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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

Given a sequence of frequencies $\{\lambda_n\}_{n\geq1}$, a corresponding generalized Dirichlet series is of the form $f(s)=\sum_{n\geq 1}a_ne^{-\lambda_ns}$. We are interested in multiplicatively generated systems, where each number $e^{\lambda_n}$ arises as a finite product of some given numbers $\{q_n\}_{n\geq 1}$, $1 < q_n \to \infty$, referred to as Beurling primes. In the classical case, where $\lambda_n = \log n$, Bohr's theorem holds: if $f$ converges somewhere and has an analytic extension which is bounded in a half-plane $\{\Re s> \theta\}$, then it actually converges uniformly in every half-plane $\{\Re s> \theta+\varepsilon\}$, $\varepsilon>0$. We prove, under very mild conditions, that given a sequence of Beurling primes, a small perturbation yields another sequence of primes such that the corresponding Beurling integers satisfy Bohr's condition, and therefore the theorem. Applying our technique in conjunction with a probabilistic method, we find a system of Beurling primes for which both Bohr's theorem and the Riemann hypothesis are valid. This provides a counterexample to a conjecture of H. Helson concerning outer functions in Hardy spaces of generalized Dirichlet series.

In a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindelöf hypothesis (LH) for general sequences which coincides with the usual Lindelöf 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., 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

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: 13 pages. An appendix on fractional integrals of modular forms was added

We present a new random approximation method that yields the existence of a discrete Beurling prime system $\mathcal{P}=\{p_{1}, p_{2}, \dotso\}$ which is very close in a certain precise sense to a given non-decreasing, right-continuous, nonnegative, and unbounded function $F$. This discretization procedure improves an earlier discrete random approximation method due to H. Diamond, H. Montgomery, and U. Vorhauer [Math. Ann. 334 (2006), 1-36], and refined by W.-B. Zhang [Math. Ann. 337 (2007), 671-704]. We obtain several applications. Our new method is applied to a question posed by M. Balazard concerning Dirichlet series with a unique zero in their half plane of convergence, to construct examples of very well-behaved generalized number systems that solve a recent open question raised by T. Hilberdink and A. Neamah in [Int. J. Number Theory 16 05 (2020), 1005-1011], and to improve the main result from [Adv. Math. 370 (2020), Article 107240], where a Beurling prime system with regular primes but extremely irregular integers was constructed., Comment: 15 pages