# williamstein/mazur-explicit-formula

 @@ -187,17 +187,26 @@ \vskip40pt {\Large Equivalently, putting:\begin{itemize} \item $\gamma_E(p)=0$ if $p$ is a bad or supersingular prime for $E$ and\vskip20pt \item $\gamma_E(p)= -1$ if $E$ has more than $p+1$ ${\bf F}_p$-rational points, and \vskip20pt \item $\gamma_E(p) = +1$ if less.\end{itemize}}\end{frame} \begin{frame} - $$\Delta_E(X): =\sum_{p\le X}\gamma_E(p).$$ + $$\Delta_E(X): =\sum_{p\le X}\gamma_E(p)$$ +{\bf 11a:} +{\bf 37a:} + +{\bf 389a:} +{\bf 5077a:} + +{\bf 234446a:} +{\bf ????a:} \end{frame} \begin{frame} - More generally we might consider weighting functions $p \mapsto g_E(p)$ that have the property that } \end{frame} + More generally we might consider weighting functions $p \mapsto g_E(p)$ that have the property that: -\begin{frame}\vskip20pt -{\Large \vskip40pt -\begin{itemize} \item for all primes $p$, $g_E(p)$ is an {\it odd} function of the value $a_E(p)$, and \vskip20pt \item the {\it sum\ of\ local\ data} $$\delta_E(X):=\sum_{p\le X}g_E(p)$$ has---or can be convincingly conjectured to have---a finite {\it mean}.\end{itemize}}\end{frame} + \begin{itemize} \item for all primes $p$, $g_E(p)$ is an {\it odd} function of the value $a_E(p)$, and \vskip20pt \item the {\it sum\ of\ local\ data} $$\delta_E(X):=\sum_{p\le X}g_E(p)$$ has---or can be convincingly conjectured to have---a finite {\it mean}. + \end{itemize} + +\end{frame} %\end{document} \begin{frame}\vskip20pt {\Large \vskip40pt