We show that the density functions of nearest neighbor spacing distributions for the zeros of the real or imaginary part of the Riemann xi-function on vertical lines are described by the M-function which appears in value distribution of the logarithmic derivative of the Riemann zeta-function on vertical lines.
Denote by Liouville’s function concerning the parity of the number of prime divisors of . Using a theorem of Allouche, Mendès France, and Peyrière and many classical results from the theory of the distribution of prime numbers, we prove that is not –automatic for any . This yields that is transcendental over for any prime . Similar results are proven (or reproven) for many common number–theoretic functions, including , , , , , and others.
Let pₘ(n) stand for the middle prime factor of the integer n ≥ 2. We first establish that the size of log pₘ(n) is close to √(log n) for almost all n. We then show how one can use the successive values of pₘ(n) to generate a normal number in any given base D ≥ 2. Finally, we study the behavior of exponential sums involving the middle prime factor function.