Papers

We prove an asymptotic formula for correlations of divisor sums with quadratic cutoff functions shifted independently by \(s,u\in[0,1/2]\). The error is \(O_g(Y/(\log Y)^2)\), uniformly for \(Y^{1/5}\leq R\leq Y^{1/3}\), including coincident shifts. The main term is an explicit bilinear form in the first and second derivatives of the cutoff functions. We obtain it by taking the residue in the outer Mellin variable and evaluating the remaining double integral by Laplace inversion. The present paper establishes the divisor-sum estimate. Its application to the sign-change problem, including the choice of the prime cutoff and the remaining parameters, is treated in Part II.

Let $C_0=\frac32-\frac1{\sqrt2}\cot\big(\frac1{\sqrt2}\big) =0.67250\ldots$ and $C_1=\frac{C_0+1}{2}= 0.83625\ldots$. Recently, it is obatained by Alpöge and Furman that more than 67.25% of the non-trivial zeros of the Riemann zeta function are simple and on the critical line, and more than 83.62% are distinct. Later, Lamzouri gave a different and more direct proof. In this article, by refining the method of Lamzouri, we slightly improve the bounds $C_0$ and $C_1$ in these two results to $C_0+\delta_0$ and $C_1+\frac{\delta_0}2$ with $\delta_0=6.66624\ldots\times10^{-8}$, respectively.

If $f$ is a tuple of functions satisfying an algebraic ODE and $P\in{\mathbb C}(x)[f]$, it is common in applications to transcendental number theory to consider upper bounds for the order of zero of $P(x,f)$ at a given point in terms of $\operatorname{deg}_x P,\operatorname{deg}_f P$. Nesterenko introduced a condition known as the D-property, which holds in many applications, and proved essentially optimal bounds under this condition. We prove a global form of this result, where the field $\mathbb C(x)$ is replaced by a number field $K$ and $\operatorname{deg}_x P$ is replaced by the logarithmic height $\operatorname{h}(P)$. Under an analogous global D-property, we prove essentially optimal bounds for the number of zeroes, counted with multiplicities, in a fixed compact set. Applications of this result to point-counting theorems are developed in a separate joint paper with Hirata-Kohno and Kawashima.

Uncu and Zudilin studied reflections of finite forms of the modulo nine Kanade–Russell identities under $q\mapsto q^{-1}$, leading to ten conjectural sum–product evaluations, including two proposed by Warnaar. Hickerson conjectured four further identities in his study of contiguous relations for the associated parameterized double sums. By constructing finite forms of Hickerson's sums and studying their reflections, Konenkov proposed two additional product identities, whose conjugates give two more. In this paper, we prove all eighteen conjectural identities by combining Nahm-sum dual evaluations established by the author with parameterized divided-difference identities. For the symmetric reflections, Schur-type decompositions and finite rational identities reduce the required evaluations to nonsingular two-by-two linear systems. The asymmetric and cyclotomic cases are treated through $q$-Airy–theta connection formulas, whereas the Hickerson identities are derived from low-order divided differences.

We prove that the solution functor from bounded good filtered complexes of coherent D-modules on a smooth rigid-analytic variety to modules over the positive almost de Rham period sheaf is fully faithful. On filtered vector bundles with flat connection satisfying Griffiths transversality, the construction recovers, up to duality and extension of scalars, Scholze's horizontal sections functor. As an application, we obtain a notion of de Rham Zariski-constructible sheaves, generalising the classical notion of de Rham local systems.

We establish a new method for the study of imaginary non-CM fields of arbitrary \(2\)-power degree and class number one. In particular, we show that for degrees greater than \(16\), dihedral fields either lie in the ray class field of one of the three imaginary quadratic fields \(\mathbb{Q}(\sqrt{-2})\), \(\mathbb{Q}(\sqrt{-3})\), \(\mathbb{Q}(\sqrt{-67})\), with the conductor of this ray class field supported only on primes dividing \(2\) and with explicit upper bounds on the relevant exponents, or are Hilbert class fields of imaginary quadratic fields. We also give a complete classification of the \(104\) imaginary non-CM fields of degree \(16\) with class number one, providing explicit tables of defining polynomials, Galois groups, base fields, and conductors. These 104 non-CM fields, together with the five CM fields classified by Louboutin and Okazaki \cite{lou7}, complete the classification of all imaginary fields of degree 16 with class number one. For CM fields, the problem reduces to computing relative class numbers via analytic class number formulae, whereas for non-CM fields the absence of a totally real subfield of index \(2\) necessitates a further approach and leads to a richer variety of Galois groups and ramification patterns.

In this article, we discuss a multivariable analogue of Lehmer's parity problem. We generalise the notions of Lehmer numbers as Lehmer $q$-tuples for a fixed odd prime $q$. We count the number of Lehmer $q$-tuples and derive an asymptotic estimate with explicit error terms. This deals with estimating various character sums and Deligne's bound for incomplete exponential sums.

Let $X$ denote the number of positive integers which are a prime plus a square. We prove that $E(X)=O(X^{4/5})$.

We abstract and prove some fundamental lemmas in universal blowups

Adapting a construction of Kurihara in the setting of Drinfeld modular forms, we define Poincaré series on the Drinfeld half-plane $\Omega$. These series are built from products of meromorphic $1$-forms that are naturally associated to geodesics on the Bruhat-Tits tree. We establish convergence under a finiteness condition on the geodesics, verify this condition in several cases, and give sufficient conditions for the resulting cusp forms to be nonzero. For the principal congruence subgroup $\Gamma(\mathfrak{n})$ of $GL_2(\mathbb{F}_q[T])$, we construct explicit linearly independent families of Poincaré series by lifting certain $k$-forms from the components of the analytic reduction of $\Gamma(\mathfrak{n})\backslash\Omega$. We formulate conjectures on the vanishing orders at cusps, and we prove the first of them for an explicit family of Poincaré series by computing the corresponding expansions at the cusps; as an application, we obtain the Drinfeld modular forms $h$ and $\Delta$ as Poincaré series (up to a sign). Finally, for cocompact groups attached to quaternion algebras over $\mathbb{F}_q(T)$ that split at $\infty$, we show that the Poincaré series span the whole space of modular forms of given weight and type.

We conjecture a positive four-shift identity for rank-two Nahm sums with denominator steps $(1,31)$, giving a mixed-base realization of the Monster McKay–Thompson series $T_{31A}$ through the classical Rogers–Ramanujan function $U(1,31)$. A theta dissection directly derives the modular target. For the underlying quadratic exponent $3r^2+31rs+93s^2$, we prove nonmodularity of every individual sum with rational linear terms and normalization, together with an exclusion from finite Fricke systems with meromorphic finite-width expansions at infinity. Exact agreement through $q^{2000}$ and six vanishing radial corrections support the proposed combination; a five-term dilogarithm certificate and an exact amplitude calculation prove its leading asymptotic behavior. We further propose a three-component Fricke law for a dual combination, with an explicit icosahedral matrix involving fifth-root constants, and a quadratic identity linking its components to $T_{31A}$. The dual translation law and matrix algebra are proved; its Fricke transformation and moonshine identity remain conjectural, supported respectively by numerical evaluations and exact finite coefficient checks.

We prove that every Voronoi cell of a unit-ball packing of $\mathbb{R}^4$ has volume at least $8$, with equality only at the $D_4$ configuration, so that the twenty-four-cell conjecture holds and the density of a sphere packing in four dimensions is at most $\pi^2/16$. The proof runs through the number of contacts of a cell. Up to twenty-two a covering estimate suffices; at twenty-three the cell is bounded through an exact volume identity inside a ball and a semidefinite certificate for one inequality between pair angles; at twenty-four the configuration is the root system, which we prove from the second level of the semidefinite hierarchy with its equality case, the positive kernel verified in exact arithmetic.

We establish graph-theoretic analogues of the Brauer-Kuroda relations for ramified Galois covers of finite graphs with arbitrary finite Galois groups. These formulas relate the numbers of spanning trees of intermediate quotient graphs and recover the corresponding formulas for unramified covers. Our proofs use $h$-functions defined on multiplicity spaces of complex representations of the Galois group.

In this note, we present an integral representation of the $k$-generalized Fibonacci sequence and provide a proof using contour integration. This representation enables the derivation of several known identities and a Binet-type formula related to the $k$-generalized Fibonacci sequence.

Florian Luca devoted the last paper of his life to the set $\mathcal K$ of odd positive integers $k$ such that $2^n k+1$ is a Carmichael number for some positive integer $n$. He proved that if $m\ge 5$ is fixed and odd, then there are only finitely many primes $p$ with $p^m\in\mathcal K$. In this note, written as a small tribute to his memory, we extend his result in three directions, using only his published work as our reference base. First, we extract from his arguments explicit quantitative bounds: every Carmichael number $N=2^n p^m+1$ with $m$ odd satisfies $n<2^{2\cdot10^7(m+1)^2m^2(\log p)^2}$, and if $p>m$ the much sharper bound $n<25m^4\log p$ holds; for $p>500m^5\log m$ we further record the uniform bounds $\omega(N)\le 26m^7$ on the number of prime factors. As an application, the number of such Carmichael numbers up to $x$ is $O_m(x^{1/m})$. Second, we generalize the multiplicative independence step of Luca's proof from base $2$ to an arbitrary even base $b$: if $b$ is even, $m\ge3$ is odd and coprime to $b$, $p>m$ is prime and $N=b^n p^m+1$ is Carmichael, then $b^n p^m$ and $b^a p^\beta$ are multiplicatively independent whenever $b^a p^\beta+1$ is a prime factor of $N$. Third, we report an exhaustive computation showing that the only Carmichael number $2^n p^m+1\le 10^{18}$ with odd $3\le m\le 37$ is $1729=2^6\cdot 3^3+1$, that no such number exists with odd $5\le m\le 37$ (extended to $10^{21}$ for $5\le m\le 15$), and that the only example with even $4\le m\le 36$ is $46657=2^6\cdot 3^6+1=13\cdot 37\cdot 97$, where $3^6=27^2$ is the square of the smallest element of $\mathcal K$.

We study the detectability of split surfaces in isogeny graphs of principally polarized superspecial abelian surfaces, a question relevant to the security analysis of dimension-$2$ isogeny-based cryptography. Our approach uses refined Humbert invariants to replace explicit isogeny computations with primitive representation problems for positive definite quadratic forms in five variables. We develop algorithms to detect $(N,N)$-splittings and to compute the minimum $(N,N)$-splitting level of a superspecial Jacobian without constructing the corresponding isogeny path. The same framework detects embeddings of real multiplication (RM) orders through primitive representations of discriminants of real quadratic orders. For RM, we use exhaustive small-prime data together with $100{,}000$ random polarizations per prime for $227\leq p\leq1619$, testing primitive representations of square-free discriminants $D\leq100$; the first nontrivial primitively represented discriminant is typically small. We apply these methods experimentally in two regimes. For $11\leq p\leq251$, where the irreducible principal polarizations are known exhaustively, the automorphism data recovered from the refined Humbert invariant reproduces the counts of Ibukiyama–Katsura–Oort for the irreducible polarizations. For large parameters, we reach primes of $1000$ bits, where the splitting degrees produced satisfy $\log_2N\approx\log_2p$ with a distribution whose shape is stable across the whole range, so the detected $(N,N)$-isogeny has degree about $p^{2}$; the smallest observed largest prime-power divisor of $N$ remains small up to $250$ bits and increases sharply from $300$ bits onward.

For any positive integer $k$, let $J_{k}$ be Jordan's totient function. We show that $J_{k}$ is not injective when $k=3,4,6$.

We prove the long-standing conjecture that, for every odd prime $p$, the only integral solutions of $a^2-2=b^p$ are $(a,b)=(\pm 1, -1)$. We also completely solve the analogous equations $y^2-D=x^p$ for $D=3$ and $D=5$ and describe a general approach for other positive squarefree values of $D \not \equiv 1 \pmod 8$. The proof refines the interpolation determinant method for linear forms in two logarithms in special cases, introducing new ideas for both the arithmetic lower bounds and the analytic upper bounds.

In this paper we study the Borel summability and quantum modularity of the free energy computed by refined topological recursion on the Weber-type refined spectral curve. On the one hand, we verify expected properties, namely, Stokes rays come in a pair and the Borel-Laplace sum of the free energy coincides with the Barnes double gamma function. Somewhat unexpectedly, on the other hand, we prove that the generating function of the Stokes constants along each ray is a Jacobi quantum modular function. Our proof suggests that quantum modularity may be a general feature of refined topological recursion, applicable beyond the Weber-type refined spectral curve.

Let $F$ be a totally real Galois number field of degree $g\leq5$ and let $\ell$ be an odd prime. Let $E/F$ be an elliptic curve with an $F$-rational point of order $\ell$. Under explicit arithmetic and local hypotheses, together with an explicit non-vanishing condition modulo $\ell$, we prove that there are $\gg_{F,E,\ell}\frac{X^{1/(2g)}}{\log X}$ totally negative square classes $d\in F^\times/(F^\times)^2$ with $\left|\mathrm{N}_{F/\mathbb{Q}}\bigl(D(F(\sqrt{d})/F)\bigr)\right|<X $ for which $\operatorname{Sel}_\ell(E^d,F)$ is trivial. Such twists have rank zero and trivial $\ell$-part of the Shafarevich–Tate group. The condition is verifiable by a finite computation, which we carry out in an example. We lift the method of James and Ono from $\mathbb{Q}$ to the totally real setting, combining a theorem of Morrow, which relates the Selmer group of such a twist to the class group of the CM extension $F(\sqrt{d})$, with an indivisibility theorem of Takai for relative class numbers. Takai twists only by quadratic Hecke characters, whereas Morrow's conditions are local; we extend the twisting argument to primitive quadratic residue-class characters to make the two results compatible.

We address the problem of determining the degeneracy of a particular energy eigenvalue corresponding to a quantum particle trapped in a cubic box. In other words, we aim to find the number of ways a specific positive integer can be written as the sum of three natural number squares. This is a classic problem in number theory. We provide a closed-form, exact solution to the problem. We do not claim to provide proofs, as we have none; instead, we build up the formulae starting from simple cases where the answer could be deduced by studying the list of degeneracies. We present ample examples that could be verified against the degeneracy calculated numerically via the brute-force method. We arrived at the solution by meticulously studying the list of degeneracies for the first five million numbers, generated by Mathematica$^{\circledR}$.

We study the relationship between the integrals of Yuan's concave transforms and the characteristic volumes of adelic line bundles on projective varieties. We establish an identity in dimension one and an inequality in higher dimensions. These results yield counterexamples to an identity conjectured by Yuan.

Let $p>5$ be an odd prime, put $n=(p-1)/2$, and let $\left(\frac{\cdot}{p}\right)$ be the Legendre symbol. Define $D_p^{\pm}$ to be the following determinants \[ \begin{aligned} D_p^{\pm}&:=\det\left[(i\pm j)\left(\frac{i\pm j}{p}\right) \right]_{1\leq i,j\leq n}. \end{aligned} \] In this article, we give explicit formulas for these determinants modulo $p$ in terms of Bernoulli numbers. As a consequence of Reinhart's recent counterexample to the Ankeny–Artin–Chowla conjecture, we find a prime $p$ such that $D_p^{\pm} \equiv 0 \pmod p$. This gives a negative answer to a question of the Sun.

In this paper, we establish two classes of refined sum formulas for finite multiple zeta values of level two, a variant of finite multiple zeta values recently introduced by M. Kaneko et al. One is an explicit formula for the sum of level-two finite multiple zeta values over indices in which one prescribed component is odd and all the remaining components are even. The other is an Ohno–Zagier type relation for level-two finite multiple zeta values, which gives a generating function for sums of fixed weight, depth and height in terms of depth-one values. As applications, we derive sum formulas for fixed weight and depth and for several special heights.

Given any norm $\|\cdot\|_{\mathbb R^2}$ on $\mathbb R^2$, we prove that the Lagrange spectrum for $2$-vectors contains a Hall ray. Equivalently, every sufficiently small $\rho\ge0$ occurs as $\liminf_{q\to\infty}q^{1/2}\min_{p\in\mathbb Z^2}\|qx-p\|_{\mathbb R^2} $ for some $x\in\mathbb R^2\setminus\mathbb Q^2$. Combined with a recent result of Kleinbock \cite{Kleinbock}, this implies that the $2$-dimensional Lagrange spectrum is either a ray, mirroring the behavior for Dirichlet spectra, or a ray preceded by a dense part.

We study the arithmetic of the elliptic curves \[ A_m:y^2=x(x-m)(x+4m) \] attached to the $3/5$ congruent number problem. A difference of ternary representation numbers controls the relevant central $L$-values. For $p\equiv11\pmod{40}$ the ordinary Cassels pairing degenerates; we construct an explicit $4$-cover and show that the next Cassels–Tate pairing is governed by the normalized representation defect, equivalently by a factorial character, a Pell symbol, and a class number congruence. For composite parameters we compute the ordinary Cassels matrices of four twists and, in the two prime case, a degree $1024$ governing field for their joint distribution. The same higher descent extends to larger radicals: a second rational pushout determines the full $\Lambda'$ row of the next pairing. We also determine two explicit Selmer towers with four dimensional ordinary radical, and all finite $2$-power Selmer groups when the ordinary radical is one dimensional.

We study the mean square of the Euler–Zagier double zeta-function when the second variable moves vertically. Mean-square formulas in and beyond the region of absolute convergence were previously obtained by Matsumoto and Tsumura and by Ikeda, Matsuoka and Nagata. In particular, the latter authors determined the leading terms in the following three boundary cases: \[ \zeta_2\left(1+a,\frac12+it\right),\qquad \zeta_2\left(1+ib,\frac12+it\right),\qquad \zeta_2\left(1-a,\frac12+a+it\right), \] where $a\in\mathbb C$ with $\Re a>0$ and $b\in\mathbb R$ are fixed. The mean squares in the first and third cases, as well as in the second case when $b\neq0$, have leading terms of order $T\log T$. When $b=0$, the second case reduces to the corner $(1,1/2)$, where the leading term is of order $T(\log T)^3$. In the present paper, we refine these results by obtaining asymptotic formulas with remainder $o(T)$. In the case $b\neq0$, an additional oscillatory term of order $T$ occurs. We also show that, at the parameter points $(s_1,\sigma_2)=(k,\ell/2)$ with integers $k\geq1$ and $\ell\geq2$, the constant multiplying $T$ in the mean-square formula is a linear combination of multiple zeta values. At the boundary points $(k,1/2)$ with integers $k\geq1$, the harmonic finite parts of the corresponding divergent sums are expressed in terms of harmonic regularized multiple zeta values.

In this paper, we apply the resonance method to exhibit large values at the central point of the family of quadratic Hecke $L$-functions with prime-related moduli of the Gaussian field under the generalized Riemann hypothesis.

Let $G_n=\sum_{k=0}^n4^k\binom{2n-2k}{n-k}^{2}\binom{2k}{k}$. Zhi-Hong Sun conjectured that, for primes $p\equiv3\pmod4$, positive odd integers $m$, and $r\ge2$, the two-step congruence $G_{(mp^r-1)/2}\equiv p^2G_{(mp^{r-2}-1)/2}\pmod {p^{2r-1}}$ holds. We prove the stronger valuation statement $G_{(p^2M-1)/2}-p^2G_{(M-1)/2}\in p^{2v_p(M)+3}\mathbb Z_p$ for every positive odd $M$. The proof converts the sum to a terminating ${}_3F_2$, constructs a cancelled digit-transfer operator, and identifies a two-dimensional analytic quotient of its cubic difference operator. The quotient operator has characteristic polynomial $X^2-p^2$; its second trace is evaluated through the Gross–Koblitz formula and Greene's finite-field Dixon identity. A logarithmic-loss Green inverse converts this spectral identity into an actual integral analytic primitive. The exceptional prime $3$ requires a finite exact PARI/GP certificate, while the infinite tail is bounded symbolically. Thus the computation is finite, reproducible, and separated from the uniform part of the proof.

Let $p$ be an odd prime and let $A,B\subseteq\mathbb{F}_p$ be nonempty and distinct, with $|A|\ge |B|$. We give a complete classification of the pairs for which the mixed restricted sumset $A\mathbin{\widehat+} B=\{x+y:x\in A,\ y\in B,\ x\ne y\}$ has size $\min\{p,|A|+|B|-2\}$. The main new contribution concerns the interior range $|A|+|B|\le p-1$. For $|B|\ge3$ and $|A|-|B|\ge3$, criticality forces $A$ and $B$ to be common-endpoint arithmetic progressions, apart from a single common affine orbit with $(p,|A|,|B|)=(13,7,4)$. For $|A|-|B|\in\{1,2\}$, criticality itself forces $B\subsetneq A$, after which the restricted self-sum determines the deletion patterns. Together with the remaining cases, this yields the complete classification.