Abstract
Let s be the complex variable σ + it, let d(n) denote the number of divisors of n, and let X be a real number greater than or equal to 2. In this paper, we establish the zero-free regions for the partial sums of the square of the Riemann zeta-function, defined by \(\zeta _{X}^{2}(s) =\sum _{ n=1}^{X}d(n)n^{-s}\) for σ > 1, and estimate the number of zeros up to a given height T.
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Acknowledgements
The authors wish to express their sincere gratitude to the anonymous referee for carefully reading the original version of this paper and for providing several very helpful comments and suggestions, including a key upper bound for the number of divisors function that greatly simplifies parts of the proofs of Theorems 3 and 4. The research by Kathryn Crosby, Jordan Eliseo, and David Mazowiecki was conducted as part of the 2013 Research Experiences for Undergraduates at The University of Tennessee at Chattanooga that was supported by the National Science Foundation Grant DMS-1261308.
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Crosby, K., Eliseo, J., Ledoan, A., Mazowiecki, D. (2015). Zeros of Partial Sums of the Square of the Riemann Zeta-Function. In: Rychtář, J., Chhetri, M., Gupta, S., Shivaji, R. (eds) Collaborative Mathematics and Statistics Research. Springer Proceedings in Mathematics & Statistics, vol 109. Springer, Cham. https://doi.org/10.1007/978-3-319-11125-4_6
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