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Let p > 3 be a prime number, ζ be a primitive p-th root of unity. Suppose that the Kummer-Vandiver conjecture holds for p , i.e., that p does not divide the class number of Q (ζ + ζ - 1) . Let λ and ν be the Iwasawa invariants of Q (ζ) and put λ = : ∑ i ∈ I λ i and ν = : ∑ i ∈ I ν i in the usual way, where I is the set of odd integers in { 3 , ... , p - 2 } . We prove for each i ∈ I that λ i ≤ 1 , so that the p-primary part of class group of each Q (exp (2 π - 1 / p n)) is determined by the set { (i , ν i) : i ∈ I } . [ABSTRACT FROM AUTHOR]

Let p be an odd prime number, Kn = Q(ζpn+1) the pn+1th cyclotomic field and the relative class number of Kn. Fixing an integer d ∈ Z with , we denote by Ln the imaginary quadratic subextension of the imaginary (2, 2)-extension with Ln ≠ Kn. When d < 0, we have . Denote by and the minus parts of the 2-adic Iwasawa lambda invariants of Kn and Ln, respectively. By a theorem of Friedman, these invariants are stable for sufficiently large n. First, under the assumption that is odd for all n ≥ 1, we give a quite explicit version of this result. Second, we show that the assumption is satisfied for all p ≤ 599. Further, using these results, we compute the invariants and with d = -1, -3 for all p ≤ 599 and all n with the help of the computer. [ABSTRACT FROM AUTHOR]

We show that for any integers n ≥≥ 2 and 0 ≤≤ k ≤≤ n, one may find k distinct nth roots of unity whose sum is 0 if and only if both k and n -- k are expressible as linear combinations of prime factors of n with nonnegative coefficients. [ABSTRACT FROM AUTHOR]

Several mathematical frameworks of phase-locking are developed. The classical Huyghens approach is generalized to include all harmonic and subharmonic resonances and is found to be connected to 1/f noise and prime number theory. Two types of quantum phase-locking operators are defined, one acting on the rational numbers, the other on the elements of a Galois field. In both cases we analyse in detail the phase properties and find that they are related to the Riemann zeta function and to incomplete Gauss sums, respectively. [ABSTRACT FROM AUTHOR]