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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]

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]

We give density results on a conjecture of MezŐ about hyperharmonic numbers. It is shown with stronger error terms that almost all hyperharmonic numbers are not integers. Other types of density results based on a new sufficient condition about the nonintegerness of hyperharmonic numbers are obtained. Quantitative estimates for the frequency of noninteger hyperharmonic numbers are deduced using the distribution of prime numbers. In particular, assuming the Riemann hypothesis or Cramér's conjecture on the gap between consecutive prime numbers, we come very close to establishing Mező's conjecture. [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]

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: 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]

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]