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Matiyasevich, Saidak, Zvengrowsk proved the following result:

Let $σ_0$ be greater than or equal to the real part of any zero of ξ. Then $|ξ(s)|$ is strictly increasing in the half-plane $σ > σ_0$.

I have a trouble interpreting this result so if for example $\sigma_0<1/2$ is the real part of a zero $z_0 =\sigma_0+it$ then does this mean $|\xi(\sigma+it)|>|\xi(\sigma_0+it)|$ for $\sigma>\sigma_0$. This would give a contradiction when one hit the root $1-\sigma_0+it$, so what is the above Theorem actually saying?

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  • $\begingroup$ I do not understand what it means for a function on the complex plane to be "increasing on a half-plane". $\endgroup$
    – Daniel Asimov
    Commented May 24 at 19:38

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The result you mention is not due to Matiyasevich-Saidak-Zvengrowsk. Instead, it appeared in Sondow-Dumitrescu: A monotonicity property of Riemann's xi function and a reformulation of the Riemann hypothesis, Period. Math. Hungar. 60 (2010), 37-40. Here is the precise statement, quoted from the paper:

Theorem. The xi function is increasing in modulus along every horizontal half-line lying in any open right half-plane that contains no zeros of xi. Similarly, the modulus decreases on each horizontal half-line in any zero-free, open left half-plane.

I hope this clarifies.

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  • $\begingroup$ The result I'm quoting is from the article "Horizontal monotonicity of the modulus of the zeta function,L-functions, and related functions" and I still don't understand it. But I'm well aware of the results in Sondow-Dumitrescu you mention. $\endgroup$
    – 12321
    Commented May 24 at 15:01
  • $\begingroup$ I see yes I didn't read the statement of the theorem correctly. $\endgroup$
    – 12321
    Commented May 24 at 15:11

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