Phase transition in a stochastic prime number generator

We introduce a stochastic algorithm that acts as a prime number generator. The dynamics of such algorithm gives rise to a continuous phase transition which separates a phase where the algorithm is able to reduce a whole set of integers into primes and a phase where the system reaches a frozen state with low prime density. We present both numerical simulations and an analytical approach in terms of an annealed approximation, by means of which the data are collapsed. A critical slowing down phenomenon is also outlined.Comment: accepted in PRE (Rapid Comm.

Zeta Function Zeros, Powers of Primes, and Quantum Chaos

We present a numerical study of Riemann's formula for the oscillating part of the density of the primes and their powers. The formula is comprised of an infinite series of oscillatory terms, one for each zero of the zeta function on the critical line and was derived by Riemann in his paper on primes assuming the Riemann hypothesis. We show that high resolution spectral lines can be generated by the truncated series at all powers of primes and demonstrate explicitly that the relative line intensities are correct. We then derive a Gaussian sum rule for Riemann's formula. This is used to analyze the numerical convergence of the truncated series. The connections to quantum chaos and semiclassical physics are discussed

Unrealistic pessimism and obsessive‐compulsive symptoms during the COVID‐19 pandemic: two longitudinal studies

OBJECTIVE: Unrealistic pessimism (UP) is an aspect of overestimation of threat (OET) that has been associated with obsessive‐compulsive disorder/symptoms (OCD/OCS). During the COVID‐19 pandemic, UP may have played an important role in the course of OCD. To investigate the relationship, we conducted two longitudinal studies assuming that higher UP predicts an increase in OCS. METHOD: In Study 1, we investigated UP in the general population (N = 1,184) at the start of the pandemic asking about overall vulnerability to infection with SARS‐CoV‐2 and UP regarding infection and outcome of severe illness. Further, OCS status (OCS+/−) was assessed at the start of the pandemic and 3 months later. In Study 2, we investigated UP in individuals with OCD (N = 268) regarding the likelihood of getting infected, recovering, or dying from an infection with SARS‐CoV‐2 at the start of the pandemic and re‐assessed OCS 3 months later. RESULTS: In Study 1, UP was higher in the OCS+ compared to the OCS− group, and estimates of a higher overall vulnerability for an infection predicted a decrease in OCS over time. UP regarding severe illness predicted an increase in symptoms over time. In Study 2, UP was found for a recovery and death after an infection with SARS‐CoV‐2, but not for infection itself. CONCLUSIONS: Exaggeration of one’s personal vulnerability rather than OET per se seems pivotal in OCD, with UP being associated with OCD/OCS+ as well as a more negative course of symptomatology over the pandemic in a nonclinical sample. PRACTITIONER POINTS: Unrealistic optimism, a bias common in healthy individuals, is thought to be a coping mechanism promoting well‐being in the face of danger or uncertainty. The current study extends findings that its inversion, unrealistic pessimism, may play an important role in obsessive‐compulsive disorder and may also be involved in the development of the disorder. This study highlights the importance that prevention programs during a pandemic should include targeting unrealistic pessimism

Event-Related Potential Correlates of Performance-Monitoring in a Lateralized Time-Estimation Task

Performance-monitoring as a key function of cognitive control covers a wide range of diverse processes to enable goal directed behavior and to avoid maladjustments. Several event-related brain potentials (ERP) are associated with performance-monitoring, but their conceptual background differs. For example, the feedback-related negativity (FRN) is associated with unexpected performance feedback and might serve as a teaching signal for adaptational processes, whereas the error-related negativity (ERN) is associated with error commission and subsequent behavioral adaptation. The N2 is visible in the EEG when the participant successfully inhibits a response following a cue and thereby adapts to a given stop-signal. Here, we present an innovative paradigm to concurrently study these different performance-monitoring-related ERPs. In 24 participants a tactile time-estimation task interspersed with infrequent stop-signal trials reliably elicited all three ERPs. Sensory input and motor output were completely lateralized, in order to estimate any hemispheric processing preferences for the different aspects of performance monitoring associated with these ERPs. In accordance with the literature our data suggest augmented inhibitory capabilities in the right hemisphere given that stop-trial performance was significantly better with left- as compared to right-hand stop-signals. In line with this, the N2 scalp distribution was generally shifted to the right in addition to an ipsilateral shift in relation to the response hand. Other than that, task lateralization affected neither behavior related to error and feedback processing nor ERN or FRN. Comparing the ERP topographies using the Global Map Dissimilarity index, a large topographic overlap was found between all considered components.With an evenly distributed set of trials and a split-half reliability for all ERP components ≥.85 the task is well suited to efficiently study N2, ERN, and FRN concurrently which might prove useful for group comparisons, especially in clinical populations