For each natural number n we determine the average order #(n) of the elements in a cyclic group of order n. We show that a large fraction of the contribution to #(n) comes from the #(n) primitive elements of order n. It is therefore of interest to study also the function #(n) = #(n)/#(n). We

. We show that the average order n of a permutation in Sn satisfies log n = C r n log n +O `# n log log n log n ' , which refines earlier results of Erdos and Turan, Schmutz, and Goh and Schmutz. 1. Introduction. For # # S n let N(#) be the order of # in the group S n . Erdos

For each natural number n we determine the average order α(n) of the elements in a cyclic group of order n. We show that more than half of the contribution to α(n) comes from the ϕ(n) primitive elements of order n. It is therefore of interest to study also the function β(n) = α(n)/ϕ(n). We

ABSTRACT. We consider the functions M(n), the maximum exponent of any prime power dividing n and m(n), the minimum exponent of any prime power dividing n. The sums ∑ n≤x M(n) and n≤x m(n) have been well investigated in the literature. In this note, we will improve known estimates of both the above sums under the assumption of the Riemann hypothesis. We will also obtain Ω-type estimates for these sums unconditionally.

We prove an asymptotic formula for the average order of the gcd-sum function by using a new convolution identity. 1 Introduction and main result In 2001, Broughan [1] studied the gcd-sum function g defined for any positive integer n by g (n) = n∑ (k,n),

-stock measures estimated with daily data, relies on the principle that order flow induces greater return reversals when liquidity is lower. Over a 34-year period, the average return on stocks with high sensitivities to liquidity exceeds that for stocks with low sensitivities by 7.5 % annually, adjusted

It is widely known that when there are errors with a moving-average root close to −1, a high order augmented autoregression is necessary for unit root tests to have good size, but that information criteria such as the AIC and the BIC tend to select a truncation lag (k) that is very small. We

. However, it would clearly be undesirable to have large queues (possibly on the order of a delay bandwidth product) that were full much of the time; this would significantly increase the average delay in the network. Therefore, with increasingly high-speed networks, it is increasingly important to have

endto-end latencies and similar failure recovery properties. More importantly, it is able to achieve these results by using orders of magnitude lower control traffic. Finally, we present results from our wide-area testbed in which we experimented with 32-100 member groups distributed over 8 different

Recent work in text classification has used two different first-order probabilistic models for classification, both of which make the naive Bayes assumption. Some use a multi-variate Bernoulli model, that is, a Bayesian Network with no dependencies between words and binary word features (e