| The main purpose of this dissertation is to study the arithmetical properties of some summations in number theory, Including the arithmetic properties of the classical Dedekind sums, Exponential sums, The shifted summations involving the generalized Fibonacci numbers and Lucas numbers, and the arithmetic properties of some Smarandache type functions. The main achievements contained in this dissertation are follows:1. Using analytical and elementary methods studied the hybrid mean value related to the Dedekind sums, and obtained some interesting formulas about as following: and2. Using the elementary methods studied the properties of theκ-power free numbers, and obtained a sharper up bound estimate for n-thκ-power free number.3. Defined the second-order linear recurrence sequence Ui= c1αi+C2βi, and obtained some exact calculating formulas for the summation This results explained the formulas about Fibonacci and Lucas number.4. Studied the value distribution of Smarandache function and the mean value of Smarandache power function. Including a mean value of multiplicative function F.Smarandache S(n), obtained the following result: where is a constant. This result explained the fact that Xu Zhefeng's estimation, that ish for any integer x≥3, we have: whereζ(s) is Riemann zeta-function, Ok denote the big-0 only concerning with k. At last, I studied the mean value of F.Smarandache LCM function. We have:5. Studied the mean value of the sum and r=2, and obtained some exact calculation formulas.
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