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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: 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]

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]