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Christopher Hughes

Publications and source records attributed to Christopher Hughes.

14 recordsLinked to original sources
arXiv2026

Generalisations of the Landau--Gonek Theorem and applications to mean values of zeta

The Landau--Gonek Theorem evaluates $X^\rho$ summed over the non-trivial zeros of the Riemann zeta function. Their result shows great sensitivity to the arithmetic nature of $X$. We prove a related result concerning the sum of $\chi(\rho) X^\rho$ over the zeros of zeta, where $\chi(s)$ is the term arising in the functional equation for the zeta function. Again, this result depends deeply on whether $X$ is an integer or not. We show the result splits into three cases, depending on whether $X$ is smaller than $T$, about the same size as $T$, or bigger than $T$. The reason this result is useful is that it easily permits the calculation of discrete moments of the Riemann zeta function via the approximate functional equation. As an application of this result, we provide an alternative proof of Shanks' conjecture.

math.NT↗
arXiv2026

The discrete second moment of mixed derivatives of the Riemann zeta function

We establish the full asymptotic for the discrete second moment of the Riemann zeta function of mixed derivatives evaluated at the zeta zeros, providing both unconditional and conditional error terms. This was first studied by Gonek, where only the leading order asymptotic was given, later extended by Conrey--Snaith and Milinovich to include the lower order terms for the first derivative. We extend the case of the first derivative to all derivatives.

math.NT↗
arXiv2025

Complex moments of the derivative of the Riemann zeta function

We conjecture results about the complex moments of the derivative of the Riemann zeta function, evaluated at the non-trivial zeros of the Riemann zeta function. We do this via two different random matrix computations. In the first, we find an exact formula for the complex moments of the derivative of the characteristic polynomials of unitary matrices averaged over Haar measure using Selberg's integral. In the second, we consider the hybrid approach for zeta, first proposed by Gonek, Hughes and Keating.

math.NT↗
arXiv2025

Integer moments of the derivatives of the Riemann zeta function

We conjecture the full asymptotic expansion of a product of Riemann zeta functions, evaluated at the non-trivial zeros of the zeta function, with shifts added in each argument. By taking derivatives with respect to these shifts, we form a conjecture for the integer moments of mixed derivatives of the zeta function. This generalises a result of the authors where they took complex moments of the first derivative of the zeta function, evaluated at the non-trivial zeros. We approach this problem in two different ways: the first uses a random matrix theory approach, and the second by the Ratios Conjecture of Conrey, Farmer, and Zirnbauer.

math.NT↗
arXiv2025

Coexisting Flux String Vacua from Numerical K\"ahler Moduli Stabilisation

We present a comprehensive study of K\"ahler moduli stabilisation in Type IIB flux compactifications, combining advanced numerical techniques with analytical methods. Our JAX-based computational framework enables efficient scanning of the UV parameter space, while incorporating $\alpha'$ corrections, loop and non-perturbative effects, as well as uplift contributions to the scalar potential. The implementation features rigorous vacuum validation protocols derived from analytic results. We apply our methods to explicit flux compactifications on more than 80,000 Calabi-Yau threefolds with $h^{1,1}\leq 6$ K\"ahler moduli. By systematically scanning over a wide range of values of the flux superpotential $W_0$ and the string coupling $g_s$, we find explicit realisations of every established K\"ahler moduli stabilisation scenario: for $10^{-15} \leq |W_0| \leq 10^{-2}$ we obtain both KKLT-like and K\"ahler uplifted vacua, while for the broader range $10^{-1} \leq |W_0| \leq 10^2$ we recover LVS as well as LVS-like hybrid solutions. Notably, we discover significant parameter regions where multiple vacua coexist within a single flux potential, including novel configurations pairing AdS, Minkowski, and dS minima with different volume hierarchies. These findings enable, for the first time, the analysis of vacuum decay processes within fixed flux configurations, complementing the established theory of transitions between distinct flux vacua and decays towards decompactification.

hep-th↗
arXiv2024

The second moment of the Riemann zeta function at its local extrema

Conrey and Ghosh studied the second moment of the Riemann zeta function, evaluated at its local extrema along the critical line, finding the leading order behaviour to be $\frac{e^2 - 5}{2 \pi} T (\log T)^2$. This problem is closely related to a mixed moment of the Riemann zeta function and its derivative. We present a new approach which will uncover the lower order terms for the second moment as a descending chain of powers of logarithms in the asymptotic expansion.

math.NT↗
arXiv2024

Back to the origins of brane-antibrane inflation

We study a new framework for brane-antibrane inflation where moduli stabilisation relies purely on perturbative corrections to the effective action. This guarantees that the model does not suffer from the eta-problem. The inflationary potential has two contributions: the tension of an antibrane at the tip of a warped throat, and its Coulomb interaction with a mobile brane. This represents the first realisation of the original idea of brane-antibrane inflation, as opposed to inflection point inflation which arises when the moduli are fixed with non-perturbative effects. Moreover, we formulate the brane-antibrane dynamics as an F-term potential of a nilpotent superfield in a manifestly supersymmetric effective theory. We impose compatibility with data and consistency conditions on control over the approximations and find that slow-roll inflation can occur in a large region of the underlying parameter space. The scalar spectral index is in agreement with data and the tensor-to-scalar ratio is beyond current observational reach. Interestingly, after the end of inflation the volume mode can, but does not need to, evolve towards a late-time minimum at larger values.

hep-th↗
arXiv2024

Moments of derivatives of the Riemann zeta function: Characteristic polynomials and the hybrid formula

We conjecture results about the moments of mixed derivatives of the Riemann zeta function, evaluated at the non-trivial zeros of the Riemann zeta function. We do this in two different ways, both giving us the same conjecture. In the first, we find asymptotics for the moments of derivatives of the characteristic polynomials of matrices in the Circular Unitary Ensemble. In the second, we consider the hybrid model approach first proposed by Gonek, Hughes and Keating.

math.NT↗
arXiv2023

A heuristic for discrete mean values of the derivative of the Riemann zeta function

Shanks conjectured that $\zeta ' (\rho)$, where $\rho$ ranges over non-trivial zeros of the Riemann zeta function, is real and positive in the mean. We present a history of this problem, including a generalisation to all higher-order derivatives $\zeta^{(n)}(s)$, for which the sign of the mean alternatives between positive for odd $n$ and negative for even $n$. Furthermore, we give a simple heuristic that provides the leading term (including its sign) of the asymptotic formula for the average value of $\zeta^{(n)}(\rho)$.

math.NT↗
arXiv2020

Probabilistic models for Gram's Law

Gram's Law describes a pattern that frequently occurs in the distribution of the non-trivial zeros of the Riemann zeta function along the critical line. Whenever Gram's Law holds true, it reduces the difficulty of computing the corresponding zeta zeros. In this paper, we provide a model that estimates how often this pattern occurs. The model is based on a conjecture that relates the statistical distribution of the zeta zeros to that of the eigenvalues of random unitary matrices.

math.NT↗
arXiv2019

SparseTrain:Leveraging Dynamic Sparsity in Training DNNs on General-Purpose SIMD Processors

Our community has greatly improved the efficiency of deep learning applications, including by exploiting sparsity in inputs. Most of that work, though, is for inference, where weight sparsity is known statically, and/or for specialized hardware. We propose a scheme to leverage dynamic sparsity during training. In particular, we exploit zeros introduced by the ReLU activation function to both feature maps and their gradients. This is challenging because the sparsity degree is moderate and the locations of zeros change over time. We also rely purely on software. We identify zeros in a dense data representation without transforming the data and performs conventional vectorized computation. Variations of the scheme are applicable to all major components of training: forward propagation, backward propagation by inputs, and backward propagation by weights. Our method significantly outperforms a highly-optimized dense direct convolution on several popular deep neural networks. At realistic sparsity, we speed up the training of the non-initial convolutional layers in VGG16, ResNet-34, ResNet-50, and Fixup ResNet-50 by 2.19x, 1.37x, 1.31x, and 1.51x respectively on an Intel Skylake-X CPU.

cs.LG↗
arXiv2018

The effect of repeated differentiation on $L$-functions

We show that under repeated differentiation, the zeros of the Selberg $Ξ$-function become more evenly spaced out, but with some scaling towards the origin. We do this by showing the high derivatives of the $Ξ$-function converge to the cosine function, and this is achieved by expressing a product of Gamma functions as a single Fourier transform.

math.NT↗
arXiv2013

Random permutation matrices under the generalized Ewens measure

We consider a generalization of the Ewens measure for the symmetric group, calculating moments of the characteristic polynomial and similar multiplicative statistics. In addition, we study the asymptotic behavior of linear statistics (such as the trace of a permutation matrix or of a wreath product) under this new measure.

math.PR↗