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Greg Hurst

Publications and source records attributed to Greg Hurst.

4 recordsLinked to original sources
arXiv2026

Practical Computations of the Mertens Function: $M(10^{24})$ and $M(10^{25})$

The Mertens function is defined as $M(x)=\sum_{n\leq x}\mu(n)$, where $\mu(n)$ is the M\"obius function. This paper describes a practical implementation of the classical $O(x^{2/3+\varepsilon})$ algorithm for computing $M(x)$ at isolated values, together with the segmented M\"obius and Mertens sieve on which it relies. The implementation was used to compute $M(10^{24}) = 7189337839$ and $M(10^{25}) = -258560632948$, taking $7.0$ days and $34.6$ days, respectively. These computations extend the previous record of $M(10^{23})$ by two orders of magnitude. Run standalone, the segmented sieve computed all Mertens values through $10^{16}$ in approximately $7.4$ days, compared with the $7.5$-month runtime of the author's 2018 computation. In pursuit of the most practical isolated-value method, an optimized implementation of the asymptotically faster Helfgott-Thompson algorithm is also presented, running roughly four times faster than the original implementation. With both implementations optimized, the choice between methods depends on input size and hardware. The contribution is a reproducible computation and implementation study, rather than a new asymptotic algorithm.

math.NT↗
arXiv2018

Freeform Assembly Planning

3D printing enables the fabrication of complex architectures by automating long sequences of additive steps. The increasing sophistication of printers, materials, and generative design promises to make geometric complexity a non-issue in manufacturing; however, this complexity can only be realized if a design can be translated into a physically executable sequence of printing operations. We investigate this planning problem for freeform direct-write assembly, in which filaments of material are deposited through a nozzle translating along a 3D path to create sparse, frame-like structures. We enumerate the process constraints for different variants of the freeform assembly process and show that, in the case where material stiffens via a glass transition, determining whether a feasible sequence exists is NP-complete. Nonetheless, for topologies typically encountered in real-world applications, finding a feasible or even optimal sequence is a tractable problem. We develop a sequencing algorithm that maximizes the fidelity of the printed part and minimizes the probability of print failure by modeling the assembly as a linear, elastic frame. We implement the algorithm and validate our approach experimentally, printing objects composed of thousands of sugar alcohol filaments with diameters of 100-200 microns. The assembly planner allows the freeform process to be applied to arbitrarily complex parts, from tissue engineering and microfluidics at the micrometer scale, to vascularized functional materials and soft robots at the millimeter scale, to structural components at the meter scale, thus opening a variety of assembly possibilities.

cs.CE↗
arXiv2016

Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture

The Mertens function is defined as $M(x) = \sum_{n \leq x} \mu(n)$, where $\mu(n)$ is the M\"obius function. The Mertens conjecture states $|M(x)/\sqrt{x}| < 1$ for $x > 1$, which was proven false in 1985 by showing $\liminf M(x)/\sqrt{x} < -1.009$ and $\limsup M(x)/\sqrt{x} > 1.06$. The same techniques used were revisited here with present day hardware and algorithms, giving improved lower and upper bounds of $-1.837625$ and $1.826054$. In addition, $M(x)$ was computed for all $x \leq 10^{16}$, recording all extrema, all zeros, and $10^8$ values sampled at a regular interval. Lastly, an algorithm to compute $M(x)$ in $O(x^{2/3+\varepsilon})$ time was used on all powers of two up to $2^{73}$.

math.NT↗