Let ae be the Riemann zeta function. Khinchine (1938) proved that the function f ó (t) ae(ó it)aae(ó ), where ó . 1 and t is real, is an in®nitely divisible characteristic function. We investigate further the fundamental properties of the corresponding distribution of f ó , the Riemann zeta distribution, including its support and unimodality. In particular, the Riemann zeta random variable is represented as a linear function of in®nitely many independent geometric random variables. To extend Khinchine's result, we construct the Dirichlet-type characteristic functions of discrete distributions and provide a suf®cient condition for the in®nite divisibility of these characteristic functions. By way of applications, we give probabilistic proofs for some identities in number theory, including a new identity for the reciprocal of the Riemann zeta function.

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FDG-PET/CT has the ability to detect recurrent ovarian cancer and second primary tumors in patients with increased levels of serum CA-125. FDG-PET/CT affects the clinical management by localizing recurrent lesions and creating a specific treatment plan for each patient, especially patients who demonstrate a low-level increase in serum CA-125 levels.

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PurposeTo evaluate the usefulness of 2-[18F] fluoro-2-deoxy-D-glucose-positron emission tomography/computed tomography (FDG-PET/CT) in the early detection of breast cancer tumor recurrences and its role in post-therapy surveillance.MethodsFDG-PET/CT was performed on patients with increased serum CA 15-3 levels and/or clinical/radiologic suspicion of recurrence. A group of asymptomatic patients who underwent FDG-PET/CT in the post-therapy surveillance of breast cancer served as the controls. The results were analyzed based on the patients' histological data, other imaging modalities and/or clinical follow-up. Recurrence was defined as evidence of recurrent lesions within 12 months of the FDG-PET/CT scan.ResultsBased on elevated serum CA15-3 levels (n = 31) and clinical/radiologic suspicion (n = 40), 71 scans were performed due to suspected recurrence, whereas 69 scans were performed for asymptomatic follow-up. The sensitivity and specificity of FDG-PET/CT were 87.5% and 87.1% in the patients with suspected recurrence and 77.8% and 91.7% in the asymptomatic patients. The positive predictive value in the patients with suspected recurrence (mainly due to elevated serum CA 15-3 levels) was higher than that in asymptomatic patients (P = 0.013). Recurrences were proven in 56.3% (40/71) of the patients with suspected recurrence and in 13% (9/69) of the asymptomatic patients (P<0.001). FDG-PET/CT resulted in changes in the planned management in 49.3% (35/71) of the patients with suspected recurrence and 10.1% (7/69) of the asymptomatic patients (P<0.001). After follow-up, 77.5% (55/71) of the patients with suspicious recurrences and 97.1% (67/69) of the asymptomatic patients were surviving at the end of the study (P<0.001).ConclusionsFDG-PET/CT was able to detect recurrence, and the results altered the intended patient management in the post-therapy surveillance of breast cancer. FDG-PET/CT should be used as a priority in patients with increased serum CA 15-3 levels, or with clinical/radiologic suspicion of recurrence, and might be useful for asymptomatic patients.

This paper shows that a normalization of the Hurwitz zeta function is a characteristic function. This generalizes the 1938 result of Khinchine about the Riemann zeta function. The paper investigates the infinite divisibility of the resulting distribution.

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In a recent paper, Matysiak and Szablowski [V. Matysiak, P.J. Szablowski, Theory Probab. Appl. 45 (2001) 711-713] posed an interesting conjecture about a lower bound of real-valued characteristic functions. Under a suitable moment condition on distributions, we prove the conjecture to be true. The unified approach proposed here enables us to obtain new inequalities for characteristic functions. We also show by example that the improvement in the bounds is significant if more information about the distribution is available.