Robert Charles Vaughan, Dissertation Advisor/Co-Advisor Robert Charles Vaughan, Committee Chair/Co-Chair Kirsten Eisentraeger, Committee Member Ae Ja Yee, Committee Member Jose Palacios, Outside Member
Keywords:
value distribution
Abstract:
Significant attention has been given to study various moments of the Riemann zeta function, $\zeta$, its logarithm and their generalizations However, not much is known about the moments of $\frac{\zeta'}{\zeta}$. and the logarithmic derivative of more general L-functions. For $\pi$, a cuspidal automorphic representation of $GL_d( \mathbb{A}_{\mathbb{Q}})$,
there is an associated L-function, $L(s, \pi)$. We study the value distribution of its logarithmic derivative on the 1-line, $\frac{L'}{L}(1+it, \pi).$ We are able to prove that for $t \in [T, 2T]$, in some sense, $\frac{L'}{L}(1+it, \pi)$ has ``almost'' normal distribution with mean 0 and variance $\sqrt{\frac{\log(y(T))}{2y(T)}}$. An essential ingredient of the proof is the fact that our function of interest can be approximated by Dirichlet polynomial with coefficients supported on prime powers. We prove similar results for $\frac{L'}{L}(1+it, \pi \times \overline{\pi})$ and $\log(L(1+it, \pi))$.
