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For 0 ≤ α ≤ 1 , let L α (x) = ∑ n ≤ x λ (n) n α , where λ (n) is the Liouville function. Then famous criteria of Pólya and Turán claim that the eventual sign constancy of each of L 0 (x) and L 1 (x) alone implies the Riemann hypothesis. However, Haselgrove disproved the eventual sign constancy hypothesis. As a remedy for this, we show that the eventual negativity of each of L α (x) , L α ′ (x) , L α ″ (x) for all 1 / 2 < α < 1 is equivalent to the Riemann hypothesis. Analogous equivalent conditions for the Riemann hypothesis are formulated in terms of partial sums of the Möbius function as well. Our criteria indicate a new tendency concerning the values of the Liouville and Möbius functions in stark contrast with their random behavior. Further generalizations are given in two essential directions, namely that equivalent criteria are developed for any quasi-Riemann hypothesis, and we allow the partial sums to be supported on semigroups after sifting out integers divisible by members of a set of prime numbers whose size is under control. Our negativity conditions for the quasi-Riemann hypothesis exhibit a curious rigidity in two respects, first it doesn't matter which particular primes we use to sift out, but only the size of the set, second the sign of the partial sums is unaltered with respect to two successive derivatives. Finally, other competing necessary or sufficient conditions for a quasi-Riemann hypothesis are studied regarding the monotonicity, zeros and sign changes of partial sums of Dirichlet series. [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]

Inspired by representations of the class number of imaginary quadratic fields, in this paper, we give explicit evaluations of trigonometric series having generalized harmonic numbers as coefficients in terms of odd values of the Riemann zeta function and special values of L -functions subject to the parity obstruction. The coefficients that arise in these evaluations are shown to belong to certain cyclotomic extensions. Furthermore, using best polynomial approximation of smooth functions under uniform convergence due to Jackson and their log-sine integrals, we provide approximations of real numbers by combinations of special values of L -functions corresponding to the Legendre symbol. Our method for obtaining these results rests on a careful study of generating functions on the unit circle involving generalized harmonic numbers and the Legendre symbol, thereby relating them to values of polylogarithms and then finally extracting Fourier series of special functions that can be expressed in terms of Clausen functions. [ABSTRACT FROM AUTHOR]

Assuming a weaker form of the Riemann hypothesis for Dedekind zeta functions by allowing Siegel zeros, we extend a classical result of Cramér on the number of primes in short intervals to prime ideals of the ring of integers in cyclotomic extensions with norms belonging to such intervals. The extension is uniform with respect to the degree of the cyclotomic extension. Our approach is based on the arithmetic of cyclotomic fields and analytic properties of their Dedekind zeta functions together with a lower bound for the number of primes over progressions in short intervals subject to similar assumptions. Uniformity with respect to the modulus of the progression is obtained and the lower bound turns out to be best possible, apart from constants, as shown by the Brun–Titchmarsh theorem. [ABSTRACT FROM AUTHOR]

The analytic continuation of a family of Dirichlet series whose coefficients are partition functions having parts in a finite set is established. The singularities arising from this continuation are shown to be finitely many simple poles, where the residues are computable positive rational numbers. Then we give several applications. First, special values of the continuation are studied in terms of algebraic combinations of the Hurwitz zeta function. Second, a transformation formula is obtained for exponentially decaying series involving these partition functions using special values of the continuation at negative integers. Finally, asymptotic behavior and orthogonality of certain convolutions with the normalization of these partition functions are investigated in the mean value sense with precise error terms. [ABSTRACT FROM AUTHOR]

It is shown that the set of odd values {ζ(3),ζ(5),...,ζ(2k +1),...} of the Riemann zeta function is rich enough to capture real numbers in an approximation aspect. Precisely, we prove that any real number can be strongly approximated by certain linear combinations of these odd values, where the coefficients belonging to these combinations are universal in the sense of being independent of ζ(n) for all integers n ≥ 2. This approximation property is reminiscent of the classical Diophantine approximation of Liouville numbers by rationals. [ABSTRACT FROM AUTHOR]

Using integral representations with carefully chosen rational functions as integrands, we find new families of transcendental numbers that are not U-numbers, according to Mahler's classification, represented by a series whose terms involve rising factorials and reciprocals of binomial coefficients analogous to Apéry type series. Explicit descriptions of these numbers are given as linear combinations with coefficients lying in a suitable real algebraic extension of rational numbers using elementary functions evaluated at arguments belonging to the same field. In this way, concrete examples of transcendental numbers which can be expressed as combinations of classical mathematical constants such as π and Baker periods are given together with upper bounds on their wn measures. [ABSTRACT FROM AUTHOR]

Abstract: Sign of averages of Ramanujan sums is studied and it is shown that these averages have a curious tendency to be positive. This in turn gives that Ramanujan sums are nearly orthogonal to a family of vectors whose entries are powers of consecutive integers. Further applications are given to the limit points of these averages along semigroups of integers, the peak size of partial sums of Ramanujan sum and an optimization problem on weighted exponential sums supported on reduced residue systems. Exact evaluations of trigonometric sums having combinatorially significant coefficients and subject to divisibility constraints are obtained in terms of Bernoulli numbers. [Copyright &y& Elsevier]

Abstract: Using an integral transform with a mild singularity, we obtain series representations valid for specific regions in the complex plane involving trigonometric functions and the central binomial coefficient which are analogues of the types of series representations first studied by Ramanujan over certain intervals on the real line. We then study an exponential type series rapidly converging to the special values of L-functions and the Riemann zeta function. In this way, a new series converging to Catalanʼs constant with geometric rate of convergence less than a quarter is deduced. Further evaluations of some series involving hyperbolic functions are also given. [Copyright &y& Elsevier]

Abstract: We obtain quantitative upper bounds on partial sums of the Möbius function over semigroups of integers in an arithmetic progression. Exploiting the cancellation of such sums, we deduce upper bounds for the discrepancy of fractions in the unit interval whose denominators satisfy the same restrictions. In particular, the uniform distribution and approximation of discrete weighted averages over such fractions are established as a consequence. [Copyright &y& Elsevier]

We obtain an exact formula for the average of values of L-series over two independent odd characters. The average of any positive moment of values at s = 1 is then expressed in terms of finite cotangent sums subject to congruence conditions. As consequences, bounds on such cotangent sums, limit points for the average of first moment of L-series at s = 1 and the average size of positive moments of character sums related to the class number are deduced. [ABSTRACT FROM AUTHOR]

The Mellin transform of a summatory function involving weighted averages of Ramanujan sums is obtained in terms of Bernoulli numbers and values of the Burgess zeta function. The possible singularity of the Burgess zeta function at s = 1 is then shown to be equivalent to the evaluation of a certain infinite series involving such weighted averages. Bounds on the size of the tail of these series are given and specific bounds are shown to be equivalent to the Riemann hypothesis [ABSTRACT FROM AUTHOR]

Abstract: Let χ be a Dirichlet character and be its L-function. Using weighted averages of Gauss and Ramanujan sums, we find exact formulas involving Jordanʼs and Eulerʼs totient function for the mean square average of when χ ranges over all odd characters modulo k and when χ ranges over all even characters modulo k. In principle, using our method, it is always possible to find the mean square average of if χ and have the same parity and χ ranges over all odd (or even) characters modulo k, though the required calculations become formidable when . Consequently, we see that for almost all odd characters modulo k, , where is any function monotonically tending to infinity. [Copyright &y& Elsevier]

For each positive integer Q, let ℱ Q denote the Farey sequence of order Q. We prove the existence of the pair correlation measure associated to the sum ℱ Q + ℱ Q modulo 1, as Q tends to infinity, and compute the corresponding limiting pair correlation function. [ABSTRACT FROM AUTHOR]

Abstract: We prove the existence of the pair correlation measure associated to torsion points on the real locus of an elliptic curve E and provide an explicit formula for the limiting pair correlation function. [Copyright &y& Elsevier]

We prove that certain powers of the gap function for the newform associated to an elliptic curve without complex multiplication are "finite" on average. In particular we obtain quantitative results on the number of large values of the gap function. [ABSTRACT FROM AUTHOR]

Abstract: We investigate local spacing problems along curves via smooth maps. Moreover, we provide explicit formulas for the nearest neighbor spacing distribution function of torsion points on elliptic curves over , and of rational points on the unit circle. [Copyright &y& Elsevier]

In this paper we present a survey of recent progress on the distribution of B-free numbers in short intervals and some of its applications. [ABSTRACT FROM AUTHOR]

Abstract: Given an irrational number α and a sequence B of coprime positive integers with the sum of inverses convergent, we investigate the problem of finding small values of nα , with n B-free. [Copyright &y& Elsevier]

In this paper we investigate the ring Ar(R) of arithmetical functions in r variables over an integral domain R. We study a class of absolute values, and a class of derivations on Ar(R). We show that a certain extension of Ar(R) is a discrete valuation ring. We also investigate the metric structure of the ring Ar(R). [ABSTRACT FROM AUTHOR]

Abstract: Given a sequence B of relatively prime positive integers with the sum of inverses finite, we investigate the problem of finding B-free numbers in short arithmetic progressions. [Copyright &y& Elsevier]

We provide new estimates for the gap function of the Delta function and for the number of nonzero values of the Ramanujan tau function in short intervals. [ABSTRACT FROM AUTHOR]

Let a(n) be the number of nonisomorphic abelian groups of order n. We obtain a short interval result for the local density of a(n). More generally, we get short interval version of results of Ivic´ on the local density of prime independent multiplicative functions. Also we prove a short interval version of the theorem of Erdo¨s and Szekeres on the summatory function of a(n) and the theorem of Greenberg and Newman on the enumeration of a certain type of finite solvable groups. [Copyright &y& Elsevier]

Presents variations of the proof on Wolstenholme's theorem.

We investigate a new type of tendency between two progressions of prime numbers which is in support of the claim that prime numbers that are congruent to 3 modulo 4 are favored over prime numbers that are congruent to 1 modulo 4. In particular, we show that the Riemann hypothesis for the corresponding L-function is equivalent to the occurrence of such a tendency. A generalization to teams of progressions of prime numbers is given, where the teams are formed by grouping according to the values of a quadratic character. In this way, it is shown that there is a tendency favoring prime numbers belonging to progressions arising from the quadratic nonresidues modulo a prime number congruent to 3 or 5 modulo 8. The scope of the tendency is extended conditionally, either by assuming the Riemann hypothesis for certain Dirichlet L-functions or by the presence of Siegel zeros. Our approach requires numerical verifications over certain ranges of the parameters, and in this respect, we freely benefit from computer software to carry out such tasks. Lastly, the divisor function is seen to be favorable over its average value along semigroups by comparing partial sums of the associated Dirichlet series. [ABSTRACT FROM AUTHOR]

We investigate a new type of tendency between two progressions of prime numbers which is in support of the claim that prime numbers that are congruent to 3 modulo 4 are favored over prime numbers that are congruent to 1 modulo 4. In particular, we show that the Riemann hypothesis for the corresponding L-function is equivalent to the occurrence of such a tendency. A generalization to teams of progressions of prime numbers is given, where the teams are formed by grouping according to the values of a quadratic character. In this way, it is shown that there is a tendency favoring prime numbers belonging to progressions arising from the quadratic nonresidues modulo a prime number congruent to 3 or 5 modulo 8. The scope of the tendency is extended conditionally, either by assuming the Riemann hypothesis for certain Dirichlet L-functions or by the presence of Siegel zeros. Our approach requires numerical verifications over certain ranges of the parameters, and in this respect, we freely benefit from computer software to carry out such tasks. Lastly, the divisor function is seen to be favorable over its average value along semigroups by comparing partial sums of the associated Dirichlet series. [ABSTRACT FROM AUTHOR]