| Summary: | The main part of the thesis is applications of the Strong Artin conjecture to number theory. We have two applications. One is generating number fields with extreme class numbers. The other is generating extreme positive and negative values of Euler-Kronecker constants.
For a given number field $K$ of degree $n$, let $\widehat{K}$ be the normal closure of $K$ with $Gal(\widehat{K}/\Bbb Q)=G.$ Let $Gal(\widehat{K}/K)=H$ for some subgroup $H$ of $G$. Then,
$$
L(s,\rho,\widehat{K}/\Bbb Q)=\frac{\zeta_K(s)}{\zeta(s)}
$$
where $Ind_H^G1_H = 1_G + \rho$.
When $L(s,\rho)$ is an entire function and has a zero-free region $[\alpha,1] \times [-(\log N)^2, (\log N)^2]$ where $N$ is the conductor of $L(s,\rho)$, we can estimate $\log L(1,\rho)$ and $\frac{L'}{L}(1,\rho)$ as a sum over small primes:
$$
\log L(1,\rho) = \sum_{p\leq(\log N)^{k}}\lambda(p)p^{-1} + O_{l,k,\alpha}(1)$$
$$
\frac{L'}{L}(1,\rho)=-\sum_{p\leq x} \frac{\lambda(p) \log{p}}{p} +O_{l,x,\alpha}(1).
$$
where $0 < k < \frac{16}{1-\alpha}$ and $(\log N)^{\frac{16}{1-\alpha}} \leq x \leq N^{\frac{1}{4}}$. With these approximations, we can study extreme values of class numbers and Euler-Kronecker constants.
Let $\frak{K}$ $(n,G,r_1,r_2)$ be the set of number fields of degree $n$ with signature $(r_1,r_2)$ whose normal closures are Galois $G$ extension over $\Bbb Q$. Let $f(x,t) \in \Bbb Z[t][x]$ be a parametric polynomial whose splitting field over $\Bbb Q (t)$ is a regular $G$ extension. By Cohen's theorem, most specialization $t\in \Bbb Z$ corresponds to a number field $K_t$ in $\frak{K}$ $(n,G,r_1,r_2)$ with signature $(r_1,r_2)$ and hence we have a family of Artin L-functions $L(s,\rho,t)$. By counting zeros of L-functions over this family, we can obtain L-functions with the zero-free region above.
In Chapter 1, we collect the known cases for the Strong Artin conjecture and prove it for the cases of $G=A_4$ and $S_4$. We explain how to obtain the approximations of $\log (1,\rho)$ and $\frac{L'}{L}(1,\rho)$ as a sum over small primes in detail. We review the theorem of Kowalski-Michel on counting zeros of automorphic L-functions in a family.
In Chapter 2, we exhibit many parametric polynomials giving rise to regular extensions. They contain the cases when $G=C_n,$ $3\leq n \leq 6$, $D_n$, $3\leq n \leq 5$, $A_4, A_5, S_4, S_5$ and $S_n$, $n \geq 2$.
In Chapter 3, we construct number fields with extreme class numbers using the parametric polynomials in Chapter 2.
In Chapter 4, We construct number fields with extreme Euler-Kronecker constants also using the parametric polynomials in Chapter 2.
In Chapter 5, we state the refinement of Weil's theorem on rational points of algebraic curves and prove it.
The second topic in the thesis is about simple zeros of Maass L-functions. We consider a Hecke Maass form $f$ for $SL(2,\Bbb Z)$. In Chapter 6, we show that if the L-function $L(s,f)$ has a non-trivial simple zero, it has infinitely many simple zeros. This result is an extension of the result of Conrey and Ghosh.
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