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Lindelöf Hypothesis

Let mu(sigma) be the least upper bound of the numbers A such that |zeta(sigma+it)|t^(-A) is bounded as t->infty, where zeta(s) is the Riemann zeta function. Then the Lindelöf hypothesis states that mu(sigma) is the simplest function that is zero for sigma>1/2 and 1/2-sigma for sigma<1/2.

The Lindelöf hypothesis is equivalent to the hypothesis that mu(1/2)=0 (Edwards 2001, p. 186).

Backlund (1918-1919) proved that the Lindelöf hypothesis is equivalent to the statement that for every sigma>1/2, the number of roots in the rectangle {T<=I[s]<=T+1,sigma<=R[s]<=1} grows less rapidly than lnT as T->infty (Edwards 2001, p. 188).

See also

Lindelöf's Theorem

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References

Backlund, R. "Über die Beziehung zwischen Anwachsen und Nullstellen der Zeta-Funktion." Ofversigt Finka Vetensk. Soc. 61, No. 9, 1918-1919.Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.Lindelöf, E. "Quelque remarques sur la croissance de la fonction zeta(s)." Bull. Sci. Math. 32, 341-356, 1908.

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Lindelöf Hypothesis

Cite this as:

Weisstein, Eric W. "Lindelöf Hypothesis." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LindelofHypothesis.html

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