Newman's conjecture (proved by Rodgers and Tao in 2018) concerns a certain family of deformations $\{ξ_t(s)\}_{t \in \mathbb{R}}$ of the Riemann xi function for which there exists an associated constant $Λ\in \mathbb{R}$ (called the de Bruijn-Newman constant) such that all the zeros of $ξ_t$ lie on the critical line if and only if $t \geq Λ$. The Riemann hypothesis is equivalent to the statement that $Λ\leq 0$, and Newman's conjecture states that $Λ\geq 0$.
In this paper we give a new proof of Newman's conjecture which avoids many of the complications in the proof of Rodgers and Tao. Unlike the previous best methods for bounding $Λ$, our approach does not require any information about the zeros of the zeta function, and it can be readily be applied to a wide variety of $L$-functions. In particular, we establish that any $L$-function in the extended Selberg class has an associated de Bruijn-Newman constant and that all of these constants are nonnegative.
Stated in the Riemann xi function case, our argument proceeds by showing that for every $t < 0$ the function $ξ_t$ can be approximated in terms of a Dirichlet series $ζ_t(s)=\sum_{n=1}^{\infty}\exp(\frac{t}{4} \log^2 n)n^{-s}$ whose zeros then provide infinitely many zeros of $ξ_t$ off the critical line.
In the paper we introduce the new approach how to use an orthonormality relation of coefficients of Dirichlet series defining given L-functions from the Selberg class to prove joint universality.
The author reviews results and conjectures of Selberg on a class of Dirichlet series functions which share properties with the Riemann zeta function, and he relates this work to the theory of Artin L-functions.
The paper is withdrawn. The reviewer pointed out that the assertion that the quantity $D_Γ$ is finite (used by the author) is still unproved for the general group, and perhaps is false.
We derive a fully explicit version of the Selberg trace formula for twist-minimal Maass forms of weight 0 and arbitrary conductor and nebentypus character, and apply it to prove two theorems. First, conditional on Artin's conjecture, we classify the even 2-dimensional Artin representations of small conductor; in particular, we show that the even icosahedral representation of smallest conductor is the one found by Doud and Moore, of conductor 1951. Second, we verify the Selberg eigenvalue conjecture for groups of small level, improving on a result of Huxley from 1985.
As the reviewer have pointed out, the proof of Roelke Conjecture contains an error. For cofinite groups, we obtain a formula connecting the discrete spectrum of Laplace operator and the resonance spectrum. Using this formula, we give a proof of the Roelke conjecture. In this version, we make several corrections in Theorem 1 and Lemma 5.
In this paper we prove that the p-adic L-function that interpolates the Rankin-Selberg product of a general modular form and a CM form of higher weight divides the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa main conjecture for the p-adic L-function. We prove this conjecture using the congruences between Klingen Eisensteinseries and cusp forms on the group GU(3; 1), following the strategy of a recent work of C. Skinner and E. Urban. This theorem can be used to deduce a converse of Gross-Zagier-Kolyvagin theorem and the precise BSD formula in …
In this paper we prove that the $p$-adic $L$-function that interpolates the Rankin-Selberg product of a general weight two modular form which is unramified and non-ordinary at $p$, and an ordinary CM form of higher weight contains the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-function. This generalizes an earlier work of the author to the non-ordinary case. The result of this paper plays a crucial role in the proof of Iwasawa main conjecture and refined Birch-Swinnerton-Dyer formula for supersi…
We prove a version of the Extra-zero conjecture formulated by the first named author for p-adic L-functions associated to Rankin-Selberg convolutions of modular forms of the same weight. The novelty of this result is to provide strong evidence in support of this conjecture in the non-critical case, which remained essentially unstudied.
We study the Fitting ideals over the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of Selmer groups attached to the Rankin--Selberg convolution of two modular forms $f$ and $g$. Inspired by the Theta elements for modular forms defined by Mazur and Tate in ``Refined conjectures of the Birch and Swinnerton-Dyer type'', we define new Theta elements for Rankin--Selberg convolutions of $f$ and $g$ using Loeffler--Zerbes' geometric $p$-adic $L$-functions attached to $f$ and $g$. Under certain technical hypotheses, we generalize a recent work of Kim--Kurihara on elliptic cu…
We show that Selberg's eigenvalue conjecture concerning small eigenvalues of the automorphic Laplacian for congruence groups is equivalent to a conjecture about the non-existence of residual eigenvalues for a perturbed system. We prove this using a combination of methods from asymptotic perturbation theory and number theory.
The conjectural newform theory for generic representations of $p$-adic ${\rm SO}_{2n+1}$ was formulated by P.-Y. Tsai in her thesis in which Tsai also verified the conjecture when the representations are supercuspidal. The main purpose of this work is to compute the Rankin-Selberg integrals for ${\rm SO}_{2n+1}\times{\rm GL}_r$ with $1\le r\le n$ attached to newforms and also oldforms under the validity of the conjecture.
J.-C. Yoccoz proposed a natural extension of Selberg's Eigenvalue Conjecture to moduli spaces of abelian differentials. We prove an approximation to this conjecture. This gives a qualitative generalization of Selberg's $\frac{3}{16}$ Theorem to moduli spaces of abelian differentials on surfaces of genus $\geq 2$.
In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg $L$-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanis…
We give a new non-trivial upper bound for the Selberg integral of the three-divisor function $d_3(n)$. Our method applies our recent conjecture together with Laporta, for the modified Selberg integral of $d_3(n)$, and a kind of modified Gallagher Lemma for the exponential sums.
Generalizing work of Polya, de Bruijn and Newman, we allow the backward heat equation to deform the zeros of quadratic Dirichlet L-functions. There is a real constant Λ_Kr (generalizing the de Bruijn-Newman constant Λ) such that for time t>=Λ_Kr all such L-functions have all their zeros on the critical line; for time t<Λ_Kr there exist zeros off the line. Under GRH, Λ_Kr<=0; we make the complementary conjecture 0<=Λ_Kr. Following the work of Csordas et. al. on Lehmer pairs of Riemann zeros, we use low-lying zeros of quadratic Dirichlet L-functions to show that -1.13* 10^{-7}<Λ_Kr. In the last …
De Bruijn and Newman introduced a deformation of the Riemann zeta function $ζ(s)$, and found a real constant $Λ$ which encodes the movement of the zeros of $ζ(s)$ under the deformation. The Riemann hypothesis (RH) is equivalent to $Λ\le 0$. Newman made the conjecture that $Λ\ge 0$ along with the remark that "the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so." Newman's conjecture is still unsolved, and previous work could only handle the Riemann zeta function and quadratic Dirichlet $L$-functions, obtaining lower bounds very close…
We construct motivic cohomology classes attached to Rankin--Selberg convolutions of modular forms of weights $\ge 2$, show that these vary analytically in p-adic families, and relate their image under the p-adic regulator map to values of L-functions. As consequences, we prove new cases of Perrin-Riou's conjecture on motivic L-values; we prove finiteness results for Tate--Shafarevich groups for twists of elliptic curves by dihedral Artin characters; and we prove one inclusion in the Iwasawa main conjecture for a single modular form over an imaginary quadratic field.
Let $f$ and $g$ be two modular forms which are non-ordinary at $p$. The theory of Beilinson-Flach elements gives rise to four rank-one non-integral Euler systems for the Rankin-Selberg convolution $f \otimes g$, one for each choice of $p$-stabilisations of $f$ and $g$. We prove (modulo a hypothesis on non-vanishing of $p$-adic $L$-fuctions) that the $p$-parts of these four objects arise as the images under appropriate projection maps of a single class in the wedge square of Iwasawa cohomology, confirming a conjecture of Lei-Loeffler-Zerbes. Furthermore, we define an explicit logarithmic matrix…
We formulate and for the most part prove a conjecture in the style of Mazur-Greenberg for the nonvanishing of central values of Rankin-Selberg $L$-functions attached to elliptic curves in abelian extensions of imaginary quadratic fields. This in particular generalizes the theorem of Rohrlich on $L$-functions of elliptic curves in cyclotomic towers to the setting of abelian extensions of imaginary quadratic fields, corresponding to families of degree-four $L$-functions given by $\operatorname{GL}(2)\times\operatorname{GL}(2)$ Rankin-Selberg $L$-functions. It also generalizes the theorems of Roh…
We establish the equivalence of conjectures concerning the pair correlation of zeros of $L$-functions in the Selberg class and the variances of sums of a related class of arithmetic functions over primes in short intervals. This extends the results of Goldston & Montgomery [7] and Montgomery & Soundararajan [11] for the Riemann zeta-function to other $L$-functions in the Selberg class. Our approach is based on the statistics of the zeros because the analogue of the Hardy-Littlewood conjecture for the auto-correlation of the arithmetic functions we consider is not available in general. One of o…
We prove a refinement of the global Gan-Gross-Prasad conjecture proposed by Ichino-Ikeda and N. Harris for unitary groups under some local conditions. We need to assume some expected properties of L-packets and some part of the local Gan-Gross-Prasad conjecture.
Following the prequel work \cite{VO3}, we prove a generalization of "Mazur's conjecture" for $L$-functions of elliptic curves in abelian extensions of imaginary quadratic fields, including the assertion that the Mordell-Weil rank of an elliptic curve in the ${\bf{Z}}_p^2$-extension is finitely generated modulo Heegner points. The novelty of the approach here is to use the existence of a suitable $p$-adic $L$-function to reduce the problem to a minimal nonvanishing criterion, which should be applicable to a broader class of problems than considered here.
For any $m \geq 1$, let $H_m$ denote the quantity $\liminf_{n \to \infty} (p_{n+m}-p_n)$. A celebrated recent result of Zhang showed the finiteness of $H_1$, with the explicit bound $H_1 \leq 70000000$. This was then improved by us (the Polymath8 project) to $H_1 \leq 4680$, and then by Maynard to $H_1 \leq 600$, who also established for the first time a finiteness result for $H_m$ for $m \geq 2$, and specifically that $H_m \ll m^3 e^{4m}$. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound $H_1 \leq 12$, improving upon the previous bound $H_1 \leq 16$ of Goldsto…
Let $π$ and $π_0$ be unitary cuspidal automorphic representations. We prove log-free zero density estimates for Rankin-Selberg $L$-functions of the form $L(s,π\timesπ_0)$, where $π$ varies in a given family and $π_0$ is fixed. These estimates are unconditional in many cases of interest; they hold in full generality assuming an average form of the generalized Ramanujan conjecture. We consider applications of these estimates related to mass equidistribution for Hecke-Maass forms, the rarity of Landau-Siegel zeros of Rankin-Selberg $L$-functions, the Chebotarev density theorem, and $\ell$-torsion…
For a $p$-adic local field $F$ of characteristic 0, with residue field $\mathfrak{f}$, we prove that the Rankin-Selberg gamma factor of a pair of level zero representations of linear general groups over $F$ is equal to a gamma factor of a pair of corresponding cuspidal representations of linear general groups over $\mathfrak{f}$. Our results can be used to prove a variant of Jacquet's conjecture.
Over the last few years Pohl (partly jointly with coauthors) developed dual `slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $Γ\backslash\mathbb{H}$ with cusps and all finite-dimensional unitary representations $χ$ of $Γ$. The eigenfunctions with eigenvalue $1$ of the fast transfer operators determine the zeros of the Selberg zeta function for $(Γ,χ)$. Further, if $Γ$ is cofinite and $χ$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue $1$ of the s…
Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove that there are infinitely many such congruences for every prime modulus exceeding 3. In addition, we provide a simple criterion guaranteeing the truth of Newman's conjecture for any prime modulus exceeding 3 (recall that Newman's conjecture asserts that the partition function hits every residue class modulo a given integer M infinitely often).
Let $F$ be a $p$-adic field and $π$ be an irreducible smooth representation of $\textrm{Sp}_{2n}(F)$. In this paper, we show that if $π$ and $π^κ$ are both generic for a common generic character of the maximal unipotent of a fixed Borel, then $π\cong π^κ$, where $π^κ$ is the representation induced by the conjugation action of an element $κ\in \textrm{GSp}_{2n}(F)$. This result is a consequence of the standard local Langlands conjecture and local Gan-Gross-Prasad conjecture. As a consequence, we extend the dependence relation of the local Rankin-Selberg gamma factors for $\textrm{Sp}_{2n}\times…
In this paper we consider Dirichlet series absolutely converging for $σ>1$ with an Euler product, natural bounds on the coefficients and satisfying orthogonality relations of Selberg type. Let $N\geq 1$, $F_1(s),...,F_N(s)$ be as above and $P(X_1,...,X_N)$ be a non-monomial polynomial with coefficients in the ring of $p$-finite Dirichlet series absolutely converging for $σ\geq 1$; then $P(F_1(s),\ldots,F_N(s))$ has infinitely many zeros for $σ>1$. Our result in particular applies to Artin $L$-functions, automorphic $L$-functions under the Ramanujan conjecture, and the elements of the Selberg c…
We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization and refinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, and Kasten-Schmidt. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric power L-functions attached to a holomorphic cusp form. Assuming Langlands functoriality one can prove similar algebraicity results for the special values of any odd symmetric power L-function. We also prove a conjecture of Blas…
For each $t \in {\bf R}$, define the entire function $$ H_t(x) := \int_0^\infty e^{tu^2} Φ(u) \cos(xu)\ du$$ where $Φ$ is the super-exponentially decaying function $$ Φ(u) := \sum_{n=1}^\infty (2π^2 n^4 e^{9u} - 3πn^2 e^{5u} ) \exp(-πn^2 e^{4u} ).$$ Newman showed that there exists a finite constant $Λ$ (the \emph{de Bruijn-Newman constant}) such that the zeroes of $H_t$ are all real precisely when $t \geq Λ$. The Riemann hypothesis is the equivalent to the assertion $Λ\leq 0$, and Newman conjectured the complementary bound $Λ\geq 0$. In this paper we establish Newman's conjecture. The argument…
This is a survey of recent work on values of Rankin-Selberg $L$-functions of pairs of cohomological automorphic representations that are {\it critical} in Deligne's sense. The base field is assumed to be a CM field. Deligne's conjecture is stated in the language of motives over $\QQ$, and express the critical values, up to rational factors, as determinants of certain periods of algebraic differentials on a projective algebraic variety over homology classes. The results that can be proved by automorphic methods express certain critical values as (twisted) period integrals of automorphic forms. …
By the unfolding method, Rankin-Selberg L-functions for ${\rm GL}(n)\times{\rm GL}(m)$ can be expressed in terms of period integrals. These period integrals actually define invariant forms on tensor products of the relevant automorphic representations. By the multiplicity-one theorems due to Sun-Zhu and Chen-Sun such invariant forms are unique up to scalar multiples and can therefore be related to invariant forms on equivalent principal series representations. We construct meromorphic families of such invariant forms for spherical principal series representations of ${\rm GL}(n,\mathbb{R})$ an…
We establish a sub-convexity estimate for Rankin-Selberg $L$-functions in the combined level aspect, using the circle method. If $p$ and $q$ are distinct prime numbers, $f$ and $g$ are non-exceptional newforms (modular or Maass) for the congruence subgroups $Γ_0(p)$ and $Γ_0(q)$ (resp) with trivial nebentypus, then for all $ε>0$ we show that there exists an $A >0$ such that $$ L\left(\frac{1}{2}+it, f \times g \right) \ll_{ε,μ_f, μ_g}(1+|t|)^A \frac{(pq)^{1/2+ε}}{\max\{p,q \}^{\frac{1}{64}}}. $$ The dependence on $μ_f$ and $μ_g$, the parameters at infinity for $f$ and $g$ respectively, is poly…
We investigate so-called "higher" Siegel theta lifts on Lorentzian lattices in the spirit of Bruinier-Ehlen-Yang and Bruinier-Schwagenscheidt. We give a series representation of the lift in terms of Gauss hypergeometric functions, and evaluate the lift as the constant term of a Fourier series involving the Rankin-Cohen bracket of harmonic Maass forms and theta functions. Using the higher Siegel lifts, we obtain a vector-valued analogue of Mertens' result stating that the Rankin--Cohen bracket of the holomorphic part of a harmonic Maass form of weight $\frac{3}{2}$ and a unary theta function, p…
Using Hecke triangle surfaces of finite and infinite area as examples, we present techniques for thermodynamic formalism approaches to Selberg zeta functions with unitary finite-dimensional representations $(V,χ)$ for hyperbolic surfaces (orbifolds) $Γ\backslash\mathbb{H}$ as well as transfer operator techniques to develop period-like functions for $(Γ,χ)$-automorphic cusp forms. This leads to several natural conjectures. We further show how to extend these results to the billiard flow on the underlying triangle surfaces, and study the convergence of transfer operators along sequences of Hecke…
