arXiv subjects

K. Srinivasa Raghava

Publications and source records attributed to K. Srinivasa Raghava.

7 recordsLinked to original sources

arXiv2026

Root-of-Unity Sections of Dilated Euler-Product Quotients

We study root-of-unity sections of quotients formed from dilated Euler products. At the boundary point $q=1$, odd sections tend to nonzero cyclotomic constants, whereas even sections have infinitely many simple real poles. We determine the locations, spacing, and residues of all sufficiently late poles. In the cubic case we find the leading oscillatory error in an elementary approximation and prove that its exponential scale is sharp. At the origin we determine exact contact, including a gap-dependent parity law, and integral divisibility with the first nonzero coefficient determined for every modulus $|k-1|\ge2$. For Euler powers $k\ge2$, a moving formal pole controls the section error at small fixed nomes; a real zero-free interval gives a further convergence regime. For the twenty-fourth power we prove explicit CM approximation bounds and a signed asymptotic as the dilation increases.

math.GM↗

arXiv2026

Arithmetic Constraints and Limit Laws for Diagonal Rational Partitions

Let $R_N(m)$ count unordered partitions of $m$ into reduced positive fractions whose numerators and denominators are at most $N$, excluding integer parts. Uniformly for $ρ$ in compact positive intervals, $\log R_N(\lfloorρN\rfloor)=\sqrt{2ρ}\,N^{3/2}-κ(ρ)N^{3/2}/\log N+o(N^{3/2}/\log N)$. The positive, continuously differentiable function $κ$ is an explicit sum of lattice-distance integrals, with $κ(1)\approx 0.00264713$. For a uniform partition of $n$, all but $o_{\mathbb{P}}(n/\log n)$ prime-denominator blocks in $(n/2,n]$ have total 2 or 3, according to whether $p/n$ lies below or above an explicit threshold near 0.7522. For fixed $N$, we give the Ehrhart numerator and determine how its poles control quasipolynomial coefficients. Its residue distribution is asymmetric for $N\ge 4$, but agrees with an independent model in every moment below order $\lceil N/2\rceil$. We identify the first discrepancy and prove a Gaussian limit. We also obtain joint denominator and size laws under two sampling rules.

math.NT↗

arXiv2026

Sampling and Moments of Reciprocal Quadratic Mellin Integrals

We study logarithmic Mellin integrals attached to a reciprocal quadratic rational function. When the rational kernel has algebraic coefficients, positive-integer samples of an entire deformation have a generating function which, after division by $π$, is algebraic for every $a\in\mathbb Q$ with $0<\vert{}a\vert{}<1$. For the principal half-weight example we determine the minimal quartic, complete finite branch locus, minimal differential equation, polynomial recurrence, and coefficient asymptotics. Throughout the saddle region $c>1$, $-2\sqrt c<b<0$, positive hyperbolic formulas for real $0<a<1$ give determinate Hausdorff moment sequences, strict total positivity, and monotone quotient limits. For real $\vert{}a\vert{}<1$, a general coefficient asymptotic in the same region shows that one maximum controls both the sampling radius and the high logarithmic moments. On the fixed-critical locus, two-term diagonal asymptotics give a normalized product tending to $π$; division by the displayed correction $1+γ/M$ gives an $O(M^{-2})$ approximation.

math.NT↗

arXiv2026

A Ramanujan Pell Elliptic K3 Surface and Primitive Five-Cube Near Misses

We construct a trace-16 Ramanujan--Pell family of primitive positive five-cube near misses \[ a_n^3+b_n^3+c_n^3+d_n^3+e_n^3=t_n^3+(-1)^{n+1}, \] obtained from a six-cube identity of quadratic forms and the negative Pell orbit generated by $8+\sqrt{65}$. The six coefficient sequences have rational recurrence generating functions with common reciprocal denominator \[ R(q)=1-257q-257q^2+q^3=(1+q)(1-258q+q^2). \] The quadratic identity is derived from a conic source, explaining the constants, while the Pell mechanism accounts for both the alternating error term and the denominator. The same construction yields the Mordell curve \[ E_K:\ y^2=x^3-432K(u)^2,\qquad K(u)=(1-13u+26u^2)^3+(6+182u^2)^3. \] We prove that its minimal smooth projective model is an elliptic K3 surface with geometric fibre configuration $6IV$; over $\mathbb Q$, the singular-fibre divisor is supported at one degree-two and one degree-four closed point. The section induced by the displayed decomposition of $K$ has canonical height $4/3$. The torsion groups over $\mathbb Q(u)$, $\mathbb Q(\sqrt{-3})(u)$, and $\overline{\mathbb Q}(u)$ are respectively $0$, $\mathbb Z/3\mathbb Z$, and $\mathbb Z/3\mathbb Z$. Over $\mathbb Q(\sqrt{-3})(u)$, the complex multiplication orbit of the section gives an Eisenstein Mordell--Weil sublattice and a visible generated rank-16 sublattice of the geometric Neron--Severi group of discriminant $-108$; no fullness or primitivity is claimed. We identify the remaining free Mordell--Weil problem with an explicit equivariant Hom module, exhibit an anti-symplectic reciprocal involution, and show that $w^3=K(u)$ has a Fermat-cubic quotient. No modularity assertion is made for the recurrence functions.

math.GM↗

arXiv2026

Finite Fourier Duality, Radial Values, and Arithmetic of a Mod Eight False Theta Quotient

We study a family of mod-eight false-theta moment quotients and determine their radial behaviour at every root of unity. The main structural result establishes an exact finite Fourier closure for the associated periodic boundary sequences. This closure yields a completed functional equation and proves the nonvanishing of the relevant periodic Dirichlet values at all negative odd integers. Consequently, each quotient exhibits either universal normalized factorial asymptotics or a finite nonzero radial value. The finite values belong to cyclotomic fields, satisfy Galois covariance, and specialize at the principal root to signed odd Springer numbers. We also determine exact coefficient divisibility, construct an optimal coefficientwise two-adic interpolation, and identify the unique dominant zero of the common denominator. This zero gives an unconditional coefficient decomposition. Finally, using exact boundary estimates and Möbius inversion, we prove positivity of every exponent in the formal Euler transform of the denominator.

math.NT↗

arXiv2026

Hyperbolic Arcsine Kernels, Finite Fourier Filters, and Quartic Central Binomial Harmonic Sums

Classical expansions of powers of the inverse sine contain central-binomial coefficients and finite repeated harmonic sums. We place the odd-square and ordinary-square coefficient families into two hyperbolic arcsine kernels and use these kernels as generating functions on which finite Fourier projection and Mellin deformation can be carried out before specialization. The quadratic projection extracts quartic subsequences and gives identities involving $\binom{4r}{2r}$, $π$, and $L=\log(1+\sqrt2)$. The same projection admits accelerated interior forms and, after Mellin deformation, denominator-power and logarithmic companions with polylogarithms at $(\sqrt2-1)^2$. The paper also records the square-law convolution between the two kernels, periodic-weight filters, finite spectral truncations at negative square parameters, and the analytic details needed for branch choices, boundary convergence, and termwise Mellin operations. A final comparison shows that direct quartic kernels lead to a different ${}_4F_3$ family.

math.NT↗

arXiv2026

Boolean Walsh Eta Units and Eisenstein Bases For Squarefree Levels

Let $M>1$ be squarefree and let $D(M)$ be its Boolean divisor cube. To each Boolean character $χ_T$ we attach the eta-quotient \[ R_T^{(M)}(τ)=\prod_{d\mid M}η(dτ)^{χ_T(d)}, \qquad χ_T(d)=(-1)^{|T\cap\supp(d)|}. \] At squarefree level, the finite Fourier transform on the divisor cube simultaneously diagonalizes the squarefree Ligozat cusp-order matrix, Fricke complementation, Atkin--Lehner action on cusp labels, and the constant-term map for logarithmic Eisenstein series. In particular, for $T\ne\varnothing$, \[ \ord_{1/c}R_T^{(M)} =\frac{Λ_T^{(M)}}{24}χ_T(c), \qquad Λ_T^{(M)}= \prod_{p\in T}(p-1) \prod_{\substack{p\mid M\\ p\notin T}}(p+1), \] and the forms $D\log R_T^{(M)}$ form a Walsh basis of the Eisenstein subspace of $M_2(Γ_0(M))$. The structural theorem also determines explicit Fricke constants, Atkin--Lehner eigenvalues, good-prime Hecke eigenvalues, local $U_p$ triangular blocks, a simultaneous bad-prime eigenbasis, and the indices of two explicit principal cuspidal divisor sublattices inside the formal degree-zero cusp-divisor lattice. As an application we specialize to the Heegner prime product \[ N=2\cdot3\cdot7\cdot11\cdot19\cdot43\cdot67\cdot163. \] The first Boolean boundary gives the eta-normalized Heegner-coloured partition product, while the top Walsh character gives the Möbius eta-unit identity \[ D\log R_{\mathcal P}^{(N)}(τ)=40415760- \sum_{n\ge1}σ_1(n^\perp)q^n. \] The same application gives algebraic modular-unit relations for the reciprocal partition product and exact Fricke-fixed logarithmic derivative identities for $1/π$, interpreted through the modular completion of $E_2$ and through accelerated paired products.

math.GM↗