Null trajectories for the symmetrized Hurwitz zeta function

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Journal Article
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2007, 463 (2077), pp. 303 - 319
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We consider the Hurwitz zeta function ζ(s, a) and develop asymptotic results for a=p/q, with q large, and, in particular, for p/q tending to 1/2. We also study the properties of lines along which the symmetrized parts of ζ(s, a), ζ+(s, a) and ζ-(s, a) are zero. We find that these lines may be grouped into four families, with the start and end points for each family being simply characterized. At values of a=1/2, 2/3 and 3/4, the curves pass through points which may also be characterized, in terms of zeros of the Riemann zeta function, or the Dirichlet functions L -3(s) and L-4(s), or of simple trigonometric functions. Consideration of these trajectories enables us to relate the densities of zeros of L-3(s) and L-4(s) to that of ζ(s) on the critical line. © 2006 The Royal Society.
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