Summary: For a positive integer \(k\), a book (with \(k\) pages) is a topological space consisting of a spine, which is a line, and \(k\) pages, which are half-planes with the spine as their boundary. We say that a graph \(G\) admits a \(k\)-page book embedding or is \(k\)-page book embeddable if there exists a linear ordering of the vertices on the spine and one can assign the edges of \(G\) to \(k\) pages such that no two edges of the same page cross. \textit{M. Yannakakis} [J. Comput. Syst. Sci. 38, No. 1, 36--67 (1989; Zbl 0673.05022)] proved that every plane graph admits a 4-page book embedding. In this paper, we improve this to graphs on the projective plane, that is, those embedded on the projective plane without edge-crossings. \textit{A. Nakamoto} and \textit{T. Nozawa} [Congr. Numerantium 225, 63--71 (2015; Zbl 1335.05050)] showed that every graph on the projective plane admits a 9-page book embedding. In this paper, we improve the latter result to 6-page embedding. Furthermore, we also prove that every graph on the projective plane admits a 3-page book embedding if it is 5-connected and a 5-page book embedding if it is 4-connected. Our idea of the proofs is to use a Tutte path, which is different from previous ones.