Let $\mathcal{L}(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ be an $L$-function in the Selberg class, and $q_{\mathcal{L}}$ its conductor. Let $\ell_0(\mathcal{L})$ be the constant term of the Laurent expansion of $\mathcal{L}'/\mathcal{L}$ at $s=1$. We show that for certain families $\mathcal{F}$ of $L$-functions in the Selberg class with polynomial Euler product:
$\bullet$ If $\mathcal{L}\in\mathcal{F}$ has no zeros $β+ iγ$ with $β> 1 - δ(\log q_{\mathcal{L}})^{-1}$, $|γ| < (\log q_{\mathcal{L}})^{-1/2}$ for some absolute $δ>0$, then $\Re(\ell_0(\mathcal{L})) \ll_{\mathcal{F}} \log q_{\mathcal{L}}$;
$\bullet$ If $\Re(\ell_0(\mathcal{L})) \ll \log q_{\mathcal{L}}$ for all $\mathcal{L}\in \mathcal{F}$, then there is some absolute $δ> 0$ such that $\mathcal{L}$ has no zeros $β+ iγ$ with $β> 1 - δ(\log q_{\mathcal{L}})^{-1}$, $|γ| < (1-β)^{1/2}(\log q_{\mathcal{L}})^{-1/2}$.
This generalizes, for instance, the case of families of Dedekind zeta functions of number fields with bounded degree.
We find an asymptotic expansion of Selberg's central limit theorem for the Riemann zeta function on $σ= \frac12 + ( \log T)^{-θ}$ and $t \in [T, 2T]$, where $ 0 < θ< \frac12$ is a constant.
In this note, we prove Selberg's announced result on $r$-gaps between zeros of the Riemann zeta-function $ζ$. Our proof uses a result on variations of $\argζ$ by Tsang based on Selberg's method. The same result with explicit constants under the Riemann Hypothesis has been obtained by Conrey and Turnage-Butterbaugh using a different method. We explain how to obtain explicit constants under the Riemann Hypothesis using our approach which is based on Selberg's and Tsang's arguments.
Let $f$ be a Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplace eigenvalue $1/4+μ_f^2$, $μ_f>0$. Let $g$ be an arbitrary but fixed holomorphic or Maass cusp form for $\rm SL_2(\mathbb{Z})$. In this paper, we establish the following uniform subconvexity bound for the Rankin-Selberg $L$-function $L(s,f\otimes g)$ $$ L\left(1/2+it,f\otimes g\right)\ll (μ_f+|t|)^{9/10+\varepsilon}, $$ where the implied constant depends only on $\varepsilon$ and $g$.
Levinson and Montgomery proved that the Riemann zeta-function $ζ(s)$ and its derivative have approximately the same number of non-real zeros left of the critical line. R. Spira showed that $ζ'(1/2+it)=0$ implies $ζ(1/2+it)=0$. Here we obtain that in small areas located to the left of the critical line and near it the functions $ζ(s)$ and $ζ'(s)$ have the same number of zeros. We prove our result for more general zeta-functions from the extended Selberg class $S$. We also consider zero trajectories of a certain family of zeta-functions from $S$.
In this paper we prove that the Generalized Riemann Hypothesis (GRH) for functions in the class $\mathcal{S}^{\sharp\flat}$ containing the Selberg class is equivalent to a certain integral expression of the real part of the generalized Li coefficient $λ_F(n)$ associated to $F\in\mathcal{S}^{\sharp\flat}$, for positive integers $n$. Moreover, we deduce that the GRH is equivalent to a certain expression of $Re(λ_F(n))$ in terms of the sum of the Chebyshev polynomials of the first kind. Then, we partially evaluate the integral expression and deduce further relations equivalent to the GRH involvin…
An arithmetic function $f$ is called a sieve function of range $Q$, if it is the convolution product of the constantly $1$ function and $g$ such that $g(q)\ll_{\varepsilon} q^{\varepsilon}$, $\forall\varepsilon>0$, for $q\leq Q$, and $g(q)=0$ for $q>Q$. Here we establish a new result on the autocorrelation of $f$ by using a famous theorem on bilinear forms of Kloosterman fractions by Duke, Friedlander and Iwaniec. In particular, for such correlations we obtain non-trivial asymptotic formulæ that are actually unreachable by the standard approach of the distribution of $f$ in the arithmetic prog…
Let $\CmZ_{Y_0(N)}$ be the constant term of the logarithmic derivative at $s=1$ of the Selberg zeta function of the modular curve $Y_0(N)$. Jorgenson and Kramer established the bound $\CmZ_{Y_0(N)}=O_ε(N^ε)$, $ε>0$ by relating it to geometric invariants. In this article we give, for $N$ prime, another proof via $L$-functions and exponential sums improving on a previous approach by Abbes-Ullmo and Michel-Ullmo. We further derive a power of $\log N$ bound along the same line.
The aim of this article is to study rank-one Eisenstein cohomology for the group GL(N)/F, where F is a totally real field extension of Q. This is then used to prove rationality results for ratios of successive critical values for Rankin-Selberg L-functions for GL(n) x GL(n') over F with the parity condition that nn' is even. The key idea is to interpret Langlands's constant term theorem in terms of Eisenstein cohomology.
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…
Closed string amplitudes at genus $h\leq 3$ are given by integrals of Siegel modular functions on a fundamental domain of the Siegel upper half-plane. When the integrand is of rapid decay near the cusps, the integral can be computed by the Rankin-Selberg method, which consists of inserting an Eisenstein series $E_h(s)$ in the integrand, computing the integral by the orbit method, and finally extracting the residue at a suitable value of $s$. String amplitudes, however, typically involve integrands with polynomial or even exponential growth at the cusps, and a renormalization scheme is required…
We give a construction of a new $p$-adic Maass-Shimura operator defined on an affinoid subdomain of the preperfectoid $p$-adic universal cover $\mathcal{Y}$ of a modular curve $Y$. We define a new notion of $p$-adic modular forms as sections of a certain sheaf $\mathcal{O}_Δ$ of "nearly rigid functions" which transform under the action of subgroups of the Galois group $\mathrm{Gal}(\mathcal{Y}/Y)$ by $\mathcal{O}_Δ^{\times}$-valued weight characters. This extends Katz's notion of $p$-adic modular forms as functions on the Igusa tower $Y^{\mathrm{Ig}}$; indeed we may recover Katz's theory by re…
Selberg's central limit theorem states that the values of $\log|ζ(1/2+i τ)|$, where $τ$ is a uniform random variable on $[T,2T]$, is distributed like a Gaussian random variable of mean $0$ and standard deviation $\sqrt{\frac{1}{2}\log \log T}$. It was conjectured by Radziwiłł that this breaks down for values of order $\log\log T$, where a multiplicative correction $C_k$ would be present at level $k\log\log T$, $k>0$. This constant should be equal to the leading asymptotic for the $2k^{th}$ moment of $ζ$, as first conjectured by Keating and Snaith using random matrix theory. In this paper, we p…
In 1946, A. Selberg proved $N(σ,T) \ll T^{1-\frac{1}{4} \left(σ-\frac{1}{2}\right)} \log{T}$ where $N(σ,T)$ is the number of nontrivial zeros $ρ$ of the Riemann zeta-function with $\Re\{ρ\}>σ$ and $0<\Im\{ρ\}\leq T$. We provide an explicit version of this estimate, together with an explicit approximate functional equation and an explicit upper bound for the second power moment of the zeta-function on the critical line.
We study the value distribution of the Riemann zeta function near the line $\Re s = 1/2$. We find an asymptotic formula for the number of $a$-values in the rectangle $ 1/2 + h_1 / (\log T)^θ\leq \Re s \leq 1/2+ h_2 /(\log T)^θ$, $T \leq \Im s \leq 2T$ for fixed $h_1, h_2>0$ and $ 0 < θ<1/13$. To prove it, we need an extension of the valid range of Lamzouri, Lester and Radziwiłł's recent results on the discrepancy between the distribution of $ζ(s)$ and its random model. We also propose the secondary main term for the Selberg's central limit theorem by providing sharper estimates on the line $\R…
This is the second in a series of two articles where we study various aspects of the spectral theory associated to families of hyperbolic Riemann surfaces obtained through elliptic degeneration. In the first article, we investigate the asymptotics of the trace of the heat kernel both near zero and infinity and we show the convergence of small eigenvalues and corresponding eigenfunctions. Having obtained necessary bounds for the trace, this second article presents the behavior of several spectral invariants. Some of these invariants, such as the Selberg zeta function and the spectral counting f…
Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many large natural numbers $n$ such that $(γ_{n+r}-γ_n)/(2π/\log γ_n) > r + Θ\sqrt{r}$ and $(γ_{n+r}-γ_n)/(2π/\log γ_n) < r - \vartheta\sqrt{r}$ for explicit absolute positive constants $Θ$ and $\vartheta$, where $γ$ denotes an ordinate of a zero of the Riemann zeta-function on the critical line. Selberg published announcements of this result several times but did not include a proof. We also suggest a general framework which might lead to stronger statements concerning the vertical distribution of nontriv…
We continue with our study of the non-critical exceptional zeros of Katz' $p$-adic $L$-functions attached to a CM field $K$, following two threads. In the first thread, we redefine our (group-ring-valued) $\mathcal{L}$-invariant associated to each $\mathbb{Z}_{p}$-extension $K_Γ$ of $K$ in terms of $p$-adic height pairings and interpolate them as $K_Γ$ varies to a universal (multivariate) group-ring-valued $\mathcal{L}$-invariant. In the second thread, we use our results to study the exceptional zeros of the non-genuine Rankin--Selberg $p$-adic $L$-functions attached to the self-products of ne…
Let $S(σ,t)=\frac{1}π\argζ(σ+it)$ be the argument of the Riemann zeta-function at the point $σ+it$ in the critical strip. For $n\geq 1$ and $t>0$, we define \begin{equation*} S_{n}(σ,t) = \int_0^t S_{n-1}(σ,τ) \,dτ+ δ_{n,σ\,}, \end{equation*} where $δ_{n,σ}$ is a specific constant depending on $σ$ and $n$. Let $0\leq β<1$ be a fixed real number. Assuming the Riemann hypothesis, we establish lower bounds for the maximum of $S_n(σ,t+h)-S_n(σ,t)$ near the critical line, on the interval $T^β\leq t \leq T$ and in a small range of $h$. This improves some results of the first author and generalizes a…
