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We show the zero-density estimate \[ N(\zeta_{\mathcal{P}}; \alpha, T) \ll T^{\frac{4(1-\alpha)}{3-2\alpha-\theta}}(\log T)^{9} \] for Beurling zeta functions $\zeta_{\mathcal{P}}$ attached to Beurling generalized number systems with integers distributed as $N_{\mathcal{P}}(x) = Ax + O(x^{\theta})$. We also show a similar zero-density estimate for a broader class of general Dirichlet series, consider improvements conditional on finer pointwise or $L^{2k}$-bounds of $\zeta_{\mathcal{P}}$, and discuss some optimality questions.

In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)= \sum_{j=1}^\infty a_j n_j^{-s}$ with positive coefficients for which the counting function $A(x) = \sum_{n_{j}\le x}a_{j}$ satisfies $A(x)=\rho x + O(x^\beta)$ for some $\rho>0$ and $\beta<1$. We prove that $\frac1T\int_0^T |f(\sigma+it)|^2\, dt \to \sum_{j=1}^\infty a_j^2n_j^{-2\sigma}$ for $\sigma>\frac{1+\beta}{2}$ and obtain an upper bound for this moment for $\beta<\sigma\le \frac{1+\beta}{2}$. We provide a number of examples indicating the sharpness of our results., Comment: To appear in Quart. J. Math

We investigate the existence of well-behaved Beurling number systems, which are systems of Beurling generalized primes and integers which admit a power saving in the error term of both their prime and integer-counting function. Concretely, we search for so-called $[\alpha,\beta]$-systems, where $\alpha$ and $\beta$ are connected to the optimal power saving in the prime and integer-counting functions. It is known that every $[\alpha,\beta]$-system satisfies $\max\{\alpha,\beta\}\ge1/2$. In this paper we show there are $[\alpha,\beta]$-systems for each $\alpha \in [0,1)$ and $\beta \in [1/2, 1)$. Assuming the Riemann hypothesis, we also construct certain families of $[\alpha,\beta]$-systems with $\beta<1/2$., Comment: 24 pages

In a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindel\"of hypothesis (LH) for general sequences which coincides with the usual Lindel\"of hypothesis for the Riemann zeta function in the case of the sequence of positive integers. They made two conjectures: that LH should hold for every admissible sequence of positive integers, and that LH should hold for the ''generic'' admissible sequence of positive real numbers. In this paper, we give counterexamples to the first conjecture, and show that the second conjecture can be either true or false, depending on the meaning of ''generic'': we construct probabilistic processes producing sequences satisfying LH with probability 1, and we construct Baire topological spaces of sequences for which the subspace of sequences satisfying LH is meagre. We also extend the main result of Gonek, Graham, and Lee, stating that the Riemann hypothesis is equivalent to LH for the sequence of prime numbers, to the context of Beurling generalized number systems., Comment: 24 pages

We study the distribution of zeros of zeta functions associated to Beurling generalized prime number systems whose integers are distributed as $N(x) = Ax + O(x^{\theta})$. We obtain in particular \[ N(\alpha, T) \ll T^{\frac{c(1-\alpha)}{1-\theta}}\log^{9} T, \] for a constant $c$ arbitrarily close to $4$, improving significantly the current state of the art. We also investigate the consequences of the obtained zero-density estimates on the PNT in short intervals. Our proofs crucially rely on an extension of the classical mean-value theorem for Dirichlet polynomials to generalized Dirichlet polynomials., Comment: 22 pages, fixed some minor errors

Distributed-order time-fractional wave equations appear in the modeling of wave propagation in viscoelastic media. The material characteristics of the medium are modeled through constitutive functions or distributions in the distributed-order constitutive law. In this work we propose to take positive Radon measures for the constitutive "functions". First, we derive a thermodynamical restriction on the constitutive measures which is easy to check, and therefore suitable for applications. Then we prove that the setting with measures in combination with the derived thermodynamical restriction guarantee existence and uniqueness of solutions for the distributed-order fractional wave equation. We further discuss the support and regularity of the fundamental solution, and conclude with a discussion on wave velocities.

We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given $\alpha\in (0,1]$ and $c>0$ (with $c\leq 1$ if $\alpha=1$), a generalized number system is constructed with Riemann prime counting function $ \Pi(x)= \operatorname*{Li}(x)+ O(x\exp (-c \log^{\alpha} x ) +\log_{2}x), $ and whose integer counting function satisfies the extremal oscillation estimate $N(x)=\rho x + \Omega_{\pm}(x\exp(- c'(\log x\log_{2} x)^{\frac{\alpha}{\alpha+1}})$ for any $c'>(c(\alpha+1))^{\frac{1}{\alpha+1}}$, where $\rho>0$ is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240]., Comment: 27 pages

We present a new and simple method for the determination of the pointwise H\"{o}lder exponent of Riemann's function $\sum_{n=1}^{\infty} n^{-2}\sin(\pi n^{2} x)$ at every point of the real line. In contrast to earlier approaches, where wavelet analysis and the theta modular group were needed for the analysis of irrational points, our method is direct and elementary, being only based on the following tools from number theory and complex analysis: the evaluation of quadratic Gauss sums, the Poisson summation formula, and Cauchy's theorem., Comment: 14 pages. An appendix on fractional integrals of modular forms was added

This paper concerns the micro-local and qualitative analysis of the fractional Zener wave equation. The classical and Gevrey-type wave front sets of the fundamental solution are determined, and questions on dispersion, dissipation, wave propagation speed and wave packet shape are addressed., Comment: 34 pages, 3 figures

Given $\beta\in(0,1)$, we show the existence of a Beurling generalized number system whose integer counting satisfies $N(x) = ax + O\bigl(x\exp(-c\log^{\beta} x)\bigr)$ for some $a>0$ and $c>0$, and whose prime counting function satisfies $\pi(x) = \mathrm{Li}(x) + \Omega\bigl(x\exp(-c'(\log x)^{\frac{\beta}{\beta+1}})\bigr)$ for some $c'>0$. This is done by generalizing a construction of Diamond, Montgomery, and Vorhauer. This Beurling system serves as additional motivation for a conjecture of Bateman and Diamond from 1969, concerning the PNT with Malliavin-type remainder., Comment: 11 pages

We study the family of Fourier-Laplace transforms $$ F_{\alpha,\beta}(z)= \operatorname*{F.p.} \int_{0}^{\infty} t^{\beta}\exp(\mathrm{i} t^{\alpha}-\mathrm{i} z t)\:\mathrm{d} t, \quad \operatorname*{Im} z<0, $$ for $\alpha>1$ and $\beta\in\mathbb{C}$, where Hadamard finite part is used to regularize the integral when $\operatorname*{Re} \beta\leq -1$. We prove that each $F_{\alpha,\beta}$ has analytic continuation to the whole complex plane and determine its asymptotics along any line through the origin. We also apply our ideas to show that some of these functions provide concrete extremal examples for the Wiener-Ikehara theorem and a quantified version of the Ingham-Karamata theorem, supplying new simple and constructive proofs of optimality results for these complex Tauberian theorems., Comment: 15 pages

We construct a Beurling generalized number system satisfying the Riemann hypothesis and whose integer counting function displays extremal oscillation in the following sense. The prime counting function of this number system satisfies $\pi(x)= \operatorname*{Li} (x)+ O(\sqrt{x})$, while its integer counting function satisfies the oscillation estimate $N(x) = \rho x + \Omega_{\pm}\bigl(x\exp(-c\sqrt{\log x\log\log x})\bigr)$ for some $c>0$, where $\rho>0$ is its asymptotic density. The construction is inspired by a classical example of H. Bohr for optimality of the convexity bound for Dirichlet series, and combines saddle-point analysis with the Diamond-Montgomery-Vorhauer probabilistic method via random prime number system approximations., Comment: 31 pages, 2 figures

We construct explicit counterexamples that show that it is impossible to get any remainder, other than the classical ones, in the Wiener-Ikehara theorem and the Ingham-Karamata theorem under just an additional analytic continuation hypothesis to a half-plane (or even to the whole complex plane)., Comment: 8 pages