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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]

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: 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]

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]

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]