arXiv subjects

Jinbo Yu

Publications and source records attributed to Jinbo Yu.

5 recordsLinked to original sources

arXiv2026

Schur Eisenstein series and Schur MacMahon series

We introduce and study two partition-indexed families of quasimodular forms obtained from Schur functions: Schur Eisenstein series and Schur MacMahon series. An explicit transition between them can be interpreted as a convolution in a Fa\`a di Bruno Hopf algebra of symmetric functions. We discuss the classical sl2-action and prove that Schur Eisenstein series for partitions with parts of size at most 3 give a basis for quasimodular forms. Further, we conjecture that the Schur MacMahon series span all quasimodular forms with integral coefficients.

math.NT↗

arXiv2026

The Young Tableaux Hopf algebra and multiple Schur series

In this paper, we introduce multiple Schur series, which are defined by Schur-type sums over semi-standard Young tableaux and generalize both Schur multiple zeta values and multiple Eisenstein series. To study their algebraic structure, we construct a connected, commutative, graded Hopf algebra of Young tableaux and identify its linearized quotient with the quasi-shuffle algebra. Within this Hopf algebra and its quotient, we establish several relations, including a hook formula and the Jacobi--Trudi formula. Furthermore, we relate this Hopf algebra to the ring of symmetric functions, which yields polynomial reduction formulas for tableaux with constant entries. As applications, we recover Schur multiple zeta values, introduce Schur multiple Eisenstein series together with a $q$-analogue of Schur multiple zeta values, and discuss their (quasi)modularity.

math.NT↗

arXiv2026

Positive density for Sun's $2^k+m$ conjecture

In 2013, Zhi-Wei Sun proposed a Romanov-type conjecture stating that every integer $n > 1$ can be written as $n = k + m$ with $k, m \ge 1$ such that $2^k + m$ is a prime. In this paper, we unconditionally prove that the natural numbers satisfying this property have a positive density. We compute this density to be at least $0.0734$. We also discuss the limitations of our method. Under a uniform Hardy-Littlewood prime pairs conjecture, we show that the lower bound of density obtained by this method cannot exceed $1/(\log 2 + 1) \approx 0.5906$.

math.NT↗

arXiv2026

Almost-primes in Sun's $x^2+ny^2$ conjecture

In 2015 Zhi-Wei Sun proposed the conjecture that any integer $n > 1$ admits a partition $n = x + y$ with integers $x, y >0$ such that $x + ny$ and $x^2 + ny^2$ are simultaneously prime. To approach this conjecture we use the method of weighted sieve as developed by Richert, Halberstam, and Diamond. In this article, we first formalize the conjecture into a sieve problem. We verify that the conditions required to use Richert's weighted sieve are satisfied and establish partial results with almost-prime solutions for sufficiently large $n$.

math.NT↗

arXiv2023

The mean square of the product of the Riemann zeta-function and a Dirichlet polynomial in the critical strip

We refine a previous work of K. Matsumoto and H. Ishikawa, obtaining an asymptotic formula for the mean square of the product of the Riemann zeta-function and a Dirichlet polynomial in the critical strip (1/4<$\sigma$<1/2), by obtaining an explicit formula of Atkinson type for its error term. This work is closely related to the generalized Dirichlet divisor problem. The form of these formulas is akin to, yet more complex than, Voronoi's formulas in the divisor problem.

math.NT↗