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Ankush Goswami

Publications and source records attributed to Ankush Goswami.

18 recordsLinked to original sources
arXiv2026

Machine-Guided Recurrence Boundary Theory for Nahm Sums

We develop a proof-carrying recurrence--boundary method for affine families of positive-definite Nahm sums. A universal coordinate-contiguous relation provides exact algebraic cells for finite certificates; a tropical face-limit theorem identifies parameter directions along which a higher-rank Nahm sum degenerates to a lower-rank theta or theta--hypergeometric boundary value; and a recurrence--boundary principle recovers the initial sums from sufficiently many independent boundary limits. Machine learning and reinforcement learning are used only for discovery. Evolutionary symbolic search proposes recurrences and asymptotic rays from exact algebraic data, while a $Q$-learning agent searches for short sequences of legal contiguous-cell identities. No learned output is accepted as proof: every successful candidate is replaced by an exact symbolic certificate. As the main application, we prove Shi and Wang's Conjecture~3.8 (arXiv:2607.23257) for the Nahm sums dual to Zagier's twelfth rank-three example. The search finds a second-order recurrence for a one-parameter family and a five-cell certificate for it, together with two asymptotic rays leading to binary and unary theta series. These give two linear equations for the two initial sums. Solving the resulting $2\times2$ system reduces the conjecture to two generalized-eta identities, certified by valence arguments on $Γ_1(300)$ and $Γ_1(100)$. Consequently the dual of Zagier's twelfth example is modular, and every member of the affine family is an explicit $\mathbb{Z}[q,q^{-1}]$-combination of the two base products. Complete verification artifacts accompany the paper.

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arXiv2026

Sign Patterns in a Two Colored Partition Companion series

We study two closely related questions arising from the recent work of Andrews and El Bachraoui on the two-color partition series \[ S_1(q)=\sum_{n\ge0}s_1(n)q^n=\sum_{a\ge0}q^a(-q^{a+1};q)_\infty^2 \] and its odd companion, denoted by $T_o(q)$. First, for the eta-normalized companion \[ C(q)=(q;q)_\infty T_o(q)=\sum_{n\ge0}c(n)q^n, \] we prove a strong form of the Andrews--El Bachraoui sign conjecture that $\limsup c(n)=+\infty$ and $\liminf c(n)=-\infty$. Second, we construct an involution using the Franklin-type involution of Chen and Liu to combinatorially explain Andrews--El Bachraoui congruence for $s_1(n)$ modulo 4.

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arXiv2026

Separable integer partition classes and Slater's list -- II

Slater's list of Rogers-Ramanujan type identities remains a central source of striking series-product formulas in the theory of partitions and basic hypergeometric series. Although many of these identities admit elegant analytic proofs through Bailey pairs, Bailey chains, or transformations of basic hypergeometric series, the partition-theoretic meaning of their series sides is often much less apparent. In this paper, which continues the program initiated in arXiv:2603.14179, we apply Andrews' theory of separable integer partition classes to further identities from Slater's list. We construct a strict overpartition class and two families of overpartitions with positional gap conditions, in which overlining is permitted only at alternating positions. Their multivariate generating functions give natural refinements of the series sides of Slater's identities (12), (28), (29), (47), (48), (50), and (51). We then use Heine-type transformations, a limiting form of Heine's transformations, Watson's $q$-analogue of Whipple's theorem, and classical theta-product identities to obtain alternative series representations and recover the associated products. In addition, we derive a new companion identity to a Slater identity and a signed companion formula. Our results further demonstrate that SIP classes provide a flexible framework for converting basic hypergeometric series into structured partition generating functions, while simultaneously producing refinements, transformations, and new Rogers-Ramanujan type identities.

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arXiv2026

Weighted partitions with interval restrictions: exact formulas and a bivariate master identity

Let $a_2''(n)$ and $b_2''(n)$ be the signed partition functions introduced by Andrews and El Bachraoui for interval-restricted partitions whose parts greater than $1$ are controlled by the smallest even part and by the number of ones. We prove two conjectures for these functions. The first gives the generating function for $a_2''(n)$ as an elementary rational term plus a false theta series with periodic signs; the second asserts that the companion coefficients $b_2''(n)$ take only the values $-1,0,1,2$. The central structural result introduces an auxiliary variable $z$ recording the number of non-compulsory parts greater than $1$. We obtain closed forms for the two resulting generating functions and prove the master identity $(1+q^2)\mathcal B(z,q)-(1+q)\mathcal A(z,q)=-q^4/(1-q^3)$ using both analytic and combinatorial techniques. At $z=-1$, this identity, together with a Rogers--Fine evaluation, gives the false theta formula for $a_2''(n)$ and an explicit generating function for $b_2''(n)$. The latter formula implies the asserted coefficient range and leads to an exact coefficient description of $b_2''(n)$. We also include a direct Heine--Rogers--Fine proof of the false theta formula, ordinary and fixed-refinement consequences of the master identity, and the resulting quantum modular interpretation.

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arXiv2026

Separable integer partition classes and Slater's list -- I

Slater's list of Rogers-Ramanujan type identities consists of 130 series-product identities whose analytic proofs rely primarily on Bailey pair techniques. Although these identities play an important role in the theory of $q$-series and partitions, combinatorial interpretations for many of them remain unknown, largely because the series sides are difficult to interpret naturally in terms of partitions. In this paper we apply Andrews' theory of separable integer partition (SIP) classes to several identities from Slater's list. By constructing suitable SIP classes, we obtain natural partition-theoretic interpretations and parameterized generalizations of their series sides. We then apply various $q$-hypergeometric transformations to these generalized series to derive alternative expressions, which in certain cases reduce to infinite products. These results illustrate how the SIP framework provides a systematic approach to understanding Rogers-Ramanujan type identities and offer new combinatorial insights into identities appearing in Slater's list.

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arXiv2025

Resurgence, Habiro elements and strange identities

We prove resurgence properties for the Borel transform of a formal power series associated to elements in the Habiro ring that come from radial limits of partial theta series via strange identities. As an application, we prove a conjecture in quantum topology due to Costin and Garoufalidis for two families of torus knots.

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arXiv2024

Parity results concerning the generalized divisor function involving small prime factors of integers

Let $ν_y(n)$ denote the number of distinct prime factors of $n$ that are $<y$. For $k$ a positive integer, and for $k+2\leq y\leq x$, let $S_{-k}(x,y)$ denote the sum \begin{eqnarray*} S_{-k}(x,y):=\sum_{n\leq x}(-k)^{ν_y(n)}. \end{eqnarray*} In this paper, we describe our recent results on the asymptotic behavior of $S_{-k}(x,y)$ for $k+2\leq y\leq x$, and $x$ sufficiently large. There is a crucial difference in the asymptotic behavior of $S_{-k}(x,y)$ when $k+1$ is a prime and $k+1$ is composite, and this makes the problem particularly interesting. The results are derived utilizing a combination of the Buchstab-de Bruijn recurrence, the Perron contour integral method, and certain difference-differential equations. We present a summary of our results against the background of earlier work of the first author on sums of the Möbius function over integers with restricted prime factors and on a multiplicative generalization of the sieve.

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arXiv2023

Asymptotics and sign patterns for coefficients in expansions of Habiro elements

We prove asymptotics and study sign patterns for coefficients in expansions of elements in the Habiro ring which satisfy a strange identity. As an application, we prove asymptotics and discuss positivity for the generalized Fishburn numbers which arise from the Kontsevich-Zagier series associated to the colored Jones polynomial for a family of torus knots. This extends Zagier's result on asymptotics for the Fishburn numbers.

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arXiv2023

Strong $q$-analogues for values of the Dirichlet beta function

An infinite class of relations between modular forms is constructed that generalizes evaluations of the Dirichlet beta function at odd positive integers. The work is motivated by a base case appearing in Ramanujan's Notebooks and a parallel construction for the Riemann zeta function. The identities are shown to be strong $q$-analogues by virtue of their reduction to the classical beta evaluations as $q\to 1^{-}$ and explicit evaluations at CM points for $|q|<1$. Inequalities of Deligne determine asymptotic formulas for the Fourier coefficients of the associated modular forms.

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arXiv2022

Combinatorial identities associated with a bivariate generating function for overpartition pairs

We obtain a three-parameter $q$-series identity that generalizes two results of Chan and Mao. By specializing our identity, we derive new results of combinatorial significance in connection with $N(r, s, m, n)$, a function counting certain overpartition pairs recently introduced by Bringmann, Lovejoy and Osburn. For example, one of our identities gives a closed-form evaluation of a double series in terms of Chebyshev polynomials of the second kind, thereby resulting in an analogue of Euler's pentagonal number theorem. Another of our results expresses a multi-sum involving $N(r, s, m, n)$ in terms of just the partition function $p(n)$. Using a result of Shimura we also relate a certain double series with a weight 7/2 theta series.

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arXiv2020

Quantum modularity of partial theta series with periodic coefficients

We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich-Zagier series $\mathscr{F}_t(q)$ which matches (at a root of unity) the colored Jones polynomial for the family of torus knots $T(3,2^t)$, $t \geq 2$, is a weight $3/2$ quantum modular form. This generalizes Zagier's result on the quantum modularity for the "strange" series $F(q)$.

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arXiv2020

On sums of coefficients of polynomials related to the Borwein conjectures

Recently, Li obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture. As a consequence, he showed the positivity of this sum. His result was based on a sieving principle discovered by himself and Wan. In fact, Li points out in his paper that his method can be generalized to prove an asymptotic formula for a general partial sum involving coefficients for any prime $p>3$. In this work, we extend Li's method to obtain asymptotic formula for several partial sums of coefficients of a very general polynomial. We find that in the special cases $p=3, 5$, the signs of these sums are consistent with the three famous Borwein conjectures. Similar sums have been studied earlier by Zaharescu using a completely different method. We also improve on the error terms in the asymptotic formula for Li and Zaharescu. Using a recent result of Borwein, we also obtain an asymptotic estimate for the maximum of the absolute value of these coefficients for primes $p=2, 3, 5, 7, 11, 13$ and for $p>15$, we obtain a lower bound on the maximum absolute value of these coefficients for sufficiently large $n$.

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arXiv2020

Some formulae for coefficients in restricted $q$-products

In this paper, we derive some formulae involving coefficients of polynomials which occur quite naturally in the study of restricted partitions. Our method involves a recently discovered sieve technique by Li and Wan (Sci. China. Math. 2010). Based on this method, by considering cyclic groups of different orders we obtain some new results for these coefficients. The general result holds for any group of the form $\mathbb{Z}_{N}$ where $N\in\mathbb{N}$ and expresses certain partial sums of coefficients in terms of expressions involving roots of unity. By specializing $N$ to different values, we see that these expressions simplify in some cases and we obtain several nice identities involving these coefficients. We also use a result of Sudler (QJMAAT 1964) to obtain an asymptotic formula for the maximum absolute value of these coefficients.

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arXiv2020

Congruences for generalized Fishburn numbers at roots of unity

There has been significant recent interest in the arithmetic properties of the coefficients of $F(1-q)$ and $\mathscr{F}_t(1-q)$ where $F(q)$ is the Kontsevich-Zagier strange series and $\mathscr{F}_t(q)$ is the strange series associated to a family of torus knots as studied by Bijaoui, Boden, Myers, Osburn, Rushworth, Tronsgard and Zhou. In this paper, we prove prime power congruences for two families of generalized Fishburn numbers, namely, for the coefficients of $(ζ_N - q)^s F((ζ_N - q)^r)$ and $(ζ_N - q)^s \mathscr{F}_t((ζ_N - q)^r)$, where $ζ_N$ is an $N$th root of unity and $r$, $s$ are certain integers.

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arXiv2018

On a partial sum related to the Euler totient function

Recently, Bordellés, Dai, Heyman, Pan and Shparlinski in \cite{Igor} considered a partial sum involving the Euler totient function and the integer parts $\lfloor x/n\rfloor$ function. Among other things, they obtained reasonably tight upper and lower bounds for their sum using the theory of exponent pairs and in particular, using a recently discovered Bourgain's exponent pair. Based on numerical evidences, they also pose a question on the asymptotic behaviour for their sum which we state here as a conjecture. The aim of this paper is to prove an asymptotic formula for the average of their sum in the interval $[1,x]$. We show via Perron's formula that this average is a certain weighted analogue of the sum that Bordellés \textit{et al} considered in their paper. Further, we show that their conjecture is true under certain conditions. Our proof involves Perron's contour integral method besides some analytic estimates of the Riemann zeta function $ζ(s)$ in the zero-free region.

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arXiv2012

Proof of Legendre's Conjecture

Legendre's conjecture states that there exists a prime between $n^2$ and $(n+1)^2$, for every positive integer $n$. Here I prove that for sufficiently large $n$, there is a prime number between $n^2$ and $(n+1)^2$. The proof relies on the idea of counting the maximum power, $o_p(n)$ of a prime $p\leq n$ such that $p^{o_p(n)}||n$.

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