arXiv subjects
Publications and source records attributed to 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 $\sum_{0<γ\le T}|ζ(2ρ)/ζ'(ρ)|^{2k}$ for every fixed rational $k\ge1$, and for every fixed real $k\ge1$. At $k=1$ we give a new proof of the explicit bound, half of the main term conjectured by Ng.
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 Russian and little known in the West, established an explicit positive proportion of such zeros. Using modern techniques we optimise the arithmetic mean value at the heart of the method, which has an exact closed form for reciprocal-square-root coefficients and is improved further by a positive-semidefinite family of mollifiers. We thereby obtain the strongest result yet known from Selberg's method, proving that at least $7\%$ of the non-trivial zeros of the Riemann zeta function lie on the critical line.
We prove that, for every sufficiently large modulus $q\not\equiv2\bmod4$, at least $3/8-o(1)$ of the primitive Dirichlet characters $χ$ modulo $q$ have $L(1/2,χ)\ne0$. We also establish stronger proportions for moduli $q$ satisfying certain arithmetic conditions.
We prove that, for either a Steinhaus random multiplicative function or a Rademacher random multiplicative function $f$, and every $\eps>0$, almost surely $$ \left|\sum_{n\le x}f(n)\right|\ll_{\eps,f}\sqrt{x}(\log\log x)^{1/4+\eps}. $$ Together with Harper's almost sure lower bound, this determines the sharp logarithmic exponent in both models. This settles Harper's conjecture on the large fluctuations of random multiplicative functions.
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 $ρ=\tfrac12+iγ$ of $L(s,χ)$ are simple, we prove lower bounds for the discrete moments $\sum_{0<γ\le T}|L'(ρ,χ)|^{-2}$ and $\sum_{0<γ\le T}|L(2ρ,χ^2)/L'(ρ,χ)|^2$, uniformly in the conductor. The bounds capture the proportion $β/(1+β)$ of the conjectured asymptotics, where $β=\log T/\log qT$: this is one half whenever $\log q=o(\log T)$, recovering for fixed $q$ the Dirichlet analogues of theorems of Milinovich and Ng and of Sinha, and degrades to $1/(2+A)$ when $q=T^{A}$. We conjecture the true leading order asymptotics and their analogues when we average over the family of primitive characters modulo $q$.
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, respectively. We show this by giving a full asymptotic expansion of these sums.
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 prove a related result concerning the sum of $χ(ρ) X^ρ$ over the zeros of zeta, where $χ(s)$ 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.
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.
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.
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.
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.
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 as Shanks' conjecture. Conrey and Ghosh studied the second moment of the Riemann zeta function evaluated at its local extrema along the critical line to leading order. In this paper we combine the two results, evaluating the first moment of the zeta function and its derivatives at the local extrema of zeta along the critical line, giving a full asymptotic. We also consider the factor from the functional equation for the zeta function at these 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 $\frac{e^2 - 5}{2 π} T (\log T)^2$. 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.
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, including a generalisation to all higher-order derivatives $ζ^{(n)}(s)$, 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 $ζ^{(n)}(ρ)$.
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-trivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height $T$, as $T \rightarrow \infty$. We also specify what the asymptotic formula becomes when $X$ is a positive integer, highlighting the differences in the asymptotic expansions as $X$ changes its arithmetic nature.