# williamstein/mazur-explicit-formula

Better sign count table

 @@ -151,14 +151,32 @@ \section{Brief Introduction} We will specifically be interested in issues of bias. This is what we mean: thanks to the recent resolution of the Sato-Tate Conjecture in this context, one knows that---roughly---half the Fourier coefficients $a_E(p)$ are positive and half negative. Indeed, the numbers of positive values and negative values look very close: \vskip20pt -\hskip40pt\begin{tabular} {l | r | r | r}\hline -Curve & Rank & Positive $a_E(p)$ for $p<107$ & Negative $a_E(p)$ for $p<107$\\ \hline\hline -11a & 0 & 332169 & 332119\\ \hline -32a (CM) & 0 & 166054 & 166126\\ \hline -37a & 1 & 332127 & 332240\\ \hline -389a & 2 & 332317 & 332022\\ \hline -5077a & 3 & 331706 & 332632 \\ \hline\hline + +\begin{center} +\begin{tabular} {r | c | c | c | r}\hline +Curve & Rank & Negative $a_E(p)$ for $p<10^9$ & Positive $a_E(p)$ for $p<10^9$ & Difference\\ \hline\hline +11a & 0 & 25422268 & 25423101 & -833 \\ \hline +14a & 0 & 25422229 & 25421074 & 1155 \\ \hline +128b & 0 & 25420641 & 25425608 & -4967 \\ \hline +816b & 0 & 25424848 & 25421229 & 3619 \\ \hline +2379b & 0 & 25417900 & 25427007 & -9107 \\ \hline +5423a & 0 & 25420479 & 25425242 & -4763 \\ \hline +29862s & 0 & 25420525 & 25425197 & -4672 \\ \hline +37a & 1 & 25423396 & 25422448 & 948 \\ \hline +43a & 1 & 25421536 & 25424196 & -2660 \\ \hline +160a & 1 & 25424446 & 25421488 & 2958 \\ \hline +192a & 1 & 25418843 & 25426859 & -8016 \\ \hline +2340i & 1 & 25425512 & 25419660 & 5852 \\ \hline +10336d & 1 & 25421245 & 25423628 & -2383 \\ \hline +389a & 2 & 25427014 & 25418738 & 8276 \\ \hline +433a & 2 & 25425902 & 25419896 & 6006 \\ \hline +2432d & 2 & 25423818 & 25421900 & 1918 \\ \hline +3776h & 2 & 25422350 & 25422750 & -400 \\ \hline +5077a & 3 & 25426985 & 25418831 & 8154 \\ \hline +11197a & 3 & 25429098 & 25416702 & 12396 \\ \hline \end{tabular} +\end{center} + \vskip20pt To study, then, the weighted sums that directly reflect finer statistical issues related to this symmetric distribution, we will be concentrating on weighting functions $p \mapsto g_E(p)$ that have the property that \begin{itemize} \item for all primes $p$, $g_E(p)$ is an {\it odd} function of the value $a_E(p)$, and \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 mean{\footnote{ See Section {\ref{mean}} below}} relative to multiplicative measure $dX/X$.\end{itemize} In such a context the mean of $\delta_E(X)$ can be interpreted as a {\it bias}! @@ -2897,4 +2915,4 @@ \subsection{ Galois Deformation Theorems and the pivotal role played by residua - \ + \