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On the Zeros of Higher Derivatives of Hardy'sZ-Function

✍ Scribed by Kohji Matsumoto; Yoshio Tanigawa

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 137 KB
Volume: 75
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.1998.2341

No coin nor oath required. For personal study only.

✦ Synopsis

Let k be any positive integer and N 0, k (T ) the number of the zeros in the interval (0, T ) of Z (k) (t), the kth derivative of Hardy's Z-function. We prove an inequality for N 0, k (T) (Theorem 1), and also prove that it can be replaced by the equality under the Riemann hypothesis (Theorem 2). The key fact of the proof is the construction of a meromorphic function ' k (s), which satisfies an appropriate recurrence formula and a functional equation.

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