arXiv subjects

Siddarth Menon

Publications and source records attributed to Siddarth Menon.

3 recordsLinked to original sources

arXiv2026

Improved bounds for multiplicative functions in almost all short intervals

We refine the Matom\"aki-Radziwi{\l}{\l} method for short averages of multiplicative functions. For the Liouville function and multiplicative functions supported on smooth numbers, we prove decay bounds that are essentially optimal with the Matom\"aki-Radziwi{\l}{\l} method. The key new ingredient is a sharper treatment of the sieve error, achieved by introducing a more widely separated final prime range when restricting to integers with typical factorizations. We additionally give a weaker but still improved bound for arbitrary 1-bounded multiplicative functions and discuss some limitations of the method. As an application, we give an improved bound for the averaged Chowla conjecture of Matom\"aki-Radziwi{\l}{\l}-Tao that seems essentially best possible via their method.

math.NT↗

arXiv2024

Moments of random multiplicative functions over function fields

Granville-Soundararajan, Harper-Nikeghbali-Radziwill, and Heap-Lindqvist independently established an asymptotic for the even natural moments of partial sums of random multiplicative functions defined over integers. Building on these works, we study the even natural moments of partial sums of Steinhaus random multiplicative functions defined over function fields. Using a combination of analytic arguments and combinatorial arguments, we obtain asymptotic expressions for all the even natural moments in the large field limit and large degree limit, as well as an exact expression for the fourth moment.

math.NT↗

arXiv2023

Classifying Tractable Instances of the Generalized Cable-Trench Problem

Given a graph $G$ rooted at a vertex $r$ and weight functions, $\gamma, \tau: E(G) \rightarrow \mathbb{R}$, the generalized cable-trench problem (CTP) is to find a single spanning tree that simultaneously minimizes the sum of the total edge cost with respect to $\tau$ and the single-source shortest paths cost with respect to $\gamma$. Although this problem is provably $NP$-complete in the general case, we examine certain tractable instances involving various graph constructions of trees and cycles, along with quantities associated to edges and vertices that arise out of these constructions. We show that given a graph in which all cycles are edge disjoint, there exists a fast method to determine a cable-trench solution. Further, we examine properties of graphs which contribute to the general intractability of the CTP and present some open questions in this direction.

math.CO↗