Large Values of the Zeta-Function on the Critical Line

Aleksandar Ivić ()
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Aleksandar Ivić: Serbian Academy of Science and Arts

A chapter in Analytic Number Theory, 2015, pp 171-194 from Springer

Abstract: Abstract This is primarily an overview article (Lecture given during the conference “Number Theory and its Applications Workshop” in Xi’an (China), October 23–28, 2014.) dealing with the large values of | ζ ( 1 2 + i t ) | $$\vert \zeta (\frac{1} {2} + it)\vert$$ . This approach allows one to obtain upper bounds for moments (mean values) of | ζ ( 1 2 + i t ) | $$\vert \zeta (\frac{1} {2} + it)\vert$$ , which is one of the fundamental problems of the theory of the Riemann zeta-function. A sketch of the upper bound for the 12th moment of D.R. Heath-Brown (Q J Math (Oxford) 29:443–462, 1978) is presented, together with some recent results of the author. They include upper bounds obtained by the use of large values of E ∗(T), a function closely related to the classical function E(T), the error term in the mean square formula for | ζ ( 1 2 + i t ) | $$\vert \zeta (\frac{1} {2} + it)\vert$$ . A new large values result involving E k (T), the general error-term function in the formula for the 2k-th moment of | ζ ( 1 2 + i t ) | $$\vert \zeta (\frac{1} {2} + it)\vert$$ , is also given.

Keywords: Riemann zeta-function; Moments; Large values; 11M06 (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-22240-0_12

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DOI: 10.1007/978-3-319-22240-0_12

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