arXiv subjects

Yochay Jerby

Publications and source records attributed to Yochay Jerby.

16 recordsLinked to original sources

arXiv2025

Variations of the Hardy Z-Function and the Montgomery Pair Correlation Conjecture

In 1973 Montgomery formulated the pair correlation conjecture, predicting that the local spacing statistics of the nontrivial zeros of the Riemann zeta function coincide with those of eigenvalues of large Hermitian matrices from the Gaussian Unitary Ensemble (GUE). The zeta function, however, is a fixed deterministic object, and the mechanism by which its zeros reproduce random matrix statistics has remained unclear. In this paper, assuming the Riemann Hypothesis, we prove Montgomery's pair correlation conjecture for the zeros of Hardy's $Z$-function. Building on earlier works, we use a finite-dimensional variational space of sections $Z_N(t;a)$ that approximate $Z(t)$ on each window $[2N,2N+2]$. Inside this space we define the real hall $\mathcal{RH}_N(\mathbb{R})$, consisting of those sections whose zeros in the corresponding critical rectangle are real, simple, and remain so along any homotopy from the core section. This real hall plays the role of a random matrix ensemble. Equipping it with an admissible probability measure with smooth, positive density on the coefficient space, we construct a Skorokhod-type stochastic differential equation with reflection at the discriminant boundary. We show that the induced dynamics of the unfolded zeros are, in the bulk, equivalent in law to Dyson Brownian motion with $β=2$, and by invoking modern universality results for Dyson Brownian motion and log-gases we obtain that, for any such measure, the ensemble-averaged local pair-correlation converges to the GUE sine-kernel law. Finally, using Selberg's probabilistic theory of the argument $S(t)$, we prove that the pair-correlation observables of the canonical approximants $Z_N(t;1)$ over disjoint windows behave asymptotically like decorrelated samples drawn from this GUE-distributed ensemble, and we upgrade the averaged GUE law to a deterministic pair-correlation law for the zeros of $Z(t)$.

math.NT↗

arXiv2025

On the approximation of the Hardy $Z$-function via high-order sections

Sections of the Hardy $Z$-function are given by $Z_N(t) := \sum_{k=1}^{N} \frac{cos(θ(t)-ln(k) t) }{\sqrt{k}}$ for any $N \in \mathbb{N}$. Sections approximate the Hardy $Z$-function in two ways: (a) $2Z_{\widetilde{N}(t)}(t)$ is the Hardy-Littlewood approximate functional equation (AFE) approximation for $\widetilde{N}(t) = \left [ \sqrt{\frac{t}{2 π}} \right ]$. (b) $Z_{N(t)}(t)$ is Spira's approximation for $N(t) = \left [\frac{t}{2} \right ]$. Spira conjectured, based on experimental observations, that, contrary to the classical approximation $(a)$, approximation (b) satisfies the Riemann Hypothesis (RH) in the sense that all of its zeros are real. We present theoretical justification for Spira's conjecture, via new techniques of acceleration of series, showing that it is essentially equivalent to RH itself.

math.GM↗

arXiv2025

On Edwards' Speculation and a New Variational Method for the Zeros of the $Z$-Function

In his foundational book, Edwards introduced a unique "speculation" regarding the possible theoretical origins of the Riemann Hypothesis, based on the properties of the Riemann-Siegel formula. Essentially Edwards asks whether one can find a method to transition from zeros of $Z_0(t)=cos(θ(t))$, where $θ(t)$ is Riemann-Siegel theta function, to zeros of $Z(t)$, the Hardy $Z$-function. However, when applied directly to the classical Riemann-Siegel formula, it faces significant obstacles in forming a robust plausibility argument for the Riemann Hypothesis. In a recent work, we introduced an alternative to the Riemann-Siegel formula that utilizes series acceleration techniques. In this paper, we explore Edwards' speculation through the lens of our accelerated approach, which avoids many of the challenges encountered in the classical case. Our approach leads to the description of a novel variational framework for relating zeros of $Z_0(t)$ to zeros of $Z(t)$ through paths in a high-dimensional parameter space $\mathcal{Z}_N$, recasting the RH as a modern non-linear optimization problem.

math.GM↗

arXiv2025

The $A$-philosophy for the Hardy $Z$-Function

In recent works we have introduced the parameter space $\mathcal{Z}_N$ of $A$-variations of the Hardy $Z$-function, $Z(t)$, whose elements are functions of the form \begin{equation} \label{eq:Z-sections} Z_N(t ; \overline{a} ) = \cos(θ(t))+ \sum_{k=1}^{N} \frac{a_k}{\sqrt{k+1} } \cos ( θ(t) - \ln(k+1) t), \end{equation} where $\overline{a} = (a_1,...,a_N) \in \mathbb{R}^N$. The $ A $-philosophy advocates that studying the discriminant hypersurface forming within such parameter spaces, often reveals essential insights about the original mathematical object and its zeros. In this paper we apply the $A$-philosophy to our space $\mathcal{Z}_N$ by introducing $ Δ_n(\overline{a} ) $ the $n$-th Gram discriminant of $ Z(t) $. We show that the Riemann Hypothesis (RH) is equivalent to the corrected Gram's law \[ (-1)^n Δ_n(\overline{1}) > 0, \] for any $n \in \mathbb{Z}$. We further show that the classical Gram's law $ (-1)^n Z(g_n) >0$ can be considered as a first-order approximation of our corrected law. The second-order approximation of $Δ_n (\overline{a})$ is then shown to be related to shifts of Gram points along the $ t $-axis. This leads to the discovery of a new, previously unobserved, repulsion phenomena \[ \left| Z'(g_n) \right| > 4 \left| Z(g_n) \right|, \] for bad Gram points $g_n$ whose consecutive neighbours $g_{n \pm 1}$ are good. Our analysis of the $A$-variation space $\mathcal{Z}_N$ introduces a wealth of new results on the zeros of $Z(t)$, casting new light on classical questions such as Gram's law, the Montgomery pair-correlation conjecture, and the RH, and also unveils previously unknown fundamental properties.

math.GM↗

arXiv2023

A New Discriminant for the Hardy Z-Function and the Corrected Gram's law

In this paper, we introduce a novel variational framework rooted in algebraic geometry for the analysis of the Hardy $Z$-function. Our primary contribution lies in the definition and exploration of $Δ_n(\overline{a})$, a newly devised discriminant that measures the realness of consecutive zeros of $Z(t)$. Our investigation into $Δ_n(\overline{a})$ and its properties yields a wealth of compelling insights into the zeros of $Z(t)$, including the corrected Gram's law, the second-order approximation of $Δ_n(\overline{a})$, and the discovery of the G-B-G repulsion relation. Collectively, these results provide compelling evidence supporting a new plausibility argument for the Riemann hypothesis.

math.NT↗

arXiv2021

The mirror Lagrangian cobordism for the Euler exact sequence

For $X = \mathbb{P}^n$ the Euler sequence is given by $$ 0 \rightarrow Ω^1_{\mathbb{P}^n} \rightarrow \mathcal{O}_{\mathbb{P}^n}^{n+1}(-1) \rightarrow \mathcal{O}_{\mathbb{P}^n} \rightarrow 0 $$ We describe the Lagrangian cobordism corresponding to this sequence via mirror symmetry, in the sense of Biran-Cornea. In particular, we describe the mirror Lagrangian of the cotangent sheaf $Ω^1_{\mathbb{P}^n} \in \mathcal{D}^b(\mathbb{P}^n)$ in the mirror Fukaya category $Fuk(U_Δ)$.

math.AG↗

arXiv2021

A dynamic approach for the zeros of the Riemann zeta function - collision and repulsion

For $N \in \mathbb{N}$ consider the $N$-th section of the approximate functional equation $$ ζ_N(s)= \sum_{n =1 }^N B_n(s),$$ where $$ B_n(s)= \frac{1}{2} \left [ n^{-s} + χ(s) \cdot n^{s-1} \right ].$$ Our aim in this work is to introduce a new approach for the Riemann hypothesis by studying the way pairs of consecutive zeros of $ζ_N(s)$ change with respect to $N$. For the initial stage, it is known that the non-trivial zeros of $ζ_1(s)$ all lie on the critical line $Re(s)=\frac{1}{2}$. In the region $2N \leq Im(s) \leq 2 π(N+1)$ the function $ζ_N(s)$ serves as an approximation of $ζ(s)$ itself, and it was conjectured by Spira that in this region $ζ_N(s)$ also admits zeros only on the critical line. We show that the appearance of zeros of a section off the critical line can be realized as the result of two consecutive zeros meeting and pushing each other off the critical line as $N$ changes, a process to which we refer to as a collision of zeros. Based on a study of the properties of $ζ_N(s)$, we suggest a way of re-arranging the order of summation of the elements $B_n(s)$ in $ζ_{N}(s)$ with $N=\left [ \frac{Im(s)}{2} \right ]$ that is expected to avoid collisions altogether, we refer to such a re-arrangement as a repelling re-arrangement. In particular, establishing that the suggested repelling re-arrangement indeed avoids collisions for any pair of zeros would imply RH.

math.NT↗

arXiv2020

An approximate functional equation for the Riemann zeta function with exponentially decaying error

It is known by a formula of Hasse-Sondow that the Riemann zeta function is given, for any $ s=σ+it \in \mathbb{C}$, by $ \sum_{n=0}^{\infty} \widetilde{A}(n,s)$ where $$ \widetilde{A}(n,s):=\frac{1}{2^{n+1}(1-2^{1-s})} \sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{(k+1)^s} .$$ We prove the following approximate functional equation for the Hasse-Sondow presentation: For $ \vert t \vert = πxy $ and $ 2y \neq (2N-1)π$ then $$ ζ(s)= \sum_{n \leq x } \widetilde{A}(n,s)+\frac{χ(s)}{1-2^{s-1}} \left (\sum_{k \leq y} (2k-1)^{s-1} \right ) +O \left (e^{-ω(x,y) t} \right ), $$ where $ 0 <ω(x,y)$ is a certain transcendental number determined by $ x$ and $ y$. A central feature of our new approximate functional equation is that its error term is of exponential rate of decay. The proof is based on a study, via saddle point techniques, of the asymptotic properties of the function $$ \widetilde{A}(u,s):= \frac{1}{2^{u+1} (1-2^{1-s}) Γ(s)} \int_{0}^{\infty} \left ( e^{-w} \left ( 1- e^{-w} \right)^u \right ) w^{s-1} dw,$$ and integrals related to it.

math.NT↗

arXiv2017

An experimental study of the monotonicity property of the Riemann zeta function

In 1970, based on newly available empiric evidence, a remarkable monotonicity property for $| ζ(z) |$ was conjectured by R. Spira. The $ζ$-monotonicity property can be written as follows: $$ | ζ(x_2 + y i ) | < | ζ\left ( x_1 +y i \right )| \hspace{0.5cm} \textrm {for any } \hspace{0.25cm} x_1 < x_2 \leq 0.5 \textrm{ and } 6.29 <y. $$ In this work we present an experimental study of the monotonicity conjecture, in the course of which new properties of $ζ(z)$ are discovered. For instance, the spectrum of semi-limits $ λ(z) \subset \mathbb{R}$ and the core function $C(z)$, which serves as a non-chaotic simplification of $ζ(z)$ to the left of the critical line

math.GM↗

arXiv2017

On Fermat curves modulo a finite number

We show that the existence of a non-trivial solution of $x^n+y^n=p^n$, with $p$ a prime number, is equivalent to the existence of a solution of a certain (over-determined) system of $(n-1)$-recursion relations ("zipper" equations) in $\mathbb{Z}_{p-1}$.

math.GM↗

arXiv2017

On Landau-Ginzburg systems and $\mathcal{D}^b(X)$ of various toric Fano manifolds with small picard group

For a toric Fano manifold $X$ denote by $Crit(X) \subset (\mathbb{C}^{\ast})^n$ the solution scheme of the Landau-Ginzburg system of equations of $X$. Examples of toric Fano manifolds with $rk(Pic(X)) \leq 3$ which admit full strongly exceptional collections of line bundles were recently found by various authors. For these examples we construct a map $E : Crit(X) \rightarrow Pic(X)$ whose image $\mathcal{E}=\left \{ E(z) \vert z \in Crit(X) \right \}$ is a full strongly exceptional collection satisfying the M-aligned property. That is, under this map, the groups $Hom(E(z),E(w))$ for $z,w \in Crit(X)$ are naturally related to the structure of the monodromy group acting on $Crit(X)$.

math.AG↗

arXiv2017

On exceptional collections of line bundles and mirror symmetry for toric Del-Pezzo surfaces

Let $X$ be a toric Del-Pezzo surface and let $Crit(W) \subset (\mathbb{C}^{\ast})^n$ be the solution scheme of the Landau-Ginzburg system of equations. Denote by $X^{\circ}$ the polar variety of $X$. Our aim in this work is to describe a map $L : Crit(W) \rightarrow Fuk_{trop}(X^{\circ})$ whose image under homological mirror symmetry corresponds to a full strongly exceptional collection of line bundles.

math.AG↗

arXiv2014

On Landau-Ginzburg Systems and $\mathcal{D}^b(X)$ of projective bundles

Let $X=\mathbb{P}(\mathcal{O}_{\mathbb{P}^s} \oplus \bigoplus_{i=1}^r \mathcal{O}_{\mathbb{P}^s}(a_i))$ be a Fano projective bundle over $\mathbb{P}^s$ and denote by $Crit(X) \subset (\mathbb{C}^{\ast})^n$ the solution scheme of the Landau-Ginzburg system of equations of $X$. We describe a map $E : Crit(X) \rightarrow Pic(X)$ whose image $\mathcal{E}= \{E(z) | z \in Crit(X) \}$ is the full strongly exceptional collection on $X$ found by Costa and Mir$\acute{\textrm{o}}$-Roig. We further show that $Hom(E(z),E(w))$ for $z,w \in Crit(X)$ can be described in terms of a monodromy group acting on $Crit(X)$.

math.AG↗

arXiv2014

On Landau-Ginzburg systems, Quivers and Monodromy

Let $X$ be a toric Fano manifold and denote by $Crit(f_X) \subset (\mathbb{C}^{\ast})^n$ the solution scheme of the corresponding Landau-Ginzburg system of equations. For toric Del-Pezzo surfaces and various toric Fano threefolds we define a map $L : Crit(f_X) \rightarrow Pic(X)$ such that $\mathcal{E}_L(X) : = L(Crit(f_X)) \subset Pic(X)$ is a full strongly exceptional collection of line bundles. We observe the existence of a natural monodromy map $$ M : π_1(L(X) \setminus R_X,f_X) \rightarrow Aut(Crit(f_X))$$ where $L(X)$ is the space of all Laurent polynomials whose Newton polytope is equal to the Newton polytope of $f_X$, the Landau-Ginzburg potential of $X$, and $R_X \subset L(X)$ is the space of all elements whose corresponding solution scheme is reduced. We show that monodromies of $Crit(f_X)$ admit non-trivial relations to quiver representations of the exceptional collection $\mathcal{E}_L(X)$. We refer to this property as the $M$-aligned property of the maps $L: Crit(f_X) \rightarrow Pic(X)$. We discuss possible applications of the existence of such $M$-aligned exceptional maps to various aspects of mirror symmetry of toric Fano manifolds.

math.AG↗

arXiv2012

The symplectic topology of projective manifolds with small dual

We study smooth projective varieties with small dual variety using methods from symplectic topology. We prove the affine parts of such varieties are subcritical, and that the hyperplane class is invertible in their quantum cohomology. We derive several topological and algebraic geometric consequences from that. The main tool in our work is the Seidel representation associated to Hamiltonian fibrations.

math.AG↗

arXiv2011

A Note On Lagrangian Vanishing Sphere Bundles

A classical way to construct a Lagrangian in a symplectic manifold $Σ$ is to let $Σ$ appear as a smooth fiber in a Lefschetz fibration. If this is possible the singularities of the fibration induce Lagrangian spheres in $Σ$ and these spheres, in turn, are representatives of the corresponding vanishing cycles in the homology of $Σ$. In this paper our aim is twofold: The first is to describe a generalization of the above mentioned construction to the "Morse-Bott" case. This leads, whenever such a degeneration exists, to the existence of Lagrangian sphere bundles rather than just spheres. In the second part of the paper we study the type of topological restrictions on such Lagrangian sphere bundles arising from the theory of Floer homology for Lagrangian intersections and to illustrate the techniques involved.

math.SG↗