Abstract
The average value of a certain normalization of Ramanujan sums is determined in terms of Bernoulli numbers and odd values of the Riemann zeta function. The distribution of values and limiting behavior of such a normalization are then studied along subsets of Beurling type integers with positive density and sequences of moduli with constraints on the number of distinct prime factors.
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The author is grateful to the referee for many helpful comments and suggestions.
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Dedicated to the celebration of one hundred twenty-fifth birthday of Srinivasa Ramanujan
The author is supported by the Distinguished Young Scholar Award, Tüba-Gebip of Turkish Academy of Sciences.
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Alkan, E. Distribution of averages of Ramanujan sums. Ramanujan J 29, 385–408 (2012). https://doi.org/10.1007/s11139-012-9424-4
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DOI: https://doi.org/10.1007/s11139-012-9424-4
Keywords
- Averages of Ramanujan sums
- Distribution of values
- Beurling type integers
- Limiting behavior
- Jordan’s totient function
- Euler’s totient function
- Möbius function
- Bernoulli numbers
- Bernoulli polynomials