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A famous conjecture of Hardy and Littlewood claims the subadditivity of the prime counting function, namely that π(x+y) ≤ π(x)+π(y) holds for all integers x; y ≥ 2, where π(x) is the number of primes not exceeding x. It is widely believed nowadays that this conjecture is not true since Hensley and Richards stunningly discovered an incompatibility with the prime k-tuples conjecture. Despite this drawback, here we generalize the subadditivity conjecture to subsets of prime numbers possessing a rich collection of preassigned structures. We show that subadditivity holds in this extended manner over certain ranges of the parameters which are wide enough to imply that it holds in an almost all sense. Under the prime k-tuples conjecture, very large values of convex combinations of the prime counting function are obtained infinitely often, thereby indicating a strong deviation of π(x) from being convex, even in a localized form. Finally, a Tauberian type condition is given for subsets of prime numbers which in turn implies an extension of a classical phenomenon, originally suggested by Legendre, about the asymptotically best fit functions to π(x) of the shape x=(log x - A). [ABSTRACT FROM AUTHOR]

Purpose: The aim of this study was to evaluate the relationship between the service style of Community Mental Health Centers (CMHC) and anxiety levels in patients with schizophrenia in the Covid-19 pandemic. Materials and Methods: Twenty-six patients with schizophrenia in remission between the ages of 22-68, who regularly attended the Erzurum Regional Training and Research Hospital TRSM, were included in the study. A form was used to evaluate the sociodemographic characteristics of the patients. The Clinical Global Impression Scale (CGI) and State and Trait Anxiety Inventory (STAI), which were applied to the patients before the pandemic, were administered for the second time during the period when the post-pandemic interviews were minimized, and administered for the third time, when regular face-to-face meetings resumed. Results: The highest STAI scores were found in the second interview and the lowest in the first interview. The scores in all 3 interviews are statistically significantly different from each other. Although the CGI values seem to be statistically significantly different from each other in all 3 interviews; this difference was not found statistically significant in post-hoc analysis. Conclusion: The results we have obtained from our study show that the Covid-19 pandemic may have negative effects on the psychological state of schizophrenia patients, and community mental health services can contribute to reducing this effect. [ABSTRACT FROM AUTHOR]

Using convexity properties of reciprocals of zeta functions, especially the reciprocal of the Riemann zeta function, we show that certain weighted Mertens sums are biased in favor of square-free integers with an odd number of prime factors. We study such type of bias for different ranges of the parameters and then consider generalizations to Mertens sums supported on semigroups of integers generated by relatively large subsets of prime numbers. We further obtain a wider range for the parameters both unconditionally and then conditionally on the Riemann Hypothesis. At the same time, we extend to certain semigroups, two classical summation formulas originating from the works of Landau concerning the behavior of derivatives of the reciprocal of the Riemann zeta function at s = 1. [ABSTRACT FROM AUTHOR]