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In diffuse interstellar clouds the chemistry that leads to the formation of the oxygen-bearing ions OH+, H2O+, and H3O+ begins with the ionization of atomic hydrogen by cosmic rays, and continues through subsequent hydrogen abstraction reactions involving H2. Given these reaction pathways, the observed abundances of these molecules are useful in constraining both the total cosmic-ray ionization rate of atomic hydrogen (ζH) and molecular hydrogen fraction (). We present observations targeting transitions of OH+, H2O+, and H3O+ made with the Herschel Space Observatory along 20 Galactic sight lines toward bright submillimeter continuum sources. Both OH+ and H2O+ are detected in absorption in multiple velocity components along every sight line, but H3O+ is only detected along 7 sight lines. From the molecular abundances we compute in multiple distinct components along each line of sight, and find a Gaussian distribution with mean and standard deviation 0.042 ± 0.018. This confirms previous findings that OH+ and H2O+ primarily reside in gas with low H2 fractions. We also infer ζH throughout our sample, and find a lognormal distribution with mean log (ζH) = –15.75 (ζH = 1.78 × 10–16 s–1) and standard deviation 0.29 for gas within the Galactic disk, but outside of the Galactic center. This is in good agreement with the mean and distribution of cosmic-ray ionization rates previously inferred from H observations. Ionization rates in the Galactic center tend to be 10-100 times larger than found in the Galactic disk, also in accord with prior studies. [ABSTRACT FROM AUTHOR]