arXiv subjects

N. Guru Sharan

Publications and source records attributed to N. Guru Sharan.

8 recordsLinked to original sources

arXiv2026

Equivalence criteria for the two-term functional equations for Herglotz-Zagier functions

For any integer $a$ and non-negative integer $b$, we define a Herglotz--Zagier (HZ) type function $F_{a,b}(x)$ by an absolutely convergent series involving the Digamma function $ψ(x)$. For each such $F_{a,b}(x)$, we associate an integer weight. In the literature, Ramanujan, Guinand, Zagier, Vlasenko-Zagier have derived two-term functional equations for some HZ type functions of positive weights. In this paper, we study a class of HZ type function associated with negative weights, and obtain their two-term functional equations. Parallelly, we associate an integer weight to the Kronecker limit type formula for the generalized Mordell--Tornheim zeta function $Θ(r,s,t,x)$. We establish that any two-term functional equation for HZ type function is equivalent to a Kronecker limit type formula of $Θ(r,s,t,x)$, preserving weight. As a consequence, we derive new Kronecker limit type formulas and obtain a new special value of the Mordell--Tornheim zeta function $ζ_{\textup{MT}}(r,s,t)$. We also obtain results of Ramanujan, Guinand, Zagier, and Vlasenko-Zagier as consequences, to show that the Mordell--Tornheim zeta function lies centrally between many known modular relations.

math.NT↗

arXiv2025

Mordell--Tornheim zeta function: Kronecker limit type formulas and Special values

In this paper, we establish Kronecker limit type formulas for the generalized Mordell--Tornheim zeta function $Θ(r,s,t,x)$ as a function of the third variable, in terms of Riemann-zeta and Gamma values. We also give series evaluations of $Θ(r,s,t,x)$ in terms of Herglotz-Zagier type functions, and their derivatives. As applications of this, we derive Kronecker limit type formula in the second variable and a new infinite family of modular relations called mixed functional equations. We also study the zeroes, special values and singularities of the above function when all its arguments $r,s$ and $t$ are equal, which builds on a few earlier results due to Romik.

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arXiv2025

Rook decomposition of the Partition function

The rook numbers are fairly well-studied in the literature. In this paper, we study the max-rook number of the Ferrers boards associated to integer partitions. We show its connections with the Durfee triangle of the partitions. The max-rook number gives a new decomposition of the partition function. We derive the generating functions of the partitions with the Durfee triangle of sizes $3$, $4$ and $5$. We obtain their exact formula and further use it to show the periodicity modulo $p$ for any $p \in \mathbb{N}$ and $p\geq2$. We also establish their parity and parity bias. We give the growth asymptotics of partitions with the Durfee triangle of sizes $3$ and $4$. We obtain a new rook analogue of the recurrence relation of the partition function.

math.CO↗

arXiv2025

Partitions with Durfee triangles of fixed size

A well-studied statistic of an integer partition is the size of its Durfee square. In particular, the number $D_k (n)$ of partitions of $n$ with Durfee square of fixed size $k$ has a well-known simple rational generating function. We study the number $R_k (n)$ of partitions of $n$ with Durfee triangle of size $k$ (the largest subpartition with parts $1, 2, \ldots, k$). We determine the corresponding generating functions which are rational functions of a similar form. Moreover, we explicitly determine the leading asymptotic of $R_k (n)$, as $n \rightarrow \infty$.

math.CO↗

arXiv2025

On a function of Ramanujan twisted by a logarithm

A two-term functional equation for an infinite series involving the digamma function and a logarithmic factor is derived. A modular relation on page 220 of Ramanujan's Lost Notebook as well as a corresponding recent result for the derivative of Deninger's function are two main ingredients in its derivation. An interesting integral $\mathscr{H}(x)$, which is of independent interest, plays a prominent role in our functional equation. Several alternative representations for $\mathscr{H}(x)$ are obtained.

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arXiv2025

The Mordell-Tornheim zeta function: Kronecker limit type formula, Series Evaluations and Applications

In this paper, we establish Kronecker limit type formulas for the Mordell-Tornheim zeta function $Θ(r,s,t,x)$ as a function of the second as well as the third arguments. As an application of these formulas, we obtain results of Herglotz, Ramanujan, Guinand, Zagier and Vlasenko-Zagier as corollaries. We show that the Mordell-Tornheim zeta function lies centrally between many modular relations in the literature, thus providing the means to view them under one umbrella. We also give series evaluations of $Θ(r,s,t,x)$ in terms of Herglotz-Zagier function, Vlasenko-Zagier function and their derivatives. Using our new perspective of modular relations, we obtain a new infinite family of results called mixed functional equations.

math.NT↗

arXiv2024

Mordell-Tornheim zeta functions and functional equations for Herglotz-Zagier type functions

The Mordell-Tornheim zeta function and the Herglotz-Zagier function $F(x)$ are two important functions in Mathematics. By generalizing a special case of the former, namely $Θ(z, x)$, we show that the theories of these functions are inextricably woven. We obtain a three-term functional equation for $Θ(z, x)$ as well as decompose it in terms of the Herglotz-Hurwitz function $Φ(z, x)$. This decomposition can be conceived as a two-term functional equation for $Φ(z, x)$. Through this result, we are not only able to get Zagier's identity relating $F(x)$ with $F(1/x)$ but also two-term functional equation for Ishibashi's generalization of $F(x)$, namely, $Φ_k(x)$ which has been sought after for over twenty years. We further generalize $Θ(z, x)$ by incorporating two Gauss sums, each associated to a Dirichlet character, and decompose it in terms of an interesting integral which involves the Fekete polynomial as well as the character polylogarithm. This result gives infinite families of functional equations of Herglotz-type integrals out of which only two, due to Kumar and Choie, were known so far. The first one among the two involves the integral $J(x)$ who special values have received a lot of attention, more recently, in the work of Muzzaffar and Williams, and in that of Radchenko and Zagier. Analytic continuation of our generalization of $Θ(z, x)$ is also accomplished which allows us to obtain transformations between certain double series and Herglotz-type integrals or their explicit evaluations.

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arXiv2023

Modular relations involving generalized digamma functions

Generalized digamma functions $ψ_k(x)$, studied by Ramanujan, Deninger, Dilcher, Kanemitsu, Ishibashi etc., appear as the Laurent series coefficients of the zeta function associated to an indefinite quadratic form. In this paper, a modular relation of the form $F_k(α)=F_k(1/α)$ containing infinite series of $ψ_k(x)$, or, equivalently, between the generalized Stieltjes constants $γ_k(x)$, is obtained for any $k\in\mathbb{N}$. When $k=0$, it reduces to a famous transformation given on page $220$ of Ramanujan's Lost Notebook. For $k=1$, an integral containing Riemann's $Ξ$-function, and corresponding to the aforementioned modular relation, is also obtained along with its asymptotic expansions as $α\to0$ and $α\to\infty$. Carlitz-type and Guinand-type finite modular relations involving $ψ_j^{(m)}(x), 0\leq j\leq k, m\in\mathbb{N}\cup\{0\},$ are also derived, thereby extending previous results on the digamma function $ψ(x)$. The extension of Guinand's result for $ψ_j^{(m)}(x), m\geq2,$ involves an interesting combinatorial sum $h(r)$ over integer partitions of $2r$ into exactly $r$ parts. This sum plays a crucial role in an inversion formula needed for this extension. This formula has connection with the inversion formula for the inverse of a triangular Toeplitz matrix. The modular relation for $ψ_j'(x)$ is subtle and requires delicate analysis.

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