Peter Humphries

activity

20112026

most citedStandard Zero-Free Regions for Rankin--Selberg L-Functions via Sieve Theory

32 citations · 177 across the 24 of their papers we have counts for

collaborators

24 papers

math.NT2026

Spectral fourth moments of Hecke--Maaß cusp forms

Edgar Assing, Peter Humphries

In this note, we establish essentially optimal bounds for certain spectral moments of automorphic forms for $GL (2)$ . More precisely, we consider the family of Hecke--Maaß…

math.NT2022★ 1 cited

Sparse Equidistribution of Geometric Invariants of Real Quadratic Fields

Peter Humphries, Asbjørn Christian Nordentoft

Duke, Imamoglu, and Tóth have recently constructed a new geometric invariant, a hyperbolic orbifold, associated to each narrow ideal class of a real quadratic field. Furthermore, t…

math.NT2022★ 1 cited

$L^{p}$ -Norm Bounds for Automorphic Forms via Spectral Reciprocity

Peter Humphries, Rizwanur Khan

Let $g$ be a Hecke-Maass cusp form on the modular surface $SL_{2} (Z) \ H$ , namely an $L^{2}$ -normalised nonconstant Laplacian eigenfunction on ${\rm SL}_…

math.NT2022★ 3 cited

The Twelfth Moment of Hecke $L$ -Functions in the Weight Aspect

Peter Humphries, Rizwanur Khan

We prove an upper bound for the twelfth moment of Hecke $L$ -functions associated to holomorphic Hecke cusp forms of weight $k$ in a dyadic interval $T \leq k \leq 2 T$ as $T$ tends…

math.NT2021★ 7 cited

Test Vectors for Archimedean Period Integrals

Peter Humphries, Yeongseong Jo

We study period integrals involving Whittaker functions associated to generic irreducible Casselman-Wallach representations of $GL_{n} (F)$ , where $F$ is an archimedean local…

math.NT2021★ 3 cited

Zeros of Rankin-Selberg $L$ -functions in families

Peter Humphries, Jesse Thorner

Let $F_{n}$ be the set of all cuspidal automorphic representations $π$ of $GL_{n}$ with unitary central character over a number field $F$ . We prove the first uncond…

math.RT2020★ 1 cited

The Newform $K$ -Type and $p$ -adic Spherical Harmonics

Peter Humphries

Let $K := GL_{n} (O)$ denote the maximal compact subgroup of $GL_{n} (F)$ , where $F$ is a nonarchimedean local field with ring of integers $O$ . We s…

math.NT2020★ 3 cited

Archimedean Newform Theory for $GL_{n}$

Peter Humphries

We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman-Wallach representation of $GL_{n}$ that quantifies the extent to which this represen…

math.NT2020★ 6 cited

Towards a $GL_{n}$ variant of the Hoheisel phenomenon

Peter Humphries, Jesse Thorner

Let $π$ be a unitary cuspidal automorphic representation of $GL_{n}$ over a number field, and let $π ~$ be contragredient to $π$ . We prove effective upper and lower boun…

math.NT2020★ 6 cited

Distributing Points on the Torus via Modular Inverses

Peter Humphries

We study various statistics regarding the distribution of the points \[\left\{\left(\frac{d}{q},\frac{\overline{d}}{q}\right) \in \mathbb{T}^2 : d \in (\mathbb{Z}/q\mathbb{Z})^{\ti…

math.NT2019★ 10 cited

Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics

Peter Humphries, Maksym Radziwiłł

We asymptotically estimate the variance of the number of lattice points in a thin, randomly rotated annulus lying on the surface of the sphere. This partially resolves a conjecture…

math.NT2019★ 13 cited

On the Random Wave Conjecture for Dihedral Maaß Forms

Peter Humphries, Rizwanur Khan

We prove two results on arithmetic quantum chaos for dihedral Maass forms, both of which are manifestations of Berry's random wave conjecture: Planck scale mass equidistribution an…

math.NT2019★ 3 cited

Test Vectors for Nonarchimedean Godement-Jacquet Zeta Integrals

Peter Humphries

Given an induced representation of Langlands type $(π, V)$ of $GL_{n} (F)$ with $F$ nonarchimedean, we show that there exist explicit choices of matrix coefficient $β$ and Sch…

math.NT2019★ 14 cited

Motohashi's fourth moment identity for non-archimedean test functions and applications

Valentin Blomer, Peter Humphries, Rizwanur Khan +1

Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic L-functi…

math.NT2017★ 21 cited

Equidistribution in Shrinking Sets and L^4-Norm Bounds for Automorphic Forms

Peter Humphries

We study two closely related problems stemming from the random wave conjecture for Maass forms. The first problem is bounding the $L^{4}$ -norm of a Maass form in the large eigenvalue…

math.NT2017★ 3 cited

Biases in prime factorizations and Liouville functions for arithmetic progressions

Peter Humphries, Snehal M. Shekatkar, Tian An Wong

We introduce a refinement of the classical Liouville function to primes in arithmetic progressions. Using this, we discover new biases in the appearances of primes in a given arith…

math.NT2017★ 32 cited

Standard Zero-Free Regions for Rankin--Selberg L-Functions via Sieve Theory

Peter Humphries

We give a simple proof of a standard zero-free region in the $t$ -aspect for the Rankin--Selberg $L$ -function $L (s, π \times π)$ for any unitary cuspidal automorphic represe…

math.NT2016★ 10 cited

Density Theorems for Exceptional Eigenvalues for Congruence Subgroups

Peter Humphries

Using the Kuznetsov formula, we prove several density theorems for exceptional Hecke and Laplacian eigenvalues of Maass cusp forms of weight 0 or 1 for the congruence subgroups $Γ_…

math.NT2016★ 3 cited

Effective Lower Bounds for L(1,χ) via Eisenstein Series

Peter Humphries

We give effective lower bounds for $L (1, χ)$ via Eisenstein series on $Γ_{0} (q) \ H$ . The proof uses the Maass-Selberg relation for truncated Eisenstein series and s…

math.NT2015★ 4 cited

Spectral Multiplicity for Maaß Newforms of Non-Squarefree Level

Peter Humphries

We show that if a positive integer $q$ has $s (q)$ odd prime divisors $p$ for which $p^{2}$ divides $q$ , then a positive proportion of the Laplacian eigenvalues of Maass newforms of w…

math.AP2012★ 7 cited

Approximation and equidistribution of phase shifts: spherical symmetry

Kiril Datchev, Jesse Gell-Redman, Andrew Hassell +1

Consider a semiclassical Hamiltonian \begin{equation*} H_{V, h} := h^{2} Δ+ V - E \end{equation*} where $h > 0$ is a semiclassical parameter, $Δ$ is the positive Laplacian on $\mat…

math.NT2012★ 4 cited

On the Mertens Conjecture for Function Fields

Peter Humphries

We study an analogue of the Mertens conjecture in the setting of global function fields. Building on the work of Cha, we show that most hyperelliptic curves do not satisfy the Mert…

math.NT2012★ 1 cited

On the Mertens Conjecture for Elliptic Curves over Finite Fields

Peter Humphries

We introduce an analogue of the Mertens conjecture for elliptic curves over finite fields. Using a result of Waterhouse, we classify the isogeny classes of elliptic curves for whic…

math.NT2011★ 21 cited

The Distribution of Weighted Sums of the Liouville Function and Pólya's Conjecture

Peter Humphries

Under the assumption of the Riemann Hypothesis, the Linear Independence Hypothesis, and a bound on negative discrete moments of the Riemann zeta function, we prove the existence of…

Spectral fourth moments of Hecke--Maaß cusp forms

Sparse Equidistribution of Geometric Invariants of Real Quadratic Fields

Lp-Norm Bounds for Automorphic Forms via Spectral Reciprocity

The Twelfth Moment of Hecke L-Functions in the Weight Aspect

Test Vectors for Archimedean Period Integrals

Zeros of Rankin-Selberg L-functions in families

The Newform K-Type and p-adic Spherical Harmonics

Archimedean Newform Theory for GLn​

Towards a GLn​ variant of the Hoheisel phenomenon

Distributing Points on the Torus via Modular Inverses

Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics

On the Random Wave Conjecture for Dihedral Maaß Forms

Test Vectors for Nonarchimedean Godement-Jacquet Zeta Integrals

Motohashi's fourth moment identity for non-archimedean test functions and applications

Equidistribution in Shrinking Sets and L^4-Norm Bounds for Automorphic Forms

Biases in prime factorizations and Liouville functions for arithmetic progressions

Standard Zero-Free Regions for Rankin--Selberg L-Functions via Sieve Theory

Density Theorems for Exceptional Eigenvalues for Congruence Subgroups

Effective Lower Bounds for L(1,χ) via Eisenstein Series

Spectral Multiplicity for Maaß Newforms of Non-Squarefree Level

Approximation and equidistribution of phase shifts: spherical symmetry

On the Mertens Conjecture for Function Fields

On the Mertens Conjecture for Elliptic Curves over Finite Fields

The Distribution of Weighted Sums of the Liouville Function and Pólya's Conjecture

$L^{p}$ -Norm Bounds for Automorphic Forms via Spectral Reciprocity

The Twelfth Moment of Hecke $L$ -Functions in the Weight Aspect

Zeros of Rankin-Selberg $L$ -functions in families

The Newform $K$ -Type and $p$ -adic Spherical Harmonics

Archimedean Newform Theory for $GL_{n}$

Towards a $GL_{n}$ variant of the Hoheisel phenomenon