On the moments of the Carmichael λ function

On the Range of the Iterated Euler Function

“…A slightly stronger lower bound appears in [1]. Stronger upper and lower bounds on #L 1 (x) will appear in [16]. While #L k (x) seems difficult to study for larger values of k, it is easy to see that the method of the present paper shows that uniformly for x large, #{λ k (n) : n ≤ x} ≤ x (log x) k exp 16k …”

Quaternionic Artin representations of ℚ

“…Then [10], p. 848, formulas (4) and (6)). We now follow [14], first of all by changing notation slightly (y in place of x). More importantly, we define a subset P * δ (y) of P δ (y) to eliminate large powerful divisors of p − 1.…”

“…Condition LS1 excludes at most O(y/(log y) c+1 ) elements of P δ (y), and using Abel summation and (56), we see that LS2 excludes at most Next let R δ (y) be the set of all products of [y δ ] distinct elements of P * δ (y). As in [14], we will estimate |R δ…”

“…Now take x 2, and as in [14], let y be the function of x defined by x = e y δ log y (note that y δ log y is strictly increasing for y > e −1/δ ). Recalling Proposition 20 as well as the definition (47) of σ(x), we see that Here α and δ are arbitrary numbers such that α > 0 and δ > δ.…”

“…[18], p. 385, Proposition 1), it seems natural to extend his conjecture to arbitrary number fields. In fact Ambrose [2] has already done so, except that he extends a slightly weaker form of Erdős's conjecture stated for example on p. 390 of [14]:…”