# Value distribution of automorphic L-functions

- Author:
- Pawelec, Krzysztof
- Published:
- [University Park, Pennsylvania] : Pennsylvania State University, 2019.
- Physical Description:
- 1 electronic document
- Additional Creators:
- Vaughan, R. C.
- Access Online:
- etda.libraries.psu.edu

- Restrictions on Access:
- Open Access.
- Summary:
- 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 $\log(L(1+it, \pi))$.
- Genre(s):
- Dissertation Note:
- Ph.D. Pennsylvania State University 2019.
- Reproduction Note:
- Microfilm (positive). 1 reel ; 35 mm. (University Microfilms 13917946)
- Technical Details:
- The full text of the dissertation is available as an Adobe Acrobat .pdf file ; Adobe Acrobat Reader required to view the file.

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