Abstract
In this paper we have given conditions on exponential polynomials \(P_{n}(s)\) of Dirichlet type to be attained the equality between each of two pairs of bounds, called essential bounds, \(a_{P_{n}(s)}\), \(\rho _{N}\) and \(b_{P_{n}(s)}\), \(\rho _{0}\) associated with \(P_{n}(s)\). The reciprocal question has been also treated. The bounds \(a_{P_{n}(s)}\), \(b_{P_{n}(s)}\) are defined as the end-points of the minimal closed and bounded real interval \(I= [ a_{P_{n}(s)},b_{P_{n}(s)} ] \) such that all the zeros of \(P_{n}(s)\) are contained in the strip \(I\times {\mathbb {R}}\) of the complex plane \({\mathbb {C}}\). The bounds \(\rho _{N}\), \(\rho _{0}\) are defined as the unique real solutions of Henry equations of \(P_{n}(s)\). Some applications to the partial sums of the Riemann zeta function have been also showed.
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Mora, G., Benítez, E. Essential bounds of Dirichlet polynomials. RACSAM 115, 107 (2021). https://doi.org/10.1007/s13398-021-01045-z
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DOI: https://doi.org/10.1007/s13398-021-01045-z
Keywords
- Dirichlet polynomials
- Zeros of exponential polynomials
- Diophantine and rational dependence
- Zeros of partial sums of the Riemann zeta function