Andrew Pearce‐Crump

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20212026

most citedA discrete mean-value theorem for the higher derivatives of the Riemann zeta function

5 citations · 5 across the 14 of their papers we have counts for

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16 papers · 1 filter

math.NT2026

The sixth moment of a degree two $L$ -function of arbitrary level

Andrew Pearce-Crump

Let $f$ be a primitive holomorphic cusp form of arbitrary weight, level and nebentypus. We prove $\int_{0}^{T} ∣ L (1/2 + i t, f) ∣^{6} d t ≪_{f, ε} T^{2 + ε}$ , extending Jutila's level-one theorem…

math.NT2026

Optimising Selberg's method for critical zeros

Andrew Pearce-Crump

We revisit Zhuravlev's 1974 quantitative form of Selberg's sign-change method for the zeros of the Riemann zeta function on the critical line. Zhuravlev's work, originally in Russi…

math.NT2026

Central non-vanishing of Dirichlet $L$ -functions

Benjamin Durkan, Andrew Pearce-Crump

We prove that, for every sufficiently large modulus $q \neq \equiv 2 mod 4$ , at least $3/8 - o (1)$ of the primitive Dirichlet characters $χ$ modulo $q$ have $L (1/2, χ) \neq = 0$ . We also esta…

math.NT2026

A sharp almost sure upper bound for partial sums of random multiplicative functions

Benjamin Durkan, Andrew Pearce-Crump

We prove that, for either a Steinhaus random multiplicative function or a Rademacher random multiplicative function $f$ , and every $ε > 0$ , almost surely $|\sum_{n\le x}f(n…

math.NT2026

Negative discrete second moments of Dirichlet $L$ -functions

Andrew Pearce-Crump

Let $χ$ be a primitive Dirichlet character modulo $q > 1$ . Assuming the Generalised Riemann Hypothesis for $L (s, χ)$ and that the non-trivial zeros $ρ = \frac{1}{2} + iγ$ of $L (s, χ)$ are sim…

math.NT2026

Lower bounds for discrete moments in a conjecture of Ng

Andrew Pearce-Crump

Assuming the Riemann Hypothesis and the simplicity of the non-trivial zeros of the Riemann zeta function $ζ (s)$ , we prove lower bounds for the discrete moments $\sum_{0<γ\le T}|ζ(2…

math.NT2026

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

Benjamin Durkan, Christopher Hughes, Andrew Pearce-Crump

The Landau--Gonek Theorem evaluates $X^{ρ}$ summed over the non-trivial zeros of the Riemann zeta function. Their result shows great sensitivity to the arithmetic nature of $X$ . We p…

math.NT2026

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

Benjamin Durkan, Christopher Hughes, Andrew Pearce-Crump

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 cond…

math.NT2025

Integer moments of the derivatives of the Riemann zeta function

Christopher Hughes, Andrew Pearce-Crump

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…

math.NT2025

Complex moments of the derivative of the Riemann zeta function

Christopher Hughes, Andrew Pearce-Crump

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…

math.NT2024

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

Christopher Hughes, Solomon Lugmayer, Andrew Pearce-Crump

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^…

math.NT2024

Moments of the Riemann zeta function at its local extrema

Andrew Pearce-Crump

Conrey, Ghosh and Gonek studied the first moment of the derivative of the Riemann zeta function evaluated at the non-trivial zeros of the zeta function, resolving a problem known a…

math.NT2024

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

Christopher Hughes, Andrew Pearce-Crump

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 differ…

math.NT2023

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

Christopher Hughes, Greg Martin, Andrew Pearce-Crump

Shanks conjectured that $ζ^{'} (ρ)$ , where $ρ$ ranges over non-trivial zeros of the Riemann zeta function, is real and positive in the mean. We present a history of this problem, incl…

math.NT2021

A further generalisation of sums of higher derivatives of the Riemann Zeta Function

Andrew Pearce-Crump

We prove an asymptotic for the sum of $ζ^{(n)} (ρ) X^{ρ}$ where $ζ^{(n)} (s)$ denotes the $n$ th derivative of the Riemann zeta function, $X$ is a positive real and $ρ$ denotes a non-t…

math.NT2021★ 5 cited

A discrete mean-value theorem for the higher derivatives of the Riemann zeta function

Christopher Hughes, Andrew Pearce-Crump

We show that the $n$ th derivative of the Riemann zeta function, when summed over the non-trivial zeros of zeta, is real and positive/negative in the mean for $n$ odd/even, respecti…

The sixth moment of a degree two L-function of arbitrary level

Optimising Selberg's method for critical zeros

Central non-vanishing of Dirichlet L-functions

A sharp almost sure upper bound for partial sums of random multiplicative functions

Negative discrete second moments of Dirichlet L-functions

Lower bounds for discrete moments in a conjecture of Ng

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

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

Integer moments of the derivatives of the Riemann zeta function

Complex moments of the derivative of the Riemann zeta function

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

Moments of the Riemann zeta function at its local extrema

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

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

A further generalisation of sums of higher derivatives of the Riemann Zeta Function

A discrete mean-value theorem for the higher derivatives of the Riemann zeta function

The sixth moment of a degree two $L$ -function of arbitrary level

Central non-vanishing of Dirichlet $L$ -functions

Negative discrete second moments of Dirichlet $L$ -functions