Papers

Assuming a suitably uniform Hardy–Littlewood prime tuples hypothesis, we obtain a second-order asymptotic for the proportion of short intervals containing a prescribed number of primes, when the interval length is comparable to the average prime spacing. The leading term is Poisson, but the arithmetic correction differs from the binomial correction in Cramér's independent model and predicts a stronger bias toward counts near the mean. The proof combines inclusion–exclusion with singular-series estimates of Montgomery and Soundararajan and a finite-sieve argument that controls the required alternating sums as the truncation order grows. We also give a refined random model that reproduces the correction and numerical comparisons that support the prediction.

Spectral zeta functions provide a standard tool for regularizing determinants of differential operators in quantum physics. In a previous paper (J. Math. Phys. 66: 083505, 2025), we introduced a $p$-adic spectral zeta function for discrete spectra via a locally analytic interpolation function. In this paper we extend the framework to continuous spectra using the inverse Stieltjes transform. Starting from the resolvent of a bounded operator, we construct a generalized distribution, called the $p$-adic spectral distribution, and define bosonic and fermionic zeta functions and functional determinants. We establish their analytic properties, special value formulas, and Stirling expansions, and show that our previous framework is embedded into the present one in the discrete case. In this approach, the bosonic and fermionic cases exhibit an interesting symmetry. As an application, we consider the position operator in $p$-adic quantum mechanics on an arbitrary compact subset of $\mathbb{C}_{p}$.

A Belyĭ function on a smooth projective algebraic curve defined over a number field determines a bipartite graph called a dessin d'enfant. We study the regularity and automorphism groups of dessins with uniform passports. In previous papers, we proved that every passport of the form $[n,n,n]$, $[n,b^{q},n]$, or $[b^{q},b^{q},n]$ of genus at least $2$ admits a dessin with trivial automorphism group. In this paper, we prove the analogous result for passports of the form $[a^{p},b^{q},n]$. The proof is mainly based on a counting argument: we compare a lower bound for the number of permutation representations having the prescribed passport with an upper bound for the number admitting a nontrivial automorphism. Together with our previous results, this shows that every uniform unicellular passport of genus at least $2$ admits a dessin with trivial automorphism group.

We study the ergodic integrals for linear flows on the infinite staircase, a translation surface with infinite area, and deduce detailed information on the bias properties of local discrepancy functions for irrational numbers of bounded type.

Dujella, Jak\v setić and Pe\v carić recently established in \cite{DJP}, a chain of power-sum inequalities, built from Hölder's and Cauchy's inequalities and their converse forms, and applied it to the Fibonacci sequence using the identities $\sum_{i=1}^nF_i^2=F_nF_{n+1}$ and $\sum_{i=1}^nF_iF_{i+1}=F_{n+1}^2-\frac{1+(-1)^n}{2}$. We show that this machinery applies uniformly across the one-parameter family of $k$-Fibonacci and $k$-Lucas numbers of Falcón and Plaza, via the generalized identities \begin{align*} \sum_{i=1}^nL_{k,i}^2=\frac{L_{k,n}L_{k,n+1}-2k}{k},\qquad \sum_{i=1}^nF_{k,i}^2=\frac{F_{k,n}F_{k,n+1}}{k}, \end{align*} and \begin{align*} \sum_{i=1}^nL_{k,i}L_{k,i+1}=\frac{L_{k,n+1}^2}{k}-k+\bigl((-1)^n-1\bigr)\left(\frac{2}{k}+\frac{k}{2}\right), \end{align*} which recover the classical Fibonacci and Lucas identities at $k=1$. We further exploit the cross-identity $F_{k,i}L_{k,i}=F_{k,2i}$, valid for every $k\ge1$, to obtain a Cauchy-Schwarz-type inequality linking the two families that has no counterpart in the Fibonacci-only setting. At $k=1$, this specializes to a fully explicit elementary inequality between ordinary Fibonacci and Lucas numbers, alongside the corresponding Hölder and Cauchy-conversion refinements we obtain for ordinary Lucas numbers.

Finite Fibonacci and Lucas sums involving zeta values at negative integers can have individual terms far larger than their total. In this paper, we study the distribution of these term magnitudes as the even degree increases. For non-negative weights of at most exponential growth, we prove a finite exponential expansion for the normalized absolute-term sum with an explicit remainder at every fixed order. The normalized magnitudes converge in total variation to a discrete distribution determined by the even exponential generating function of the weights. For the Fibonacci and Lucas sums, this distribution has its unique maximum at index eight and we prove that the same index gives the largest finite-sum term at every even degree at least twenty-six. We also derive the factorial growth of the summation condition number. The signed identities are placed within the established Bernoulli-polynomial reflection framework and extended to a trace-one recurrence family, including repeated roots and a vanishing scale. Exact rational calculations and precision-refined evaluations illustrate the proved bounds and distinguish a fixed cancellation profile from the growing length of the sum.

We study the elementary continued fractions of Ramanujan. The entries considered are precursors of his hypergeometric and $q$-hypergeometric continued fractions. They appear in the first third of Chapter 12 of Ramanujan's second notebook.

We study the sequence $a(0)=0$, $a(1)=1$ and $a(n)=n-a(n-a(n-a(n-1)))$ for $n\ge 2$, listed as A076502 in the On-Line Encyclopedia of Integer Sequences. We identify $a(n)$ as a two-position shift in the greedy Padovan numeration system, with a finite-state correction. The proof constructs an addition automaton from an exact integer-carry invariant and certifies its completeness by finite-language inclusion; a synchronized automaton then verifies the nested recurrence. We establish bounded discrepancy from the line of slope $c$, where $c^3-c^2+2c-1=0$, and show that the exact set of offsets from $\lfloor cn\rfloor$ is $\{-1,0,1,2\}$. We construct an explicit 26-letter non-erasing morphic presentation of the first-difference word, prove that its least balance constant is 4, and give an effective procedure for enclosing the global discrepancy extrema to arbitrary accuracy. We formalize the recurrence identification, six-decimal discrepancy bound, exact offset set, concrete morphic identity, least balance constant, and an effective extrema algorithm in Lean. Separate exact computations refine the numerical enclosures.

We prove an incidence bound for bipartite graphs on finite subsets of $\mathbb{F}^2\times \mathbb{F}^2$ defined by Boolean combinations of polynomial equations of bounded degree. If such a graph is $K_{k,k}$-free and its vertex classes have sizes $m$ and $n$, then it has $O_{t,k}((mn)^{2/3}+m+n+mn/p)$ edges, where $t$ bounds the description complexity, $p$ is the characteristic of $\mathbb{F}$, and $1/p=0$ in characteristic zero. We also prove this bound for incidences between points and distinct geometrically irreducible components of a two-parameter polynomial family, allowing singular and nonreduced members. The proof extends Lewko's interpolation and contact-multiplicity method from lines to algebraic families. Applications include rich components, polynomial values on difference sets, and polynomial expansion.

We investigate the distribution of zeros of the function $\zeta_2(s,s)$, namely the function defined by putting $s_1=s_2=s$ in the Euler double zeta-function $\zeta_2(s_1,s_2)$. Our theorems provide refinements of observations, obtained numerically in a previous paper by the first author and M. Shōji, with rigorous theoretical proofs.

For each integer k >= 1, let sigma_k(n) = sum_{d|n} d^k, let sigma_k^{-1} denote its Dirichlet inverse, and let J_k denote the kth Jordan totient function. Using established prime-power values of sigma_k^{-1}, we first obtain the elementary criterion sigma_k^{-1}(n) + J_k(n) + 2 = 0 precisely when n is prime. At k = 1, this involves Euler's totient J_1 = phi and provides a prime-only comparison for the next equation. Our main result classifies the zeros of sigma_k^{-1}(n) + mu(n) + J_k(n) + 3: for k = 1, they are precisely the primes and 18; for k >= 2, they are precisely the primes. The proof separates integers according to their prime-exponent patterns. Sign and divisibility arguments exclude composite zeros outside the cube-free nonsquarefree case, where an additional divisibility condition isolates 18 at k = 1 and precludes composite zeros for higher k. Standard convolution and Lambert-series identities provide the arithmetic framework, while an intermediate-divisor relation explains the exceptional zero. These criteria are structural characterizations rather than efficient primality tests, since direct evaluation of the multiplicative formulas generally presupposes a factorization of n.

We compute the Beilinson–Bloch height of the generalized Ceresa cycle $\mathrm{Ce}(V)=\frac 1 2(V-[-1]_*V)$ for any cycle $V$ of pure dimension $d$ on an abelian variety $A$, in terms of an arithmetic intersection number of a certain adelic line bundle. As an application, we prove that this height is given by an adelic line bundle over algebraic families.

Let \(X\subset \mathbb P^n_{\mathbf w}\) be a weighted hypersurface over \(\mathbb F_q\), with associated stack \(\mathfrak X\). For \(\mathbf x\in X(\mathbb F_{q^r})\), let \(g_{\mathbf x}(r)=\gcd(k_{S(\mathbf x)},q^r-1)\), where \(k_S=\gcd(w_i:i\in S)\), and define \(\Theta_r(s)=\sum_{\mathbf x}g_{\mathbf x}(r)^{1-s}\) and \(Z_H^{\mathrm{can}}(\mathfrak X,s;t)=\exp(\sum_{r\ge1}\Theta_r(s)t^r/r)\). We prove the finite sector factorization \[ Z_H^{\mathrm{can}}(\mathfrak X,s;t)=\prod_{e\in E_{\mathbf w}} Z(X^{(e)}_{\mathbb F_{q^{o_e}}},t^{o_e})^{J_{1-s}(e)/o_e}, \] where \(X^{(e)}\) is the closed isotropy sector, \(o_e=\operatorname{ord}_e(q)\) for \(e>1\) and \(o_1=1\), and \(J_{1-s}\) is the Jordan totient function. Hence the specializations \(s=1,0,-1,-2,\ldots\) are rational in \(t\): \(s=1\) gives the Hasse–Weil zeta function of the coarse space, \(s=0\) the twist zeta function, and \(s=1-k\) the masses of the \(k\)-fold inertia stack. The factorization yields a functional equation when the nonempty sectors are self-dual and equidimensional; if they are also pure and geometrically irreducible, equidimensionality is necessary for every integral \(s\le0\). For weighted diagonal hypersurfaces of degree \(D\) with \(p\nmid D\), a finite Fermat cover gives purity and self-duality. We also prove a Lang–Weil asymptotic for isotropy entropy, with periodic leading term governed by isotropy periods and Frobenius orbits, and identify the function-field height with the stable stack height of Ellenberg–Satriano–Zureick-Brown, while the degree-based height remains a coordinate-complexity statistic.

We study the family of functions $V_z:s \mapsto \zeta(s) \exp (zs)$ with numerical experiments. For selected sequences $z_n=x_nu$ on rays, our experiments suggest that suitable fixed points of $V_{z_n}$ approach nontrivial zeros of $\zeta$ in a nearly logarithmic pattern. We formulate this behavior as conjectures with explicit restrictions on the ray. We prove a conditional result identifying finite limits as zeros, and give sufficient hypotheses for the convergence of refinement diagnostics. Substantive AI assistance in the preparation of this draft is disclosed in Section 5.

The lower asymptotic proportion of distinct nontrivial zeros of the Riemann zeta function, relative to the total number counted with multiplicity, is at least $0.8369928814\ldots$. Earlier work proves $0.83625\ldots$, and a report we have not verified claims $0.83672\ldots$. As in the proof of the bound $0.83625$, an unconditional version of Montgomery's pair-correlation theorem gives an asymptotic energy estimate. The new ingredient is a short matrix inequality with a free clipping parameter. It strengthens the lower bound for this energy in terms of the number of distinct zeros. The gain is a nonnegative correction from overlaps between different nearby zeros on the critical line, which is retained even when some of these zeros are double. The matrix inequality, the threshold lemma, the block dichotomy, the counting assembly and the exact arithmetic are proved formally in Lean 4. The constant relies on a computer-assisted local inequality from recent work that has not yet been refereed. That computation was re-run independently, and every imported input is listed. This paper is primarily an experiment in AI-assisted mathematical research (Section 4).

Let p be a prime, q = p, A = F_p[T], and let P be a monic irreducible of degree d with cyclotomic function field K_P and mod-P Teichmuller character omega_P. For a character chi = omega_P^i, Goss defined g(X, chi), the "congruent to one modulo p" part of the Artin L-function L(X, chi), and conjectured that deg_X g(X, omega_P^i) <= 1 for every q-magic index i; this is an analogue for function fields of Vandiver's conjecture, and was raised as an open problem by Angles. We disprove it. For every a in F_p^*, put P_a = T^p - T - a, irreducible of degree p, and let i = p^n - 1 with 0 <= n <= p-1, a q-magic index. We prove an exact congruence showing that the L-polynomial reduces to (1-X)^n modulo P_a, whence deg_X g = n-1. For p >= 5 and 3 <= n <= p-1 this gives deg_X g >= 2, contradicting Goss's conjecture; moreover deg_X g = n-1 is unbounded, attaining p-2.

In this paper, we study a comparison between $\infty$-adic and finite multiple zeta values in positive characteristic. We construct a surjective comparison map from the algebra of $\infty$-adic multiple zeta values to a quotient of the algebra of finite multiple zeta values. This map factors through the quotient by $\zeta_A(q-1)$, thereby establishing the well-definedness of a Kaneko-Zagier-type conjectural map. We further show that the comparison map is compatible with special values of Carlitz multiple polylogarithms, as well as with the corresponding $\star$-inverse values. As an application, we prove the upper bound predicted by Shi's dimension conjecture for finite multiple zeta values.

We study the distribution of monogenic integral binary cubic forms $f(x,y)$ with bounded discriminant and bounded monogenizer inside a fundamental domain for the action of $\mathrm{GL}_2(\mathbb{Z})$ on the space of real binary cubic forms. We show that when the monogenizer for $f$ is $[\pm1:0]$ (i.e., $f(\pm1,0)=1$), the corresponding set is fully concentrated in the cusp. We also prove a complementary result, namely, that when the monogenizers $[\pm1:0]$ (with height bounded in terms of the discriminant bound) are excluded, the resulting set has no escape of mass to the cusp. However, we surprisingly show that despite this non-escape of mass, the points in the fundamental domain are not equidistributed.

We introduce a formal multiple polylogarithm function (FMPF) and general formal Arakawa-Kaneko zeta values (GFAKZVs), and establish several useful properties analogous to those of the ordinary Arakawa-Kaneko zeta values by employing the method of iterated integrals, a powerful tool originating in algebraic topology. In particular, we prove various duality and sum formulas involving FMPFs and GFAKZVs. This allows us to derive, in a uniform manner, a number of known results concerning Arakawa-Kaneko zeta values and Kaneko-Tsumura $\psi$-values that were previously obtained by different approaches, thereby highlighting the unifying power of the iterated integral formalism.

We give a strict generalization of the LLL algorithm over number fields, based on the reduction theory of $\GL(n)$ over the adele ring of a number field. Our algorithm is free of heuristics, with rigorous bounds on output quality and complexity. As a consequence, we obtain a hierarchy of reductions from module-(H)SVP to ideal-HSVP, an example of which has runtime and approximation factors subexponential in the field degree. More importantly, we uncover a close connection between structured lattice reduction and a Diophantine approximation over number fields.

We construct skew cyclic codes in the rank metric from the torsion of the supersingular Drinfeld module $\phi_T=t+\tau^n$ over $\mathbb{F}_{q^r}$, where $t$ is a root of a monic irreducible $\mathfrak{p}_0\in\mathbb{F}_q[T]$ of degree $r$ coprime to $n$. For every prime $\mathfrak{p}\neq\mathfrak{p}_0$ of degree $d$ and every divisor $m$ of $n$, the torsion $\phi[\mathfrak{p}]$ is a free module over the subring $R_0=\kappa[\pi^m]$ generated by a power of the Frobenius $\pi=\tau^r$, $\kappa=\mathbb{F}_q[T]/\mathfrak{p}$, and the codes are the left ideals of the centralizer $R\cong\mathrm{Mat}_m(R_0)$ of $R_0$. We prove a lattice anti-isomorphism between codes and $R_0$-submodules of $\phi[\mathfrak{p}]$, and a rank-BCH bound whose designed distance is the true distance.

The Skolem–Mahler–Lech Theorem asserts that a non-degenerate linear recurrence sequence (LRS) has only finitely many zeros, but is ineffective in the sense that no general procedure is known to determine them. In this paper a pseudo-effective approach is introduced: rather than seeking effective bounds for every sequence, one obtains explicit bounds valid for all but a controlled number of sequences of bounded height, up to a natural equivalence. Such results are established for order $3$ LRS's, and extended to all higher orders subject to suitable height constraints on the characteristic roots. The underlying mechanism is a quantitative theory for exponential-polynomial equations subject to low-height multiplicative perturbations. In particular, it is shown that the finiteness phenomenon underlying the Skolem–Mahler–Lech Theorem persists under arithmetic perturbations whose height is permitted to grow linearly at a sufficiently small rate. As a further application, a moving-target finiteness result is established for linear orbits entering expanding arithmetic neighbourhoods of proper linear subspaces.

In this paper, we study the infinite reciprocal cubic sum $\sum_{k=n}^{\infty}\frac{1}{B_k^3}$ of balancing numbers. We obtain explicit bounds for this sum and use them to determine the integer part of its reciprocal.

In this article, we describe a method to obtain lower bounds for the 2kth moment of values of L-functions localised at the central point for all real k > 0. In particular, we handle the families of quadratic twists of self-dual irreducible unitary cuspidal automorphic representations of GL_2(A_Q) and GL_3(A_Q).

In recent work, authors defined $\bar{b}^k_{r,s}(n)$ to be the number of overcolored partition $k$-tuples wherein both even and odd parts are colored with $r$ and $s$ colors, respectively. This function generalizes the overcubic partition $k$-tuples studied by Chacon and Sellers. The present work gives a uniform framework for studying congruences for $\bar{b}^k_{r,s}(n)$ modulo odd primes. For example, we prove that, for $n\ge 1$, \begin{align*} \bar{b}_{4,2}^4(5n)\equiv 0\pmod{5}. \end{align*}

Let $r\geq11$ be a prime. We show that there are infinitely many integers $C$ for which the Fermat-type equation $$ x^r+y^r=Cz^p $$ has no non-trivial primitive solutions for all sufficiently large (in terms of $r$ and~$C$) prime exponents~$p$. The proof uses several Frey curves to force simultaneous Frobenius trace equalities at the primes above~$3$; a new trace separation argument shows that these equalities imply $3\mid x+y$, after which a further Frey curve and level lowering give a contradiction with the Ramanujan bound.

Motivated by the modern characterization of the Markoff spectrum via mechanical words and the Markoff Uniqueness Conjecture, we study the Lagrange constants of continued fractions generated by a parameterized family of substitutions $\phi_{a,b}$ defined by $0 \mapsto aa$ and $1 \mapsto bb$ with $b = a + 1$. Based on a three-fold arithmetic classification of pairs of rational numbers $0 \le x < y \le 1$, we investigate the order relations between the corresponding Lagrange constants $\mathcal{L}([\phi_{a,a+1}(G(x))])$ associated with the mechanical words $G(x)$. While an inequality is established for pairs of the second type (type (2)) whenever $a \ge 1$, the comparisons for type (1) and type (3) pairs hold for $a \ge 2$. Consequently, for any integer $a \ge 2$, where the order relations across all three types are completely determined, we establish an analogue of the Markoff Uniqueness Conjecture: the map $x \mapsto \mathcal{L}([\phi_{a,a+1}(G(x))])$ is injective on $\mathbb{Q} \cap [0, 1]$. Furthermore, supported by extensive numerical computations, we propose a general conjecture on the order relations for arbitrary $b > a$, highlighting a sharp phase transition in ordering behavior at the boundary $b = a^2 + 2$.

The present study explores integer and rational matrix solutions for the Diophantine equation $X^{4}+Y^{4}=2Z^{4}$, establishing that infinitely many solutions exist over different matrix rings, including $M_2(\mathbb{Z})$, $M_3(\mathbb{Z})$, $M_4(\mathbb{Z}).$ Furthermore, expanding the scope beyond the integer domain, the analysis demonstrates that an infinite number of matrix solutions can be found in degree-4 and degree-16 field extensions, specifically within $M_4(\mathbb{Q}(\sqrt[4]{l}))$ and $M_4(\mathbb{Q}(\sqrt[4]{l_1}, \sqrt[4]{l_2}))$ when the rational numbers involved lack perfect fourth powers and are not perfect squares.

Andrews and Newman introduced the minimal excludant $\mathrm{mex}(\lambda)$ of an integer partition $\lambda$ and studied the summatory function $\sigma\mathrm{mex}(n)$, and Baruah et al. refined this to the odd- and even-restricted functions $\sigma_o\mathrm{mex}(n)$ and $\sigma_e\mathrm{mex}(n)$. In this paper, we introduce and study the overpartition analogue $\overline{\sigma_o\mathrm{mex}}(n)$, defined as the sum of odd minimal excludants over all overpartitions of $n$. We first derive the exact generating function for $\overline{\sigma_o\mathrm{mex}}(n)$, and relate it to the bivariate generating function of Aricheta and Donato for the overpartition minimal excludant. Using elementary $q$-series arguments, together with a weight-one eta-quotient identity for $\varphi(q)^2$ verified via the Gordon–Hughes–Newman–Ligozat criterion, we establish an infinite family of congruences satisfied by $\overline{\sigma_o\mathrm{mex}}(n)$. Consequently, we obtain that $\overline{\sigma_o\mathrm{mex}}(0)=1$ and, for $n\ge1$, $\overline{\sigma_o\mathrm{mex}}(n) \equiv 0 \pmod{4}$ if and only if $n$ is a perfect square. We further obtain congruences for the partial sums and self-convolution of $\overline{\sigma_o\mathrm{mex}}(n)$; in particular, an infinite family of congruences modulo $8$ for the self-convolution, expressed in terms of the divisor functions $d_1(n)$ and $d_3(n)$. We conclude the paper by establishing a Hardy–Ramanujan-type asymptotic formula for $\overline{\sigma_o\mathrm{mex}}(n)$ via the Wright circle method.

When studying the dynamical Mordell-Lang conjecture by using the $p$-adic method, experts have found that return sets of the $p$-adic version of the DML problem can be very sparse. In this article, we investigate the similar phenomenon for a complex analytic version of the DML problem and the DML problem for backward orbits.