Number Theory
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Showing new listings for Friday, 12 June 2026
- [1] arXiv:2606.12484 [pdf, html, other]
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Title: Lower bounds on expressions depending on the functions {\boldmath$φ(n)$}, {\boldmath$ψ(n)$} and {\boldmath$σ(n)$}, IIISubjects: Number Theory (math.NT)
This work is concerned with the study of lower bounds for various expressions related to the arithmetic functions $\varphi(n)$, $\psi(n)$ and $\sigma(n)$. Several explicit estimates are established.
- [2] arXiv:2606.12660 [pdf, html, other]
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Title: Root Clusters and Multiclusters over Imperfect Hilbertian FieldsComments: 35 pagesSubjects: Number Theory (math.NT); Commutative Algebra (math.AC); Group Theory (math.GR)
We extend the theory of root clusters from perfect fields to general fields which are not necessarily perfect. We introduce the following notions for field extensions over any given base field and study their interesting properties: root cluster size, multicluster size and their generalizations root capacity, multiroot capacity; ascending index, ascending normal index and their generalizations intersection indicium, intersection normal indicium; compositum indicium and compositum normal indicium. We establish our results on the Inverse problems for these generalized notions over Hilbertian fields which generalizes our earlier results which were over number fields. In particular, we show over a given Hilbertian field, the existence of a polynomial for given degree, cluster size and multicluster size and existence of an extension for given root capacity and multiroot capacity with respect to that polynomial.
- [3] arXiv:2606.12776 [pdf, html, other]
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Title: An AI Proof of 18-Variable Undecidability for Diophantine Equations over $\mathbb Z[i]$Comments: Part of the proofs in this paper have been verified by LeanSubjects: Number Theory (math.NT)
This paper presents an AI proof that there is no algorithm deciding whether a polynomial equation over the Gaussian integers in $18$ unknowns has a solution. The proof improves the $20$-unknown theorem of Matiyasevich and Sun. It follows their rationality criterion and integer test, but saves two variables: the clearing-denominator variable is avoided by imposing two integer conditions, and the remaining nonzero condition is absorbed into the relation-combining lemma by the one-variable gadget $(2R+1)(3R+1)$.
- [4] arXiv:2606.12810 [pdf, html, other]
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Title: Splitting of Polynomial Families via Galois TheorySubjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
We study the splitting behavior of parametrized families of polynomials over finite fields through a geometric and Galois-theoretic approach. While the underlying techniques are widely considered folklore in arithmetic geometry, they have rarely been written down explicitly. To maximize accessibility, we develop a framework based on classical Galois theory and the Chebotarev Density Theorem over an affine normal variety, avoiding the heavy machinery of Grothendieck's étale topology.
The primary goal is to extend and conceptually explain a recent result by Slavov, which established the condition for square values of several polynomials over a finite field to be independent. In the case where $q\equiv 1 \pmod n$, we generalize this phenomenon to $n$-th power residues, and reframe this independence condition as the natural condition on Kummer extensions to be mutually linearly disjoint. Finally, we briefly mention how these results can be translated into the modern language of étale fundamental groups, generalizing the base to geometrically integral, normal schemes of finite type over $\mathbb{F}_q$. - [5] arXiv:2606.12907 [pdf, html, other]
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Title: Effective Estimates for a Class of Farey Fraction Sums and Bounds for Mundici-Type ConstantsComments: 27 pagesSubjects: Number Theory (math.NT)
Let $D_{2}(Q)$ denote the sum of squared distances between consecutive Farey fractions in the full interval $(0, 1]$. Daniele Mundici conjectured that $C(Q):=D_{2}(Q)\cdot Q^2/\log Q$ is less than 3 for all $Q\geq 2$, which is confirmed true in \cite{DLN2026}. In this paper, we generalize this result to subintervals of $(0, 1]$ and to $h$-spacings. As applications, we obtain Mundici-type bounds in these two settings, extending the full-interval consecutive-spacing case of Mundici's conjecture.
- [6] arXiv:2606.13018 [pdf, html, other]
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Title: Multi-target hyperbolic sieves and elliptic trace obstructionsComments: 23 pagesSubjects: Number Theory (math.NT)
Let $N=pq$ be a semiprime and let $\ell\nmid Na$ be an odd prime. The hyperbolic sieve set $H_a(N;\ell)=\{ax+Nx^{-1}:x\in\mathbb F_\ell^*\}$ contains the residue of the linear form $ap+q$ modulo $\ell$ and has exact cardinality $(\ell+\chi(aN))/2$, where $\chi$ is the Legendre symbol modulo $\ell$. We study simultaneous sieving for several linear forms and give a complete local analysis of the two-target primitive-root case proposed in connection with deterministic integer factorization. For two distinct coefficients $a,b$, with $A=4aN$ and $B=4bN$, we prove an exact formula for $|H_a(N;\ell)\cup H_b(N;\ell)|$ in terms of the degree-four character sum \[ K(A,B)=\sum_{z\in\mathbb F_\ell}\chi((z^2-A)(z^2-B)).\] For a smooth projective genus-one curve $E/\mathbb{F}_\ell$, we write $t_E=\ell+1-\#E(\mathbb{F}_\ell)$ for its Frobenius trace. With this convention, $K(A,B)$ is the Frobenius trace, up to sign and an additive constant, of the genus-one curve $Y^2=(X^2-A)(X^2-B)$. Hence Hasse--Weil gives a uniform $O(\sqrt\ell)$ error from the main term $3\ell/4$, and negative traces explain the counterexamples to the pointwise bound $3\ell/4+1$. We also prove a multi-target estimate \[\left|\left|\bigcup_{j=1}^k H_{a_j}(N;\ell)\right|-\ell(1-2^{-k})\right|\le (k-1+2^{-k})\sqrt\ell+k\] for distinct coefficients $a_1,\ldots,a_k$, together with the corresponding CRT product bound. Finally, for special-shape inputs $N=u^rv$, we study the $r$-power-constrained image $H_{a,r}(N;\ell)$ and determine its exact size by an elementary involution argument. These results recast the proposed local sieve questions as explicit finite-field statements with verified local tests.
- [7] arXiv:2606.13173 [pdf, html, other]
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Title: Ramanujan's and Lim's Identities and Harmonic Maass--Jacobi FormsComments: 19 pages, comments welcomeSubjects: Number Theory (math.NT)
We study an extension of Ramanujan's identities for odd zeta values by Lim and introduce Jacobi analogues of classical Eichler integrals of Eisenstein series. In negative weight we construct explicit completions and embed these objects into a modular framework by showing that they are (singular) harmonic Maass--Jacobi forms. We further describe their non-holomorphic parts in terms of Eichler integrals, establish Ramanujan-type inversion formulas, and study their behavior under the Maass raising and lowering operators and at torsion points.
- [8] arXiv:2606.13307 [pdf, html, other]
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Title: Discrete Fourier Transform Approach to Cyclically Covering Subspaces of $\mathbb{F}^n_q$Comments: 38 pagesSubjects: Number Theory (math.NT)
Let $q$ be a prime power and $n$ a positive integer. A subspace \( U \subseteq \mathbb{F}_q^n \) is called cyclically covering if the union of all its cyclic shifts covers the whole space \( \mathbb{F}_q^n \). Let \( h_q(n) \) denote the maximum possible codimension of such a subspace. When \(\gcd(q,n)=1\), we derive necessary and sufficient conditions for \(h_q(n)=0\) via Discrete Fourier Transforms, and prove this equality is equivalent to the existence of full-weight codewords in cyclic codes of \(\mathbb{F}_q^n\). We also characterize codimension-$k$ cyclically covering subspaces.
Based on these results, we give a unified characterization of \(h_q(n)\) in the case where $q$ and $n$ are primes with \(n>q\) and $q$ being a primitive root modulo $n$. Specifically, \(h_2(n) \geq 2\) and \(h_q(n) = 0\) for \(q \neq 2\). We prove that \(h_3(n) \ge 1\) for every prime \(n > 3\) with odd \(\operatorname{ord}_n(3)\). Moreover, for any prime \(q > 3\), the Generalized Riemann Hypothesis implies the existence of infinitely many primes \(n > q\) such that $q$ is not a primitive root modulo $n$ and \(h_q(n) = 0\). We provide algebraic interpretations for the inequalities \(h_q(mn)\ge\max\{h_q(m),h_q(n)\}\) and \(h_q(mn)\ge h_q(m)+h_q(n)\). Using Galois descent, we prove \(h_{q^m}(n)\le h_q(n)\). Furthermore, we generalize a class of constructions that achieve the upper bound \(\lfloor\log_q(n)\rfloor\). Finally, under the Generalized Riemann Hypothesis, we obtain average lower bounds of \(h_q(n)\) for $q=2,3$. - [9] arXiv:2606.13590 [pdf, html, other]
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Title: Some new modular Nahm sums of ranks 3 and 4Comments: 22 pagesSubjects: Number Theory (math.NT); Combinatorics (math.CO)
We discover six new families of modular Nahm sums in ranks 3 and 4. Two of them are rank three sums obtained by modifying two of Zagier's rank three examples. Three rank four families are derived by applying the lift-dual operation to the rank three tadpole Nahm sums studied by Milas and Wang, while the other rank four family is found by the constant term method. To prove modularity, we establish Rogers-Ramanujan type identities that express these Nahm sums as infinite products which are modular.
- [10] arXiv:2606.13619 [pdf, html, other]
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Title: Split primes and the Elekes-Rónyai problemSubjects: Number Theory (math.NT); Combinatorics (math.CO)
There exist an absolute constant $c>0$ and arbitrarily large finite sets $A\subset \mathbb{R}$ with $$\left| \left\{x+y+(x-y)^2:\ x, y \in A\right\}\right| \le|A|^{2-c}.$$ Since $x+y+(x-y)^2 \in \mathbb{R}[x,y]$ is a polynomial which is neither additive nor multiplicative, this provides a counterexample for the Elekes-Rónyai problem.
New submissions (showing 10 of 10 entries)
- [11] arXiv:2606.12947 (cross-list from math.DS) [pdf, html, other]
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Title: Trace spectra of simplices in large setsComments: 26 pagesSubjects: Dynamical Systems (math.DS); Combinatorics (math.CO); Number Theory (math.NT)
Given an ordered tuple $\mathbf v=(v_0,\ldots,v_d)$ of vectors in $\mathbb{R}^d$, let $A_{\mathbf v}=[\,v_1-v_0\ \cdots\ v_d-v_0\,]$ be its edge matrix. We prove that, in every finite colouring of $\mathbb{R}^d$, one colour class realizes every prescribed value of the higher characteristic coefficients \[
(c_2(A_{\mathbf v}),\ldots,c_d(A_{\mathbf v})). \] This extends Graham's theorem on volumes, which corresponds to the last coefficient $c_d(A_{\mathbf v})=\det(A_{\mathbf v})$. We also prove a discrete analogue: if $E\subseteq\mathbb{Z}^d$ has positive upper Banach density, then, for some $q\geq 1$, the set of coefficient tuples realized by ordered tuples in $E$ contains \[
q^2\mathbb{Z}\times q^3\mathbb{Z}\times\cdots\times q^d\mathbb{Z}. \] Finally, we show that the ordinary trace $c_1(A_{\mathbf v})$ cannot be added to these conclusions. The proof combines a quantitative directional expansion result for ergodic actions of free abelian groups with a trace calculation for a family of model edge matrices. - [12] arXiv:2606.13313 (cross-list from math.MG) [pdf, html, other]
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Title: Sphere Packings in Higher Dimension (after Boaz Klartag)Comments: This text is an English translation of the notes prepared (in French) for the Bourbaki seminar given by the author in June 2026Subjects: Metric Geometry (math.MG); Number Theory (math.NT); Probability (math.PR)
Let $\delta_n^L$ be the maximal density of a lattice sphere packing in the $n$-dimensional Euclidean space. We explain how Boaz Klartag proved the inequality $\delta_n^L \geq c n^2 2^{-n}$ where $c>0$ is a universal constant. In higher dimension, even for non-lattice sphere packings, this new lower bound is a substantial improvement.
Klartag's proof uses the probabilistic method in two different ways. The first, very standard, relies on the statistical properties of a uniformly chosen random lattice. The second, completely new, studies the stochastic evolution of an ellipsoid constrained to contain non nonzero lattice points in the interior.
Cross submissions (showing 2 of 2 entries)
- [13] arXiv:2504.14291 (replaced) [pdf, html, other]
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Title: The first moment of central value of primitive quartic $L$-functions with fixed genusSubjects: Number Theory (math.NT)
We investigate the mean value of the first moment of primitive quartic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Specifically, we study the sum
\begin{equation*}
\sum_{\substack{\chi\ primitive\ quartic\\ \chi^2 primitive\\ genus(\chi)=g}}L_q(\frac{1}{2}, \chi),
\end{equation*} where $L_q(s,\chi)$ denotes the $L$-function associated with primitive quartic character $\chi$. Using double Dirichlet series, we derive an error term of size $q^{(\frac{3}{5}+\varepsilon)g}$. - [14] arXiv:2504.20286 (replaced) [pdf, html, other]
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Title: Integers Having $F_{2k}$ in Both Zeckendorf And Chung-Graham DecompositionsComments: 17 pages, 2 tablesSubjects: Number Theory (math.NT)
Zeckendorf's theorem states that every positive integer can be uniquely decomposed into nonadjacent Fibonacci numbers. On the other hand, Chung and Graham proved that every positive integer can be uniquely written as a sum of even-indexed Fibonacci numbers with coefficients $0,1$, or $2$ such that between two coefficients $2$, there is a coefficient $0$. We discover a correspondence between a lexicographically ordered sublist of Zeckendorf decompositions and letters in the golden string $\mathcal{S}$. Likewise, we identify a dual correspondence for Chung-Graham decompositions. We then use these correspondences to give the set of all positive integers having $F_{2k}$ in both of their Zeckendorf and Chung-Graham decompositions.
- [15] arXiv:2508.07288 (replaced) [pdf, html, other]
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Title: Cup product of inhomogeneous Tate cochains, and Galois cohomology of tori over local fields that split over cyclic extensionsComments: v1 9 pages, v2 10 pages, v3 11 pagesSubjects: Number Theory (math.NT); Group Theory (math.GR); Representation Theory (math.RT)
In this note we give formulas for cup product in Tate cohomology in terms of inhomogeneous cochains. Using one of these formulas, for a torus T defined over a non-archimedean local field K and splitting over a cyclic extension of K, we compute explicit cocycles representing all cohomology classes in H^1(K,T).
- [16] arXiv:2508.21237 (replaced) [pdf, html, other]
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Title: The Carlitz module and a differential Ax-Lindemann-Weierstrass theorem for the Euler gamma functionComments: 32 pagesSubjects: Number Theory (math.NT); Complex Variables (math.CV)
We prove a differential transcendence result of type "Ax-Lindemann-Weierstrass" for Euler's gamma function. Given meromorphic functions $\zeta_1,\dots,\zeta_n$ of a complex variable $\nu$ that are pairwise distinct modulo $\mathbb Z$ and algebraic over the field $k$ of meromorphic $1$-periodic functions, the functions $ \Gamma(\nu-\zeta_1(\nu)),\dots,\Gamma(\nu-\zeta_n(\nu))$ are differentially independent over the field $k(\nu)$.
We determine the structure of certain difference field extensions related to the torsion of an avatar of the Carlitz module over meromorphic functions. These extensions are abelian and purely transcendental, the latter property being crucial in our main result, and obtained applying a criterion of differential algebraicity of Hardouin and Singer. - [17] arXiv:2509.00667 (replaced) [pdf, html, other]
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Title: On Triple Quadratic Residue Symbols in Real Quadratic FieldsComments: 18 pagesJournal-ref: Research in the Mathematical Sciences 13, 14 (2026), 22 ppSubjects: Number Theory (math.NT)
We introduce triple quadratic residue symbols $[\mathfrak{p}_1, \mathfrak{p}_2, \mathfrak{p}_3]$ for certain finite primes $\mathfrak{p}_i$'s of a real quadratic field $k$ with trivial narrow class group. For this, we determine a presentation of the Galois group of the maximal pro-2 Galois extension over $k$ unramified outside $\mathfrak{p}_1, \mathfrak{p}_2, \mathfrak{p}_3$ and infinite primes, from which we derive mod 2 arithmetic triple Milnor invariants $\mu_2(123)$ yielding the triple symbol $[\mathfrak{p}_1, \mathfrak{p}_2, \mathfrak{p}_3] = (-1)^{\mu_2(123)}$. Our symbols $[\mathfrak{p}_1, \mathfrak{p}_2, \mathfrak{p}_3]$ describes the decomposition law of $\mathfrak{p}_3$ in a certain dihedral extension $K$ over $k$ of degree 8, determined by $\mathfrak{p}_1, \mathfrak{p}_2$. The field $K$ and our symbols $[\mathfrak{p}_1, \mathfrak{p}_2, \mathfrak{p}_3]$ are generalizations over real quadratic fields of Rédei's dihedral extension of $\mathbb{Q}$ and Rédei's triple symbol of rational primes. We give examples of Rédei type extensions $K$ over real quadratic fields. We also give a cohomological interpretation of our symbols in terms of Massey products.
- [18] arXiv:2510.27581 (replaced) [pdf, html, other]
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Title: Sárközy's theorem in $\mathbb{F}_q[t]$ via the van der Corput propertyComments: 37 pagesSubjects: Number Theory (math.NT)
Fix a positive prime power $q$, and let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$ with $\text{char}(\mathbb{F}_q)>2$. Suppose $A \subseteq \{f \in mathbb{F}_q[t]: °f \leq N\}$ contains no pair of elements whose difference is of the form $P-1$ with $P$ irreducible. Adapting Green's approach to Sárközy's theorem for shifted primes in $\mathbb{Z}$ using the van der Corput property, we show that \[ |A| \ll q^{(N+1)(11/12+o(1))}, \] improving upon the bound $O\big(q^{(1-c/\log N)(N+1)}\big)$ due to Lê and Spencer. An important distinction between Green's argument and ours lies in the properties of exponential sums over function fields, which differ in several interesting ways from their number-field counterparts.
- [19] arXiv:2512.05023 (replaced) [pdf, other]
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Title: On inertial types of elliptic curvesComments: 31 pagesSubjects: Number Theory (math.NT)
We classify the inertial Weil-Deligne types arising from elliptic curves over all finite extensions $F/\mathbb Q_p$. Based on this classification, we give a fully explicit description of the types and implement an algorithm that computes all inertial types of elliptic curves defined over a given $F$. As an application, we determine all inertial types arising from elliptic curves over any extension $F/\mathbb Q_p$ of degree at most 3.
- [20] arXiv:2605.00590 (replaced) [pdf, html, other]
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Title: The Hurwitz sum-of-squares problem depends on the base fieldComments: 12 pages; added applications to Adem's conjecture and Shapiro's signed-formula problemSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG); Combinatorics (math.CO)
We show that the Hurwitz problem for sums of squares can depend on the base field. More precisely, we construct an explicit formula of type $[12,12,18]$ over every field of characteristic different from $2$ in which $-1$ is a square, whereas no such formula exists over any formally real field. In particular, a formula of this type exists over $\mathbb Q(i)$ and over $\mathbb C$, but not over $\mathbb Q$ or over $\mathbb R$. This settles, in the negative, a longstanding conjecture of Shapiro from 1984, a conjecture of Adem from 1975, and answers a signed-formula problem raised by Shapiro in 2000.
- [21] arXiv:2606.00525 (replaced) [pdf, html, other]
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Title: Polylogarithmic Analogues of Euler's ConstantComments: Revised version. Improved exposition, expanded proofs, added examples, and updated to the journal submission versionSubjects: Number Theory (math.NT)
We introduce a family of constants \[ C_m := \lim_{n\to\infty} \left( \sum_{k=1}^n \operatorname{Li}_m\!\left(\frac1k\right) - \log n \right), \] which may be regarded as polylogarithmic analogues of Euler's constant. We study their basic properties and derive representations in terms of iterated logarithmic integral structures associated with the gamma function. We further introduce associated polylogarithmic zeta potentials and polylogarithmic gamma functions, establish differential relations and integral representations, and describe logarithmic branch asymptotics near the singular points. As an application, we relate the constants \(C_m\) to special values of certain Dirichlet series involving the Riemann zeta function.
- [22] arXiv:2606.08089 (replaced) [pdf, html, other]
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Title: Waring's problem involving D.H. Lehmer numbersSubjects: Number Theory (math.NT)
For every positive integer $a$ which is coprime with $p$, $p$ is an odd prime, we denote by $\overline{a}$ the unique integer satisfying $1\leq \overline{a}\leq p$ and $a\overline{a}\equiv 1(\mathrm{mod}~p)$. Put $$L(p)=\{a\in Z^+:(a,p)=1,2\nmid a+\overline{a}\}.$$ The elements of $L(p)$ are called D.H. Lehmer numbers. The main purpose of this paper is to prove that for $p$ is a fixed odd prime, every sufficiently large number unless it is congruent to 15 or 16$(\mathrm{mod}~{16})$ is representable as the sum of 14 fourth powers of D.H. Lehmer numbers. Furthermore, every sufficiently large number is representable as the sum of 16 fourth powers of D.H. Lehmer numbers.
- [23] arXiv:2407.01952 (replaced) [pdf, html, other]
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Title: Groupoid homology and K-theory for algebraic actions from number theoryComments: Final revised version; 57 pages. To appear in the Journal of Functional AnalysisSubjects: Operator Algebras (math.OA); Algebraic Topology (math.AT); Dynamical Systems (math.DS); K-Theory and Homology (math.KT); Number Theory (math.NT)
We compute the groupoid homology for the ample groupoids associated with algebraic actions from rings of algebraic integers and integral dynamics. We derive results for the homology of the topological full groups associated with rings of algebraic integers, and we use our groupoid homology calculation to compute the K-theory for ring C*-algebras of rings of algebraic integers, recovering the results of Cuntz and Li and of Li and Lück without using Cuntz-Li duality. Moreover, we compute the K-theory for C*-algebras attached to integral dynamics, resolving the conjecture by Barlak, Omland, and Stammeier in full generality.