Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.21291

Modularity criteria for multiple Eisenstein series

Jinbo Yu
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give two criteria for a linear combination of multiple Eisenstein series of fixed weight to be a modular form, one in terms of its Fourier coefficients and one in terms of its $L$-function. The first criterion follows from an exact transformation formula under $SL_2(\mathbb{Z})$, which describes the modular defect of multiple Eisenstein series explicitly. The second criterion is a Hecke converse theorem for multiple Eisenstein series, which shows that the functional equation of the $L$-function alone already implies modularity.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jinbo Yu. 2026-10-15. Modularity criteria for multiple Eisenstein series. https://arxiv.org/abs/2610.21291

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Solutions of $x_1^2+x_2^2-x_3^2=n^2$ with small $x_3$, revisited

Let $r(N)$ be the number of representations of $N$ as a sum of two squares. For every integer $n\ge 2$ we prove an asymptotic formula for $\sum_{x\equiv e \bmod 2}Φ(x/M)\,r(n^2+x^2)$, that is, for the number of integral solutions of $x_1^2+x_2^2-x_3^2=n^2$ with $x_3\equiv e \bmod 2$ in a range $|x_3|\ll M$, with smooth weight $Φ$. The formula has an explicit secondary term and a power saving in the error term, and it is non-trivial as soon as $M\ge n^{5/6+\varepsilon}$, that is, $M\ge D^{1/2-1/12+\varepsilon}$ with $D=n^2$. The leading term is $\frac{4}π(\log n)\intΦ(x/M)\,dx$, independently of the arithmetic of $n$; this corrects the main term stated in the published version (Ramanujan J. 63 (2024), 293-337). The proof factorises $n+ix_3$ in $\mathbb{Z}[i]$, which leads to Kloosterman sums. These are estimated by Weil's bound, combined with a Plancherel argument that exploits the curvature of the underlying quadric. This paper was written in collaboration with Claude (Opus 5.5, High), an AI model developed by Anthropic.

math.NT↗
arXiv2026

A Ceiling Continued Fraction Approach to the Erdős-Straus Conjecture: Heuristic finiteness of counterexamples

We introduce the Ceiling Continued Fractions (FCT) framework for constructing three-term Egyptian fraction representations in the Erdős--Straus conjecture from divisors of the shifted integers \(p+i\). A probabilistic model for their arithmetic structure leads to an asymptotic upper-bound heuristic for cumulative failure. The resulting bound is summable, and the first Borel--Cantelli lemma assigns probability zero to the event that infinitely many primes survive the full FCT block. Computational tests on \(10^9\) Mordell-type primes at each of the scales \(10^{17}\), \(10^{52}\), and \(10^{131}\) found no counterexamples.

math.NT↗
arXiv2026

Arcsine kernels, inverse-binomial harmonic sums, and classical hypergeometric transformations

We study inverse-central-binomial series with finite reciprocal-square factors and harmonic weights involving $H_n$, $H_{2n}$, and $1/n$. An arcsine integral gives explicit odd-zeta evaluations and a classification of coefficientwise cancellation within a polynomial ansatz. The scalar examples include an evaluation of weight seven and a direct proof of Sun's Conjecture 3.10. An entire generating relation organizes a second family with finite rational coefficient formulas. We then derive that relation, a supplementary generating family, and the complete two-order construction from Bailey's cubic transformation, contiguity, Gauss's quadratic and connection formulas, Euler's transformation, and Clausen's identity. The derivation retains all three infinity branches and resolves the vanishing denominators in the lower contiguous value. A doubling relation with a unique analytic solution and an explicit algebraic identity complete the proof. Thus the evaluations have a proved classical origin; no independent new hypergeometric summation principle is claimed. An application of a known complete-monotonicity property gives strict signs and global Taylor bounds. All endpoint limits, parameter derivatives, and coefficient extractions are justified.

math.NT↗