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Caroline L. Turnage-Butterbaugh

Publications and source records attributed to Caroline L. Turnage-Butterbaugh.

14 recordsLinked to original sources
arXiv2026

Pair Correlation of Zeros of the Riemann Zeta Function I: Proportions of Simple Zeros and Critical Zeros

Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem in 1973 concerning the pair correlation of zeros of the Riemann zeta-function and applied this to prove that at least 2/3 of the zeros are simple. In this paper, we investigate the versatility of the pair correlation method and show, for the first time, that it can be used to prove results on the horizontal distribution of zeros of the Riemann zeta-function. In earlier work we showed how to remove RH from Montgomery's theorem and, in turn, obtain results on simple zeros assuming conditions on the zeros that are weaker than RH. Here we assume a more general condition, namely that all the zeros $ρ= β+iγ$ with $T<γ\le 2T$ are in a narrow vertical box centered on the critical line with width $b/\log T$. We prove that under this assumption with $b=0.3185$ that at least $2/3$ of zeros are simple and on the critical line.

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arXiv2026

A decades-long breakthrough in zero-density estimates and primes in short intervals

The Riemann Hypothesis (RH) asserts that every nontrivial zero of the Riemann zeta-function has real part equal to $1/2$. A zero-density theorem provides evidence towards RH by bounding the number of zeros of the zeta-function with real part greater than $1/2$. In 2024, Larry Guth and James Maynard announced a new zero-density theorem which, for a key location in the critical strip, strengthens previous work of Ingham and is the first such improvement in over 80 years. This expository paper places this remarkable achievement in the context of the rich history of zero-density theorems and explores its implications on the distribution of primes in short intervals.

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arXiv2026

A guide to Tauberian theorems for arithmetic applications

A Tauberian theorem deduces an asymptotic for the partial sums of a sequence of non-negative real numbers from analytic properties of an associated Dirichlet series. Tauberian theorems appear in a tremendous variety of applications, ranging from well-known classical applications in analytic number theory, to new applications in arithmetic statistics, group theory, and the intersection of number theory and algebraic geometry. The goal of this article is to provide a useful reference for practitioners who wish to apply a Tauberian theorem. We explain the hypotheses and proofs of two types of Tauberian theorems: one with and one without an explicit remainder term. We furthermore provide counterexamples that illuminate that neither theorem can reach an essentially stronger conclusion unless its hypothesis is strengthened.

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arXiv2026

Critical Zeros and Unconditional Mean Value Theorems for twisted $\hbox{PGL}(2)$ and $\hbox{PGL}(3)$ $\mathrm{L}$-functions

Let $Π_{0}$ be a cuspidal automorphic representation of $\mathrm{PGL}_{3}(\mathbb{A}_{\mathbb{Q}})$. In this paper, we use Levinson's method to prove that, as $Q\to \infty$, at least $1/9$ of the zeros of the $L$-functions $L(s, Π_{0}\,\times\, χ)$ lie on the critical line, where $χ$ ranges over the family of primitive Dirichlet characters of conductor up to $Q$. This result is unconditional when $Π_{0}$ is self-dual, and otherwise holds under a mild condition. The key technical input is a new asymptotic formula with a power-saving error term for the mean square of the product of $L(s, Π_{0}\times χ)$ and a Dirichlet polynomial with arbitrary coefficients in both the $T$- and $Q$-aspects for the range $Q^ε\le T \le Q^{1/3-ε}$. When $T=Q^ε$, our asymptotic formula allows Dirichlet polynomials of length $θ<1/2-ε$; when $θ=0$, it gives a strong error term of size $O_ε(Q^{7/4+ε})$. Furthermore, our result provides evidence for the CFKRS conjectures for large twists and large vertical shifts. We also obtain corresponding results for $\mathrm{PGL}_{2}(\mathbb{A}_{\mathbb{Q}})$, which are fully unconditional, quantitatively stronger, and also appear to be new. This work develops a refined, flexible, and uniform version of the Asymptotic Large Sieve for $L$-functions that does not require any unproven progress toward the Generalized Ramanujan Conjecture. The arithmetic of $Π_{0}$ plays a crucial and delicate role in our argument. This work also makes extensive use of Mathematica to handle various elaborate Hecke algebra computations. Our mean value theorem is readily applicable to many other problems in analytic number theory.

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arXiv2025

The Alternative Hypothesis for Zeros of the Riemann Zeta-Function

In 2016, the first-named author introduced a formulation of the Alternative Hypothesis that assumes that consecutive zeros of the Riemann zeta-function are spaced at multiples of half of the average spacing, but does not assume that the zeros are simple. In this paper, we assume the Riemann Hypothesis and a similar formulation of the Alternative Hypothesis, and for each integer $k$ we obtain constraints on the density of pairs of zeros whose normalized differences are at $k/2$ times the average spacing. These constraints, in turn, restrict the density of (possible) multiple zeros. We also formulate a stronger version of the Alternative Hypothesis and show that it implies the Essential Simplicity Hypothesis.

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arXiv2025

Short mollifiers of the Riemann zeta-function

We apply the calculus of variations to construct a new sequence of linear combinations of derivatives of the Riemann $ζ$-function adapted to Levinson's method, which yield a positive proportion of zeros of the $ζ$-function on the critical line, regardless of how short the mollifier is. Our construction extends readily to modular $L$-functions. Even with Levinson's original choice of mollifier, our method more than doubles the proportions of zeros on the critical line for modular $L$-functions previously obtained by Bernard and Kühn--Robles--Zeindler, while relying on the same arithmetic inputs. This indicates that optimizing the linear combinations, an approach that has received relatively little attention, has a more pronounced effect than refining the mollifier when it is short. Curiously, our linear combinations provide non-trivial smooth approximations of Siegel's $\mathfrak{f}$-function in the celebrated Riemann--Siegel formula.

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arXiv2023

An unconditional Montgomery Theorem for Pair Correlation of Zeros of the Riemann Zeta Function

Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem concerning pair correlation of zeros of the Riemann zeta-function. One consequence of this theorem is that, assuming RH, at least $67.9\%$ of the nontrivial zeros are simple. Here we obtain an unconditional form of Montgomery's theorem and show how to apply it to prove the following result on simple zeros: Assuming all the zeros $ρ=β+iγ$ of the Riemann zeta-function such that $T^{3/8}<γ\le T$ satisfy $|β-1/2|<1/(2\log T)$, %lie in the thin box $\{s=σ+it: |σ-1/2|<1/(2\log T),\ T^{3/8}<t\le T\}$, then, as $T$ tends to infinity, at least $61.7\%$ of these zeros are simple. The method of proof neither requires nor provides any information on whether any of these zeros are on or not on the critical line where $β=1/2$. We also obtain the same result under the weaker assumption of a strong zero-density hypothesis.

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arXiv2023

On the Montgomery-Odlyzko method regarding gaps between zeros of the zeta-function

Assuming the Riemann Hypothesis, it is known that there are infinitely many consecutive pairs of zeros of the Riemann zeta-function within 0.515396 times the average spacing. This is obtained using the method of Montgomery and Odlyzko. We prove that this method can never find infinitely many pairs of consecutive zeros within 0.5042 times the average spacing.

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arXiv2022

Twisted $2k$th moments of primitive Dirichlet $L$-functions: beyond the diagonal

We study the family of Dirichlet $L$-functions of all even primitive characters of conductor at most $Q$, where $Q$ is a parameter tending to $\infty$. For an arbitrary positive integer $k$, we approximate the twisted $2k$th moment of this family by using Dirichlet polynomial approximations of $L^k(s,χ)$ of length $X$, with $Q<X<Q^2$. Assuming the Generalized Lindelöf Hypothesis, we prove an asymptotic formula for these approximations of the twisted moments. Our result agrees with the prediction of Conrey, Farmer, Keating, Rubinstein, and Snaith for this family of $L$-functions, and provides the first rigorous evidence beyond the diagonal terms for their conjectured asymptotic formula for the general $2k$th moment of this family.

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arXiv2021

On a conjecture for $\ell$-torsion in class groups of number fields: from the perspective of moments

It is conjectured that within the class group of any number field, for every integer $\ell \geq 1$, the $\ell$-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of the field). In nearly all settings, the full strength of this conjecture remains open, and even partial progress is limited. Significant recent progress toward average versions of the $\ell$-torsion conjecture has crucially relied on counts for number fields, raising interest in how these two types of question relate. In this paper we make explicit the quantitative relationships between the $\ell$-torsion conjecture and other well-known conjectures: the Cohen-Lenstra heuristics, counts for number fields of fixed discriminant, counts for number fields of bounded discriminant (or related invariants), and counts for elliptic curves with fixed conductor. All of these considerations reinforce that we expect the $\ell$-torsion conjecture is true, despite limited progress toward it. Our perspective focuses on the relation between pointwise bounds, averages, and higher moments, and demonstrates the broad utility of the "method of moments."

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arXiv2020

An effective Chebotarev density theorem for families of number fields, with an application to $\ell$-torsion in class groups

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree.

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arXiv2015

Gaps between zeros of GL(2) $L$-functions

Let $L(s,f)$ be an $L$-function associated to a primitive (holomorphic or Maass) cusp form $f$ on GL(2) over $\mathbb{Q}$. Combining mean-value estimates of Montgomery and Vaughan with a method of Ramachandra, we prove a formula for the mixed second moments of derivatives of $L(1/2+it,f)$ and, via a method of Hall, use it to show that there are infinitely many gaps between consecutive zeros of $L(s,f)$ along the critical line that are at least $\sqrt 3 = 1.732...$ times the average spacing. Using general pair correlation results due to Murty and Perelli in conjunction with a technique of Montgomery, we also prove the existence of small gaps between zeros of any primitive $L$-function of the Selberg class. In particular, when $f$ is a primitive holomorphic cusp form on GL(2) over $\mathbb{Q}$, we prove that there are infinitely many gaps between consecutive zeros of $L(s,f)$ along the critical line that are at most $< 0.823$ times the average spacing.

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arXiv2014

Consecutive primes in tuples

In a recent advance towards the Prime $k$-tuple Conjecture, Maynard and Tao have shown that if $k$ is sufficiently large in terms of $m$, then for an admissible $k$-tuple $\mathcal{H}(x) = \{gx + h_j\}_{j=1}^k$ of linear forms in $\mathbb{Z}[x]$, the set $\mathcal{H}(n) = \{gn + h_j\}_{j=1}^k$ contains at least $m$ primes for infinitely many $n \in \mathbb{N}$. In this note, we deduce that $\mathcal{H}(n) = \{gn + h_j\}_{j=1}^k$ contains at least $m$ consecutive primes for infinitely many $n \in \mathbb{N}$. We answer an old question of Erd\H os and Turán by producing strings of $m + 1$ consecutive primes whose successive gaps $δ_1,\ldots,δ_m$ form an increasing (resp. decreasing) sequence. We also show that such strings exist with $δ_{j-1} \mid δ_j$ for $2 \le j \le m$. For any coprime integers $a$ and $D$ we find arbitrarily long strings of consecutive primes with bounded gaps in the congruence class $a \bmod D$.

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