Errata for Distribution theorems for L-functions

Errata to [J]

There is a gap in [J], Ch. 4, as was brought to my attention by D. Hejhal. To prove the results in [J], pp. 117-127, in addition to those axioms listed above, one must assume the following:

Missing Axiom: We have, for $H>T^{1/2 +\epsilon}$ , and $\sigma \geq 1/2$ ,

$\begin{displaymath} N_L(\sigma ,T+H)-N_L(\sigma ,T)<<_{L,\epsilon} \left({\sqrt{H}\over T}\right)^{{1-2\sigma\over 4}}H\log H, \end{displaymath}$

(1)

where

$\begin{displaymath} N_L(\sigma ,T):=\char93 \{\rho=\beta +i\gamma\ \vert\ L(\beta +i\gamma)=0,\ 0<\gamma <T,\ \beta\geq \sigma\}. \end{displaymath}$

First, this ``missing axiom'' is a well-known result of A. Selberg [S] in the case $L(s)=\zeta(s)$ . (It is also seems to be known for Dirichlet L-series, L-functions of quadratic fields and Langlands standard L-functions for [S'].) By this and another result of Selberg [S], the results of [J] are theorems in case is a Dirichlet L-function.

There is a typo on

, page 84, line 4, $C\log r/\sqrt{r}$ should be ${1\over 50}{\sqrt{\log r}\over r}$ , and on page 84, line 9, $C\log r/{r}$ should be $C\sqrt{\log r}/r$ .
on page 158, should be ; in the remark there, $\log k$ should be $\log x$ .

David Joyner 2003-09-14