arXiv subjects

Mahipal Gurram

Publications and source records attributed to Mahipal Gurram.

6 recordsLinked to original sources

arXiv2026

Bell Transforms of Arithmetic Functions: Euler Products, Congruences, and Polynomial Sequences

We present a unified algebraic framework utilizing the formal Bell transform to bridge the Dirichlet convolution of arithmetic functions with the combinatorial structure of infinite Euler-type products. By analyzing the logarithmic derivative of exponential generating functions, we establish explicit mappings between Bell exponents and M\"obius inversions. We apply this framework to derive exact vanishing properties and congruence inheritances for classical sequences, including Ramanujan's tau function and prime-colored partitions. Furthermore, we demonstrate that the inverse Bell transform seamlessly recovers classical partition recurrences and provides a discrete combinatorial engine for generating special polynomial families, including classical Appell and Sheffer sequences.

math.NT↗

arXiv2025

Explicit Lower Bounds for Dirichlet Series of Higher Power Representation Functions

We investigate Dirichlet-type series generated by representation functions that count the number of ways an integer can be expressed as a sum of 'k' signed higher even powers. By combining generalized theta generating functions with a family of generalized cotangent series introduced in previous work, we derive two distinct explicit lower bounds for these series. The first estimate arises from a geometric restriction of the lattice to its diagonal, while the second utilizes Holder's inequality on the integral representation of the series. The methods presented here avoid modular techniques and offer a flexible analytic framework for higher-power representation problems.

math.NT↗

arXiv2025

A Unified Transformation Formula for Ramanujan's Theta Function

In this paper, we derive a unified generalization of Ramanujan's transformation identities for the theta function $f(a,b)$, originally appearing in Ramanujan's Notebooks, Parts~III and IV. Using an approach based on residue-class dissections and modular substitutions, we obtain a closed transformation formula for $f(\zeta a, \zeta b)$, where $\zeta$ is a primitive root of unity $m$. As special cases, we recover and systematically prove Ramanujan's classical results for $m=2,3,$ and $4$, including even odd dissections, cubic transformation and the compact quartic form involving complex coefficients.

math.GM↗

arXiv2025

Bailey-Zeta Limits: A $q$-Series Bridge to Dirichlet $L$-Functions and the Riemann Zeta Function

We introduce a family of deformed Bailey pairs whose $q$-series, which converge in a two-step limit ($q \to 1^-$ followed by $n \to \infty$) to Dirichlet $L$-functions scaled by $1/\sqrt{\pi}$. This construction generalizes to arbitrary bounded arithmetic progressions via character weights, providing a unified $q$-series asymptotic for $L(s,\chi)$. Our approach unveils deep connections between the combinatorial machinery of Bailey chains and analytic number theory, with applications to special values like Euler-Mascheroni constant.

math.GM↗

arXiv2025

A New Representation of the Riemann Zeta Function

In this paper,we develop a novel representation of the zeta function expressed as the limiting difference between two structured double sums. This approach leads to a new and elegant identity involving maximum functions and additive terms, providing theoretical insights. The derivation relies on generalized harmonic series and polygamma functions, linking classical analysis with contemporary summation techniques.

math.NT↗

arXiv2025

Generalized Cotangent Series and Links Zeta and Theta Functions

This paper develops a generalized cotangent-type series, extending classical expansions to higher-order lattice sums. By introducing a new family of series indexed by integer powers, we derive closed form representations that combine trigonometric and hyperbolic structures, uncover recursive patterns, and establish integral connections to generalized Jacobi theta functions. The analysis reveals deep structural relationships between lattice summations, values of the Riemann zeta function, and modular kernels of theta type. Using techniques such as contour integration, Mellin transforms, and factorization identities, we obtain new expressions for even zeta values and formulate integral identities linking these series to generalized theta functions. These results unify classical expansions of trigonometric and hyperbolic cotangent functions within a broader analytic framework, offering new insights into modular forms.

math.NT↗