arXiv subjects

Kilian Rausch

Publications and source records attributed to Kilian Rausch.

4 recordsLinked to original sources

arXiv2026

The difference of the sums of odd and even parts of restricted partitions

For a subset $D \subset \mathbb{N}$, denote by $S\left(D,n\right)$ the total sum of all odd parts minus the sum of all even parts of all partitions of $n$ in which parts from $D$ do not repeat. In this paper, we derive the generating function for $S\left(2\mathbb{N},n\right)$ and use it to give some congruences modulo 4. We also derive the generating function for $S\left(2\mathbb{N}-1,n\right)$ and give two congruences for these functions, one of which was previously proved by Garvan and Sarma.

math.NT↗

arXiv2026

A Rademacher exact type formula for pod$_2(n)$

In this paper, we calculate an exact formula for the number of partitions of a natural number $n$, where the largest part is even and no odd parts appears more than two times. The generating functions of the number of these partitions is a mixed mock modular form of weight 0. In order to obtain the formula we apply an extended version of the circle method, during which we need to bound Kloosterman sums and similar exponential sums as well as Mordell-type integrals.

math.NT↗

arXiv2025

Mock modular forms from the k-rank moments

In this paper, the generating functions of Garvans so-called $k$-ranks are used, to define a family of mock Eisenstein series. The $k$-rank moments are then expressed as partition traces of these functions. We explore the modular properties of this new family, give recursive formulas for them involving divisor like sums, and prove that their Fourier coefficient are integral. Furthermore, we show that these functions lie in an algebra that is generated only by derivatives up to a finite order but is nevertheless closed under differentiation. In the process, we also answer a question raised by Bringmann, Pandey and van Ittersum by showing that the divisor like sum $$\left(1-2^{\ell-1} \right) \frac{B_\ell}{2\ell}+ \sum_{2n-1 \geq bm \geq b} (2n-bm)^{\ell-1} q^{mn} - \sum_{m-1 \geq 2bn \geq 2b} (m-2bn)^{\ell-1} q^{mn},$$ has a quasi-completion, when $b\geq 3$ is odd.

math.NT↗

arXiv2022

Front-Running Protection for Distributed Exchanges using Tamper-Resistant Round Trip Time Measurements

In this paper we present ODIN, a front-running protection system that uses a novel algorithm to measure Round-Trip-Time (RTT) to untrusted servers. ODIN is the decentralized equivalent of THOR, a RTT-aware front-running protection system for trading on centralized exchanges. Unlike centralized exchanges, P2P exchanges have potentially malicious peers which makes reliable direct RTT measurement impossible. In order to prevent tampering by an arbitrarily malicious peer, ODIN performs an indirect RTT measurement that never interacts directly with the target machine. The RTT to the target is estimated by measuring the RTT to a randomized IP address that is known to be close to the target's IP address in the global routing network. We find that ODIN's RTT estimation algorithm provides an accurate, practical, and generic solution for collecting network latency data in a hostile network environment.

cs.DC↗