii) of \cite{ZWSun}, Conjecture 4.8(i) of \cite{Sun2019} and some strengthened forms. All results follow from a single unified approach: for a prime $p$, we diagonalize the nonzero-residue matrix indexed by $\mathbb{F}_p^\times$ and reduce the vanishing of determinants to the supersingular reduction of certain CM elliptic curves. The same framework extends naturally to further families of parameters, suggesting a general mechanism behind identities of this type.

In this paper, we generalize the classical addition theorems for the Jacobi elliptic functions, giving numerous implicit and explicit formulas. We first present determinant identities involving several of the Jacobi functions. Then, we prove explicit addition formulas for a single function. Novel 4-,5-,6- and 7-term addition formulas are also given alongside a general $n$-term addition formula.

We establish a Kneser-based anticoncentration estimate for uniform subset sums in composite cyclic groups. The estimate contains a periodic loss and is weaker than its prime-modulus counterpart. Nevertheless, together with known small- and large-set results, it proves that, for every fixed $t\geq2$ such that $\mathbb{Z}_t$ is strongly sequenceable and every sufficiently large prime $p$, every subset of $\mathbb{Z}_{tp}\setminus\{0\}$ has a valid ordering, thus establishing the analogue of Graham's rearrangement conjecture for this family of composite cyclic groups. We then identify the structural source of this loss. An inverse theorem shows that failure of the stabilizer-free growth underlying prime-type anticoncentration forces almost all of the set into a proper subgroup or one of its cosets. We exploit this structure by reverse absorption. Iterating the resulting dichotomy between non-periodic anticoncentration and structured concentration proves that every subset of \[ \mathbb{Z}_k\setminus\{0\}, \qquad k=\prod_{i=1}^{s}p_i^{e_i}, \qquad \sum_{i=1}^{s}e_i\leq L, \qquad p_1<\cdots<p_s\leq\gamma p_1, \] admits a valid ordering whenever $L$ and $\gamma$ are fixed and the primes $p_i$ are sufficiently large.

We develop a proof-carrying recurrence–boundary method for affine families of positive-definite Nahm sums. A universal coordinate-contiguous relation provides exact algebraic cells for finite certificates; a tropical face-limit theorem identifies parameter directions along which a higher-rank Nahm sum degenerates to a lower-rank theta or theta–hypergeometric boundary value; and a recurrence–boundary principle recovers the initial sums from sufficiently many independent boundary limits. Machine learning and reinforcement learning are used only for discovery. Evolutionary symbolic search proposes recurrences and asymptotic rays from exact algebraic data, while a $Q$-learning agent searches for short sequences of legal contiguous-cell identities. No learned output is accepted as proof: every successful candidate is replaced by an exact symbolic certificate. As the main application, we prove Shi and Wang's Conjecture~3.8 (arXiv:2607.23257) for the Nahm sums dual to Zagier's twelfth rank-three example. The search finds a second-order recurrence for a one-parameter family and a five-cell certificate for it, together with two asymptotic rays leading to binary and unary theta series. These give two linear equations for the two initial sums. Solving the resulting $2\times2$ system reduces the conjecture to two generalized-eta identities, certified by valence arguments on $\Gamma_1(300)$ and $\Gamma_1(100)$. Consequently the dual of Zagier's twelfth example is modular, and every member of the affine family is an explicit $\mathbb{Z}[q,q^{-1}]$-combination of the two base products. Complete verification artifacts accompany the paper.

We completely classify the possible complexity classes of non-closed operator ranges on separable Banach spaces over a Polish non-Archimedean non-trivially valued field. These are precisely $\boldsymbol{\Pi }_{1+\lambda +n+2}^{0}$ for a countable ordinal $\lambda $ that is either zero or a limit ordinal, and a finite ordinal $n$. Considering Fréchet spaces produces the additional complexity classes $\boldsymbol{\Pi }_{\lambda }^{0}$ for a countable limit ordinal $\lambda $.

Kühne established equidistribution for canonical adelic line bundles on semiabelian varieties, a setting which need not lie in the quasi-canonical range of Yuan-Zhang's theorem. We study adelic line bundles on semiabelian compactifications whose toric part is governed by a toric metrized divisor. The main unconditional result is generic equidistribution when the toric metric is monocritical and arithmetically T-effective. The proof isolates the asymptotic estimates in Kühne's argument and reinterprets them as estimates along explicit compression paths. This gives a comparison mechanism between canonical, quasi-canonical, and more general toric metrics. For the Bogomolov application, the quasi-canonical case is handled by a Kühne local-trivialization transport package. After fixing a single theta-factor convention for the local trivializations, the proof checks the Picard-zero theta factors under Kühne's operations and obtains the Bogomolov theorem for the metric class treated in this paper, namely the monocritical and arithmetically T-effective toric metrics, through the quasi-canonical replacement argument.

In previous work with Harman and Snowden, we found explicit formulas for units of rank ideals of the matrix monoid algebra $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$, where $\mathbb{K}$ is an algebraically closed field with characteristic not dividing $q$. In this work, we use these explicit formulas to establish a regularized Godement–Jacquet functional equation valid for irreducible representations of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$ and of the algebra $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$. Using this functional equation we give an explicit expression for primitive central idempotents corresponding to irreducible representations of $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$, where the characteristic of $\mathbb{K}$ does not divide $\left|\operatorname{GL}_n\left(\mathbb{F}_q\right)\right|$. Interestingly, the coefficients in this formula are closely related to degenerate non-abelian Gauss sums.

Let $f$ be a holomorphic cusp form for $SL(2,\mathbb{Z})$. In this paper, we prove for $T^{\frac{1}{3}} \ll M \ll T^{\frac{1}{2}}$, the following short interval second moment \begin{equation*} \int_T^{T+M}\left|L\left(\frac12+it,f\right) \right|^2 dt \ll_{f,\epsilon} T^\epsilon \left( M + \frac{T}{M}+\sqrt{TM}\right). \end{equation*} The proof uses the trivial delta method along with conductor lowering technique.

It is a well known result by Singmaster that any integer $d$ divides almost all binomial coefficients. The study of the structure and the size of the exceptional set has been of interest for many decades. In the case where $d$ is a prime power, these sets are well understood and their size is known. However, if $d$ has at least two distinct prime factors, no non-trivial bounds are known. In this paper, we will provide the first non trivial bound on the number of binomial coefficients coprime to 6.

Let $p\neq 2$ be a prime. We determine the set of numbers that occur as the $u$-invariant of perfectoid subfields of $\mathbb{C}_p$. Given any number that is the $u$-invariant of a field of characteristic $p$, we give an explicit construction of a perfectoid field with the same $u$-invariant. In particular, if a number is the $u$-invariant of a field of characteristic $p$, it is also the $u$-invariant of a field of characteristic zero.

On the occasion of George Andrews' and Bruce Berndt's combined 170th birthday, we prove two conjectures of Sir Frederick Pollock that are more than 170 years old. In 1843, Pollock conjectured that every positive integer is the sum of at most 13 icosahedral numbers, or at most 21 dodecahedral numbers. We establish refined versions of both conjectures.

We first classify the potentially semistable Galois representations of a given tame inertial type using some variants of Breuil-Kisin modules. Then we construct some “model" of such Breuil-Kisin modules. From here, we build a stack of our version of Breuil-Kisin modules, and show that it is isomorphic to the Emerton-Gee stack of potentially semistable representations. This also makes it possible to construct a map from the rigid generic fiber of the potentially semistable EG stack to the analytification of the stack of Weil-Deligne representations, which “interpolates" Fontaine's recipe of associating Weil-Deligne representations to potentially semistable Galois representations. We prove the smoothness of this map, and obtain the generic formal smoothness property of the potentially semistable EG stack. This can be regarded as the global version of Kisin's regularity results on the corresponding Galois deformation rings.

Let $A$ be the smallest set of positive integers containing $2$ and $3$ such that $ab-1\in A$ whenever $a,b\in A$ are distinct. We prove that $A$ has positive lower density, answering a problem of Erdős attributed to Hofstadter.

For a prime $p$ and $k\in\mathbb{F}_p$, the generalized Markoff graph $G_{p,k}$ is an undirected graph whose vertices are the solutions over the finite field $\mathbb{F}_p$ of the normalized Markoff equation \[ x_1^2+x_2^2+x_3^2=x_1x_2x_3+k, \] where two vertices are adjacent if they differ by a Vieta involution. The Markoff graph $G_p$ is obtained from $G_{p,0}$ by removing the origin. The structure of $G_p$ has been the subject of extensive study; in particular, a major breakthrough of Bourgain, Gamburd, and Sarnak established that $G_p$ contains a giant connected component. Combined with Chen's remarkable divisibility theorem, this implies that $G_p$ is connected for all sufficiently large primes $p$. In the same paper, Bourgain, Gamburd, and Sarnak further asked whether the family $\{G_p\colon \text{primes }p\geq 5\}$ forms an expander family. This motivates us to investigate the robustness of connectivity in the Markoff graphs. The main result of this paper is proved in the general setting: for every prime $p\geq5$ and every $k\in\mathbb{F}_p\setminus\{4\}$, each connected component $C$ of $G_{p,k}$ with $|V(C)|\geq 3$ is $2$-connected. Reducing to the case $k=0$, we conclude that if the Markoff graph $G_p$ is connected, then it is in fact $2$-connected. Consequently, the Markoff graph $G_p$ is $2$-connected for all sufficiently large primes $p$. This is sharp in the sense that $G_p$ is not $3$-connected for any prime $p\geq 7$.

It is proved that for every sufficiently large odd integer $n$, the equation $$ n=p_1^2+p_2^2+p_3^3+p_4^4+p_5^5+p_6^6+p_7^8 $$ is solvable in primes. This refines the work of Cai and Mu in 2015.

We propose a heuristic for the growth rate of the average class number of curves in $\mathcal M_g(\mathbb F_q)$ as $g \to \infty$. We show that this heuristic implies that the zeros of the zeta functions of such curves become equidistributed on the circle as $g\to\infty$. Our heuristic is based on the assumption that the stable cohomology of the universal Jacobian dominates a point count using the trace formula, following the method of Achter et al.

The set $A$ is a $B_h$-set if every element of the sumset $hA$ has a unique representation as a sum of $h$ elements of $A$. A $B_2$-set is also called a Sidon set. A $B_{h,\Delta}$-set is a $B_h$-set $A$ whose sumset $hA$ is $\Delta$-separated, that is, $|x-x'| \geq \Delta$ for all $x,x' \in hA$ with $x\neq x'$. Upper and lower bounds are obtained for the cardinality of the largest $B_{2,\Delta}$-sets contained in $\{1,2,\ldots, n\}$, that is, sets $A \subseteq \{1,2,\ldots, n\}$ such that, if $a,b,c,d \in A$ and $\{a,b\} \neq \{c,d\}$, then $|(a+b)-(c+d)| \geq \Delta$.

On assuming the Riemann Hypothesis, we show that Siegel zeros imply the existence of gaps between the zeros of the Riemann zeta function less than 1/2 the normalised length. Specifically, we show that an infinite family of Siegel zeros implies $\liminf_{n\to\infty}(\gamma_{n+1}-\gamma_n)\log(\gamma_n)/2\pi< 0.4733$ on RH. This refutes the existence of certain strong alternative hypotheses under these assumptions. Our arguments incorporate long Dirichlet polynomials of length $T^{17/14-\varepsilon}$ into the Montgomery–Odlyzko method.

We investigate the mixed Stirling numbers of the second kind, $\mathcal{S}(n; \mathbf{c})$, which count partitions of $n$ distinct elements into $m$ unlabeled and $k$ labeled non-empty blocks encoded by $\mathbf{c} = (m, 1^k)$. We establish their recurrence relations and exponential generating functions, and analyze their behavior modulo a prime $p$ and $p^2$. In particular, we extend the classical Touchard congruence to this mixed framework. Our results show that these configurations possess unique number-theoretic signatures distinct from classical set partitions.

Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity. We prove that if $A\subset\mathbb{N}^3$ has positive upper Banach density, then there are $(x,y,z)\in\mathbb{N}^3$ and $d\in\mathbb{N}$ such that $$(x,y,z),(x+d,y,z),(x,y+d,z),(x,y,z+\Omega(d))\in A.$$ To establish the above result, we give an $L^2$-decoupling theorem for the triple ergodic averages $$ \frac1N\sum_{n=1}^N T_1^n f_1\,T_2^n f_2\,S^{\Omega(n)}g $$ associated with three commuting transformations by isotropy factors and nilpotent structures in $\mathbb{Z}^2$-actions.

In this paper, we investigate the existence of primitive polynomials over $\mathbb{F}_{q^n}$ whose constant term is a primitive element of $\mathbb{F}_{q^n}$. We prove that such polynomials do not exist if $q$ is odd, $q^n\equiv3\pmod{4}$, and the degree $m$ of the polynomial is odd. In particular, the polynomials $f(x)=g(x)+\lambda$, where $g(x)\in\mathbb{F}_q[x]$ satisfies $g(0)=0$ and $\lambda\in\mathbb{F}_{q^n}$ is primitive, cannot be primitive under the same conditions. Further, for the cubic polynomial $x^3+x^2+x+\lambda$, we establish non-existence results in characteristics $2$ and $3$. These results, in particular, provide counterexamples to previously proposed existence conjectures. We also study the family $x^p+x+\lambda$ over $\mathbb{F}_{p^n}$. For an odd prime $p$, we prove that, provided $\sum_{i=0}^{n-1}(-1)^i\lambda^{p^i}\neq0$, the polynomial $x^p+x+\lambda$ is irreducible over $\mathbb{F}_{p^n}$ if and only if $n$ is even. Motivated by this result and supported by computational evidence, we formulate conjectures concerning the existence of such primitive polynomials, including the stronger assertion that $x^p+x+\lambda$ is primitive for every primitive $\lambda\in\mathbb{F}_{p^2}$.

We construct finite Heath-Brown-type identities for the fractional powers $\zeta(s)^{\pm a/b}$ of the Riemann zeta-function, for every reduced fraction $a/b$ with $0 < a/b < 1$, from Newton's binomial series in the algebra of arithmetic functions, and we use them to prove the Vinogradov-quality bound $\sum_{n \le x} d_{\pm a/b}(n)e(n\alpha) \ll_{a,b} ( x q^{-1/2} + x^{4/5} + x^{1/2} q^{1/2} ) (\log 2x)^{C}$, for some constant $C=C(a,b)>0$, whenever $|\alpha - r/q| \le 1/q^2$ with $(r, q) = 1$. The bound carries no $x^{\varepsilon}$ loss, and the same machinery gives the endpoint case of the Möbius function with an absolute constant. As applications we determine the major-arc expansion of $S_z(x, \alpha) = \sum_{n \le x} d_z(n)e(n\alpha)$ to arbitrary logarithmic precision for real $0<|z|<1$. For rational $z \in (-1,1)$, we determine the order of magnitude of $\sup_{\alpha} |S_z(x, \alpha)|$ and prove an asymptotic formula for the moments $\int_0^1 |S_z(x, \alpha)|^{s}\, d\alpha$ for every fixed real $s > 2$. The minor-arc analysis avoids the theory of $L$-functions entirely, and all constants are effective except those inherited from the Siegel-Walfisz theorem.

We combine a new type of modularity result with the geometry of numbers in order to prove some new cases of the Fontaine–Mazur conjecture for GL(2).

We prove the Modularity Conjecture for higher theta series on moduli stacks of Hermitian shtukas. For general linear shtukas, we establish a more refined phenomenon that we call supermodularity. As a key input, we prove the Trace Conjecture for Hitchin stacks of low corank, realizing virtual fundamental classes of special cycles as categorical traces.

We construct (potentially) semi-stable integral models for a class of Shimura varieties with maximal parahoric level at an odd prime $p$. The main input is an explicit construction of semi-stable equivariant modifications of the relevant canonical local models, where the underlying groups are Weil restrictions of unramified unitary similitude groups, symplectic similitude groups, or even orthogonal similitude groups. Via the local model diagram, these constructions give (potentially) semi-stable integral models of the corresponding Shimura varieties. In particular, the resulting models are regular and have reduced special fiber with normal crossings. As an application, we deduce the unipotence of the inertia action on nearby cycles and the $\ell$-adic cohomology of the geometric generic fibers.

On page 253 of his Lost Notebook, Ramanujan recorded an intriguing identity relating a generalized divisor function and a modified Bessel function, which was later rediscovered by Guinand and is now known as the Ramanujan-Guinand identity. In this paper, we establish a number field analogue of this identity for the Dedekind zeta function. In particular, we recover Ramanujan-Guinand identity, Ramanujan-Koshliakov identity, and also obtain the transformation formula for the logarithm of the Dedekind eta function. In 1986, Zagier obtained a formula for $\zeta_\mathbb{F}(2)$ for any number field $\mathbb{F}$. Quite surprisingly, as an application of our main theorem, we also obtain a new formula for $\zeta_\mathbb{F}(2)$ for arbitrary number field $\mathbb{F}$. Moreover, we derive an elegant identity for $\zeta_\mathbb{F}(2m+1)$ for any positive integer $m$ and any number field $\mathbb{F}$.

This article introduces crystallographic wraiths, new combinatorial invariants attached to Euclidean lattices and linked to the geometry of Voronoï cells. They are based on geometric properties of shortest linear dependencies among relevant Voronoï vectors. Moreover, it explores the new theory of crystallographic wraiths, proves that only finitely many occur in each dimension, and extends the construction to a larger class of combinatorial objects. As an application, it provides a new combinatorial proof of Voronoï's finiteness theorem for perfect lattices.

We propose an integral framework for the anticyclotomic Iwasawa theory of CM elliptic curves $E$ at primes $p$ ramified in the CM field. The $\varepsilon$-constants of the geometric specialisations of the associated $p$-adic conjugate symplectic self-dual deformation equidistribute between $\pm 1$ within every layer of the anticyclotomic tower, and none of the geometric specialisations are trianguline at $p$. We define signed Selmer groups via the Lagrangian local conditions arising from the local sign decomposition established in the prequel \cite{BKNO}, and our central result is the formulation and proof of an integral Iwasawa main conjecture relating one of them to the $p$-adic $L$-function $\mathscr{L}_p(E)$ constructed there. We further show that $\mathscr{L}_p(E)$ interpolates the central Hecke $L$-values of the twists with $\varepsilon$-constant $+1$, including twists of arbitrary infinity type, and relate its values at twists with $\varepsilon$-constant $-1$ to the $p$-adic logarithm of certain Selmer elements. This provides the first Iwasawa main conjecture in terms of a $p$-adic $L$-function and Selmer groups for a $p$-adic deformation admitting no trianguline geometric specialisation. The proofs rest on our resolution of a Rubin-type conjecture for the underlying local deformation, together with a theory of plus/minus local points along the anticyclotomic tower, based on the Gaussian plus/minus cyclotomic polynomials rooted in Gauss' Disquisitiones Arithmeticae.