We present an equivalent formulation of the Riemann Hypothesis in terms of the continuous Newton flow: All non-trivial zeros of the Riemann zeta function have real part 1/2 iff the lines of constant phase of the Riemann function approach the critical line from the right and left edges of the complex plane without U-turns. Moreover, we prove that this formulation is identical to the Hinkkanen equivalence.

In previous papers the authors established the prime avoidance property of k-th powers of prime numbers and of prime numbers within Beatty sequences. In this paper the authors consider k-th powers of Piatetski-Shapiro primes.

We provide an introduction for physicists into the Riemann Hypothesis. For this purpose, we first introduce, and then compare and contrast the Riemann function and the Dirichlet L-functions, with the Titchmarsh counterexample. Whereas the first two classes of functions are expected to satisfy the Riemann Hypothesis, the Titchmarsh counterexample is known to violate it. Throughout our article we employ elementary mathematical techniques known to every physicist. Needless to say, we do not verify the Riemann Hypothesis but suggest heuristic arguments in favor of it. We also build a bridge to quantum mechanics by interpreting the Dirichlet series central to this field as a superposition of probability amplitudes leading us to an unusual potential with a logarithmic energy spectrum opening the possibility of factoring numbers.

Let γ^*:=8/9+2/3 log(10/9)/log 10 (≈ 0.919…) , γ^*<1/c_0≤ 1 . Let γ^*<γ_0≤ 1, c_0=1/γ_0 be fixed. Let also a_0∈{0,1,…, 9}. In [23] we proved on assumption of the Generalized Riemann Hypothesis (GRH), that each sufficiently large odd integer N_0 can be represented in the form N_0=p_1+p_2+p_3 , where for i=2, 3 the primes p_i are Piatetski-Shapiro primes - primes of the form p_i=[n_i^c_0], n_i∈ℕ - whereas the decimal expansion of p_1 does not contain the digit a_0. In this paper we replace one of the Piatetski-Shapiro primes p_2 and p_3 by primes of the type p=x^2+y^2+1 .

Let $$\gamma^*:=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:,\ \gamma^*<\frac{1}{c_0}\leq 1\:.$$ Let $\gamma^*<\gamma_0\leq 1$, $c_0=1/\gamma_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$. In [23] we proved on assumption of the Generalized Riemann Hypothesis (GRH), that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where for $i=2, 3$ the primes $p_i$ are Piatetski-Shapiro primes - primes of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$ - whereas the decimal expansion of $p_1$ does not contain the digit $a_0$. In this paper we replace one of the Piatetski-Shapiro primes $p_2$ and $p_3$ by primes of the type $$p=x^2+y^2+1\:.$$

Cotangent sums play a significant role in the Nyman-Beurling criterion for the Riemann Hypothesis. Here we investigate the maximum of the values of these cotangent sums over various sets of rational numbers in short intervals.

One of the approaches to the Riemann Hypothesis is the Nyman-Beurling criterion. Cotangent sums play a significant role. Here we investigate the values of these cotangent sums for various shifts of the argument.

A certain category of cotangent sums has been proven of importance in the Nyman–Beurling criterion for the Riemann Hypothesis. In previous work ( [12,13] ) the authors proved the existence of a unique positive measure μ on R , with respect to which certain normalized cotangent sums, evaluated at rational numbers with fixed denominators are equidistributed. The tools applied in this paper belong to various fields of Mathematics, for instance the relation between the equidistribution mod 1 of the multiples of a number and the Diophantine approximation properties of that number. In this paper we prove an analogous result for the case that the denominator of the rational numbers is a fixed prime number and that the numerator is also prime.

In Maier and Tenenbaum (Invent Math 76(1):121–128, 1984), Tenenbaum and the first author proved an old conjecture of Paul Erdős about the propinquity of divisors of integers. In this paper, we prove an analogous results for Gaussian integers.

In [5], the first author and A. Sankaranarayanan established an upper bound for the sum Sigma(n <= x) f (n), where f (n) is a multiplicative function for which the sum Sigma(p is an element of I) f(p)p(-it) is small for certain intervals I. In this paper we consider functions f(p), for which vertical bar Sigma(p is an element of I) f(p)p(-it )+ integral(I)u(-it)/log u du vertical bar is small. (C) 2018 Elsevier Inc. All rights reserved.

In various papers the authors have derived asymptotics for moments of certain cotangent sums related to the Riemann Hypothesis. S. Bettin has given an upper bound for the error term in these asymptotic results. In the present paper the authors establish a lower bound for the error term for the second moment.

The Winter Colloquium on the Physics of Quantum Electronics (PQE) has been a seminal force in quantum optics and related areas since 1971. It is rather mindboggling to recognize how the concepts presented at these conferences have transformed scientific understanding and human society. In January, 2017, the participants of PQE were asked to consider the equally important prospects for the future, and to formulate a set of questions representing some of the greatest aspirations in this broad field. The result is this multi-authored paper, in which many of the world's leading experts address the following fundamental questions: (1) What is the future of gravitational wave astronomy? (2) Are there new quantum phases of matter away from equilibrium that can be found and exploited - such as the time crystal? (3) Quantum theory in uncharted territory: What can we learn? (4) What are the ultimate limits for laser photon energies? (5) What are the ultimate limits to temporal, spatial, and optical resolution? (6) What novel roles will atoms play in technology? (7) What applications lie ahead for nitrogen-vacancy centers in diamond? (8) What is the future of quantum coherence, squeezing, and entanglement for enhanced superresolution and sensing? (9) How can we solve (some of) humanity's biggest problems through new quantum technologies? (10) What new understanding of materials and biological molecules will result from their dynamical characterization with free electron lasers? (11) What new technologies and fundamental discoveries might quantum optics achieve by the end of this century? (12) What novel topological structures can be created and employed in quantum optics?

We give an estimate for sums appearing in the Nyman–Beurling criterion for the Riemann Hypothesis containing the Möbius function. The estimate is remarkably sharp in comparison to estimates of other sums containing the Möbius function. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.

Abstract A crucial role in the Nyman-Beurling-Báez-Duarte approach to the Riemann Hypothesis is played by the distance $$d_{N}^{2}:=\underset{{{A}_{N}}}{\mathop{\inf }}\,\frac{1}{2\pi }\int _{-\infty }^{\infty }{{\left| 1-\zeta {{A}_{N}}\left( \frac{1}{2}+it \right) \right|}^{2}}\frac{dt}{\frac{1}{4}+{{t}^{2}}},$$ where the infimum is over all Dirichlet polynomials $${{A}_{N}}\left( s \right)\,=\,\sum\limits_{n=1}^{N}{\frac{{{a}_{n}}}{{{n}^{s}}}}$$ of length $N$ . In this paper we investigate $d_{N}^{2}$ under the assumption that the Riemann zeta function has four nontrivial zeros off the critical line.