Abstract
If f 0(z) = Σc n z n is any integral function of finite non-zero order ϱ, consider the class of functions where r n(t) are Rademacher’s functions, representing a ‘random’ factor of the form ± 1. Littlewood and Offord [1] have shown that ‘most’ f(z) behave with great crudity and violence. If we erect an ordinate |f(z)| at the point z of the z-plane, then the resulting surface is an exponentially rapidly rising bowl, approximately of revolution, with exponentially small ‘pits’ going down to the bottom. The zeros of f, more generally the w-points where f = w, all lie in the pits for |z| > R(w). Finally the pits are very uniformly distributed in direction, and as uniformly distributed in distance as is compatible with the order ϱ.
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References
J. E. Littlewood and A. C. Offord, On the distribution of the zeros and α-values of a random integral funktion I, Journal London Math. Soc. 20 (1965), 130–136; II, Annals of Math. 49 (1948), 885-952.
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M. Nassif, On the behavior of the function Proc. London Math. Soc. (2)54 (1950), 201–214.
S. R. Tims, Note on a paper by M. Nassif, Proc. London Math. Soc. 54 (1950), 215. [Added at proof. I have now found a proof of the above conjecture, and it will appear in due course.]
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© 1969 Springer Science+Business Media New York
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Littlewood, J.E. (1969). The “Pits Effect” for the Integral Function\(f\left( z \right) = \sum {\exp \left\{ { - {\vartheta ^{ - 1}}\left( {n\log n - n} \right) + \pi i\alpha {n^2}} \right\}{z^n},\alpha = \tfrac{1}{2}\left( {\sqrt 5 - 1} \right)} \) . In: Turán, P. (eds) Number Theory and Analysis. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-4819-5_13
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DOI: https://doi.org/10.1007/978-1-4615-4819-5_13
Publisher Name: Springer, Boston, MA
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