The objectives of the whole project consist of those from each subproject, which are:
Subproject 1.
1. To derive criteria for linear dependence of arithmetic functions, based on the notion of Wronskian, similar to that of algebraic dependence based on Jacobian.
2. To analyze to what extent the zeta function can be replaced by an arbitrary arithmetic function in the log-series expansion.
3. To investigate the properties of those arithmetic functions related to the generalized M?bius function.
4. To generalize Smith's result using the generalized M?bius function.
Subproject 2.
1. To determine what kind of expansions are possible and to compute the lengths and periods of finite, respectively, periodic expansions of elements when the degree condition of Scheicher-Thuswaldner is violated.
2. To prove basic arithmetic properties, to represent functions by polynomials over a subset of a finite field, to count the number of quasi-permutation polynomials, and to derive characterizations of quasi-permutation polynomials using a method of Carlitz-Lutz, the concept of character, and specific forms of polynomials.
Subproject 3.
1. To derive characterizations of rational elements in function fields with prime-adic valuation.
2. To extend Lee's algorithm to non-simple continued fractions and to extend the results of Riyapan et al to function fields with zero-characteristic base field.
3. To derive closure properties of Liouville numbers, to establish the result of Erd?s and to verify the equivalence between Liouville numbers and Liouville continued fractions in three non-archimedean settings of p-adic numbers, function field with degree valuation and function field with prime-adic valuation.
Subproject 4.
1. To analyze how Brun's and Badea's criteria are related.
2. To derive identities for sums of reciprocals of elements satisfying recurrences with non-constant coefficients extending those of Hong, Zhi-Wei and Jian-Xin.
3. To derive characterizations of rational numbers via their p-adic Sylvester and Cantor series expansions.
Subproject 5.
1. To solve the universal Cauchy's functional equation for solutions whose domain is the set of positive real numbers and range is the complex field, and to check dependence relations among the solutions of the universal Cauchy's functional equation.
2. To analyze those functional equations characterizing the cotangent and hyperbolic tangent functions and to obtain closed form solutions for rational recursive equations whose solutions are trigonometric functions other than the cotangent function.
