Motivated by the works of Liu, we provide a unified approach to find Appell-Lerch series and Hecke-type series representations for mock theta functions. We establish a number of parameterized identities with two parameters $a$ and $b$. Specializing the choices of $(a,b)$, we not only give various known and new representations for the mock theta functions of orders 2, 3, 5, 6 and 8, but also present many other interesting identities. We find that some mock theta functions of different orders are related to each other, in the sense that their representations can be deduced from the same $(a,b)$-parameterized identity. Furthermore, we introduce the concept of false Appell-Lerch series. We then express the Appell-Lerch series, false Appell-Lerch series and Hecke-type series in this paper using the building blocks $m(x,q,z)$ and $f_{a,b,c}(x,y,q)$ introduced by Hickerson and Mortenson, as well as $\overline{m}(x,q,z)$ and $\overline{f}_{a,b,c}(x,y,q)$ introduced in this paper. We also show the equivalences of our new representations for several mock theta functions and the known representations.

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Let $${f \in \mathbb{Z}[x]}$$ , $${\deg f =3}$$ . Assume that$f$does not have repeated roots. Assume as well that, for every prime$q$, $${f(x)\not\equiv 0}$$ mod$q$^{2}has at least one solution in $${(\mathbb{Z}/q^2 \mathbb{Z})^*}$$ . Then, under these two necessary conditions, there are infinitely many primes$p$such that$f$($p$) is square-free.

We establish a general Kronecker limit formula of arbitrary rank over global function fields with Drinfeld period domains playing the role of upper-half plane. The Drinfeld-Siegel units come up as equal characteristic modular forms replacing the classical $\Delta$. This leads to analytic means of deriving a Colmez-type formula for "stable Taguchi height" of CM Drinfeld modules having arbitrary rank. A Lerch-Type formula for "totally real" function fields is also obtained, with the Heegner cycle on the Bruhat-Tits buildings intervene. Also our limit formula is naturally applied to the special values of both the Rankin-Selberg L-functions and the Godement-Jacquet L-functions associated to automorphic cuspidal representations over global function fields.

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