Ramanujan Conjecture Research Articles

The twisted second moment of modular half-integral weight $L$-functions

Given a half-integral weight holomorphic Kohnen newform f on \Gamma_{0}(4) , we prove an asymptotic formula for large primes p with power saving error term for \sum_{\chi \pmod{p}}^{\qquad{}{}_{\scriptsize{*}}}| L(1/2,f,\chi) |^{2}. Our result is unconditional, it does not rely on the Ramanujan–Petersson conjecture for the form f . This gives a very sharp Lindelöf-on-average result for Dirichlet series attached to Hecke eigenforms without an Euler product. The Lindelöf hypothesis for such series was originally conjectured by Hoffstein. There are two main inputs. The first is a careful spectral analysis of a highly unbalanced shifted convolution problem involving the Fourier coefficients of half-integral weight forms. The second input is a bound for sums of products of Salié sums in the Pólya–Vinogradov range. Half-integrality is fully exploited to establish such an estimate. We use the closed form evaluation of the Salié sum to relate our problem to the sequence \alpha n^{2} \pmod{1} . Our treatment of this sequence is inspired by work of Rudnick–Sarnak and the second author on the local spacings of \alpha n^{2} modulo 1.

Sums of coefficients of general L-functions over arithmetic progressions and applications

In this paper, we study the asymptotic distribution of coefficients of general L-functions over arithmetic progressions without the Ramanujan conjecture. As an application, we consider the high mean of Fourier coefficients of holomorphic forms or Maass forms for Γ=SL(2,Z) over arithmetic progressions, and improve the results of Jiang and Lü [10]. Our new results remove the restriction to prime module and improve the interval length of module q.

Summation formulae of multiplicative functions over arithmetic progressions and applications

Abstract In this paper, we investigate the asymptotic distribution of a class of multiplicative functions over arithmetic progressions without the Ramanujan conjecture. We also apply these results to some interesting arithmetic functions in automorphic context, such as coefficients of automorphic L-functions, coefficients of their Rankin–Selberg.

Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues

By assuming Vinogradov–Korobov-type zero-free regions and the generalized Ramanujan–Petersson conjecture, we establish nontrivial upper bounds for almost all short sums of Fourier coefficients of Hecke–Maass cusp forms for [Formula: see text]. As applications, we obtain nontrivial upper bounds for the averages of shifted sums involving coefficients of the Hecke–Maass cusp forms for [Formula: see text]. Furthermore, we present a conditional result regarding sign changes of these coefficients.

On signs of Fourier coefficients of Hecke-Maass cusp forms on 𝐺𝐿₃

We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group S L 3 ( Z ) \mathrm {SL}_3(\mathbb Z) . When the underlying form is self-dual, we show that there are ≫ ε X 5 / 6 − ε \gg _\varepsilon X^{5/6-\varepsilon } sign changes among the coefficients { A ( m , 1 ) } m ≤ X \{A(m,1)\}_{m\leq X} and that there is a positive proportion of sign changes for many self-dual forms. Similar result concerning the positive proportion of sign changes also hold for the real-valued coefficients A ( m , m ) A(m,m) for generic G L 3 \mathrm {GL}_3 cusp forms, a result which is based on a new effective Sato-Tate type theorem for a family of G L 3 \mathrm {GL}_3 cusp forms we establish. In addition, non-vanishing of the Fourier coefficients is studied under the Ramanujan-Petersson conjecture.

Riesz-type criteria for L-functions in the Selberg class

AbstractWe formulate a generalization of Riesz-type criteria in the setting of L-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the grand Riemann hypothesis (GRH) for L-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for L-functions in this subclass. Identities of Ramanujan–Hardy–Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer G-function of the type ${G^{n , 0}_{0 , n}}$ .

Potential automorphy over CM fields

Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and furthermore satisfy the Sato--Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for $\mathrm{GL}_2(\mathbb{A}_F)$.

Divisor-bounded multiplicative functions in short intervals

We extend the Matomäki–Radziwiłł theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a function f in typical intervals of length h(log X)^c, with h = h(X) rightarrow infty and where c = c_f ge 0 is determined by the distribution of {|f(p) |}_p in an explicit way. We give three applications. First, we show that the classical Rankin–Selberg-type asymptotic formula for partial sums of |lambda _f(n) |^2, where {lambda _f(n)}_n is the sequence of normalized Fourier coefficients of a primitive non-CM holomorphic cusp form, persists in typical short intervals of length hlog X, if h = h(X) rightarrow infty . We also generalize this result to sequences {|lambda _{pi }(n) |^2}_n, where lambda _{pi }(n) is the nth coefficient of the standard L-function of an automorphic representation pi with unitary central character for GL_m, m ge 2, provided pi satisfies the generalized Ramanujan conjecture. Second, using recent developments in the theory of automorphic forms we estimate the variance of averages of all positive real moments {|lambda _f(n) |^{alpha }}_n over intervals of length h(log X)^{c_{alpha }}, with c_{alpha } > 0 explicit, for any alpha > 0, as h = h(X) rightarrow infty . Finally, we show that the (non-multiplicative) Hooley Delta -function has average value gg log log X in typical short intervals of length (log X)^{1/2+eta }, where eta >0 is fixed.

On Joint Universality in the Selberg–Steuding Class

The famous Selberg class is defined axiomatically and consists of Dirichlet series satisfying four axioms (Ramanujan hypothesis, analytic continuation, functional equation, multiplicativity). The Selberg–Steuding class S is a complemented Selberg class by an arithmetic hypothesis related to the distribution of prime numbers. In this paper, a joint universality theorem for the functions L from the class S on the approximation of a collection of analytic functions by shifts L(s+ia1τ),…,L(s+iarτ), where a1,…,ar are real algebraic numbers linearly independent over the field of rational numbers, is obtained. It is proved that the set of the above approximating shifts is infinite, its lower density and, with some exception, density are positive. For the proof, a probabilistic method based on weak convergence of probability measures in the space of analytic functions is applied together with the Backer theorem on linear forms of logarithms and the Mergelyan theorem on approximation of analytic functions by polynomials.

One-level density of quadratic twists of $L$-functions

In this paper, we investigate the one-level density of low-lying zeros of quadratic twists of automorphic $L$-functions under the generalized Riemann hypothesis and the Ramanujan-Petersson conjecture. We improve upon the known results using only functional equations for quadratic Dirichlet $L$-functions.

Optimal strong approximation for quadrics over [formula omitted

Suppose q is a fixed odd prime power, F(x) is a non-degenerate quadratic form over Fq[t] of discriminant Δ in d≥5 variables x, and f,g∈Fq[t], λ∈Fq[t]d. We show that whenever deg⁡f≥(4+ε)deg⁡g+Oε,F(1), gcd⁡(Δ∞,fg)=O(1), and the necessary local conditions are satisfied, we have a solution x∈Fq[t]d to F(x)=f such that x≡λmodg. For d=4, we show that the same conclusion holds if we instead have deg⁡f≥(6+ε)deg⁡g+Oε,F(1). This gives us a new proof (independent of the Ramanujan conjecture over function fields proved by Drinfeld) that the diameter of any k-regular Morgenstern Ramanujan graph G is at most (2+ε)logk−1⁡|G|+Oε(1). In contrast to the d=4 case, our result is optimal for d≥5. Our main new contributions are a stationary phase theorem over function fields for bounding oscillatory integrals, and a notion of anisotropic cones to circumvent isotropic phenomena in the function field setting.

Density theorems for {textrm{GL}}(n)

Strong bounds—going beyond Sarnak’s density hypothesis—are obtained for the number of automorphic forms for the group Gamma _0(q) subseteq textrm{SL}(n, {mathbb {Z}}) violating the Ramanujan conjecture at any given unramified place. The proof is based on a relative trace formula of Kuznetsov type and best-possible bounds for certain Kloosterman sums for textrm{GL}(n). Further applications are given.

On some estimates involving Fourier coefficients of Maass cusp forms

Let [Formula: see text] be a Hecke–Maass cusp form for [Formula: see text] with Laplace eigenvalue [Formula: see text] and let [Formula: see text] be its [Formula: see text]th normalized Fourier coefficient. It is proved that, uniformly in [Formula: see text], [Formula: see text] where the implied constant depends only on [Formula: see text]. We also consider the summation function of [Formula: see text] and under the Ramanujan conjecture we are able to prove [Formula: see text] with the implied constant depending only on [Formula: see text].

Jutila's circle method and [formula omitted] shifted convolution sums

Let λf(n) be the normalized Fourier coefficients of a Hecke-Maass or Hecke holomorphic cusp form f for congruence group Γ0(N) with level N and nebentypus χN. Let Q(x) be a positive definite integral quadratic form, and r(n,Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply Jutila's circle method to derive a sharp bound for the shifted convolution sum of Fourier coefficients λf(n) and r(n,Q). We generalize and improve previous results without the Ramanujan conjecture.

Ramanujan Complexes and Golden Gates in PU(3)

In a seminal series of papers from the 80’s, Lubotzky, Phillips and Sarnak applied the Ramanujan–Petersson Conjecture for \(GL_{2}\) (Deligne’s theorem), to a special family of arithmetic lattices, which act simply-transitively on the Bruhat–Tits trees associated with \(SL_{2}({\mathbb {Q}}_{p})\). As a result, they obtained explicit Ramanujan Cayley graphs from \(PSL_{2}\left( {\mathbb {F}}_{p}\right) \), as well as optimal topological generators (“Golden Gates”) for the compact Lie group PU(2). In higher dimension, the naive generalization of the Ramanujan Conjecture fails, due to the phenomenon of endoscopic lifts. In this paper we overcome this problem for \(PU_{3}\) by constructing a family of arithmetic lattices which act simply-transitively on the Bruhat–Tits buildings associated with \(SL_{3}({\mathbb {Q}}_{p})\) and \(SU_{3}({\mathbb {Q}}_{p})\), while at the same time do not admit any representation which violates the Ramanujan Conjecture. This gives us Ramanujan complexes from \(PSL_{3}({\mathbb {F}}_{p})\) and \(PSU_{3}({\mathbb {F}}_{p})\), as well as golden gates for PU(3).

A density theorem for Sp(4)$\operatorname{Sp}(4)$

Strong bounds are obtained for the number of automorphic forms for the group $\Gamma_0(q) \subseteq \operatorname{Sp}(4,\mathbb{Z})$ violating the Ramanujan conjecture at any given unramified place, which go beyond Sarnak's density hypothesis. The proof is based on a relative trace formula of Kuznetsov type, and best-possible bounds for certain Kloosterman sums for $\operatorname{Sp}(4)$.

A Bombieri–Vinogradov Theorem for Higher-Rank Groups

Abstract We establish a result of Bombieri–Vinogradov type for the Dirichlet coefficients at prime ideals of the standard $L$-function associated to a self-dual cuspidal automorphic representation $\pi $ of $\operatorname {GL}_n$ over a number field $F$ when $\pi$ is not a quadratic twist of itself. Our result does not rely on any unproven progress towards the generalized Ramanujan conjecture or the nonexistence of Landau–Siegel zeros. In particular, when $\pi $ is fixed and not equal to a quadratic twist of itself, we prove the first unconditional Siegel-type lower bound for the twisted $L$-values $|L(1,\pi \otimes \chi )|$ in the $\chi $-aspect, where $\chi $ is a primitive quadratic Hecke character over $F$. Our result improves the levels of distribution in other works that relied on these unproven hypotheses. As applications, when $n=2,3,4$, we prove a $\textrm {GL}_n$ analogue of the Titchmarsh divisor problem and a nontrivial bound for a certain $\textrm {GL}_n\times \textrm {GL}_2$ shifted convolution sum.

Zeros of Rankin–Selberg $L$-functions at the edge of the critical strip (with an appendix by Colin J. Bushnell and Guy Henniart)

Let \pi (respectively \pi_0 ) be a unitary cuspidal automorphic representation of \mathrm{GL}_m (respectively \mathrm{GL}_{m_0} ) over \mathbb{Q} . We prove log-free zero density estimates for Rankin–Selberg L -functions of the form L(s,\pi\times\pi_0) , where \pi varies in a given family and \pi_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 in class groups of number fields.

Joint equidistribution on the product of the circle and the unit cotangent bundle of the modular surface

We use spectral method to prove a joint equidistribution of primitive rational points and the same along expanding horocycle orbits in the products of the circle and the unit cotangent bundle of the modular surface. This result explicates the error bound in a recent work of Einsiedler, Luethi, and Shah [3, Theorem 1.1]. The error is sharp upon the best known progress towards the Ramanujan conjecture at the finite places for the modular surface.

On the exceptional set of the generalized Ramanujan conjecture for GL(3)

Recently Luo and Zhou showed that for a fixed GL(2) Hecke-Maass cusp form, the natural density of primes at which the Satake parameters fail the Ramanujan Conjecture does not exceed 1/35. In this short note, we investigate the GL(3) case and obtain two similar (conditional) results.