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We discuss questions related to the non-existence of gaps in the series defining modular forms and other arithmetic functions of various types, and improve results of Serre, Balog & Ono and Alkan using new results about exponential sums and the distribution of B-free numbers.

BackgroundIn acute-care hospitals, no evidence of a protective effect of healthcare worker (HCW) vaccination on hospital-acquired influenza (HAI) in patients has been documented. Our study objective was to ascertain the effectiveness of influenza vaccination of HCW on HAI among patients.MethodsA nested case-control investigation was implemented in a prospective surveillance study of influenza-like illness (ILI) in a tertiary acute-care university hospital. Cases were patients with virologically-confirmed influenza occurring ≥ 72 h after admission, and controls were patients with ILI presenting during hospitalisation with negative influenza results after nasal swab testing. Four controls per case, matched per influenza season (2004-05, 2005-06 and 2006-07), were randomly selected. Univariate and multivariate conditional logistic regression models were fitted to assess factors associated with HAI among patients.ResultsIn total, among 55 patients analysed, 11 (20%) had laboratory-confirmed HAI. The median HCW vaccination rate in the units was 36%. The median proportion of vaccinated HCW in these units was 11.5% for cases vs. 36.1% for the controls (P = 0.11); 2 (20%) cases and 21 (48%) controls were vaccinated against influenza in the current season (P = 0.16). The proportion of ≥ 35% vaccinated HCW in short-stay units appeared to protect against HAI among patients (odds ratio = 0.07; 95% confidence interval 0.005-0.98), independently of patient age, influenza season and potential influenza source in the units.ConclusionsOur observational study indicates a shielding effect of more than 35% of vaccinated HCW on HAI among patients in acute-care units. Investigations, such as controlled clinical trials, are needed to validate the benefits of HCW vaccination on HAI incidence in patients.