Journal d’Analyse Mathématique

, Volume 23, Issue 1, pp 237–268

# The “pits effect” for functions in the unit circle

• J. E. Littlewood
Article

## Keywords

Unit Circle Integral Function Straightforward Calculation Finite Order Partial Quotient
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## References

1. 1.
M. Nassif, On the behaviour of the function$$f(z) = \sum e^{\surd \bar 2\pi in^2 } \frac{{z^{2_n } }}{{n!}}$$ Proc. London Math. Soc.,54 (1954), 201–214.
2. 2.
J. E. Littlewood, The “pits effect” for the integral function$$f(z) = \sum \exp \left\{ { - e^{ - 1} (n \log n - n) + \pi i\alpha n^2 } \right\}z, \alpha = \raise.5ex\hbox{\scriptstyle 1}\kern-.1em/ \kern-.15em\lower.25ex\hbox{\scriptstyle 2} (\sqrt 5 - 1)$$ Abhandlungen aus Zahlentheorie und Analysis zur Erinnerung an Edmund Landau (1877–1938), VEB Deutscher Verlag der Wissenschaften, Berlin 1968, 195–215.Google Scholar
3. 3.
—, A “pits effect” for all smooth enough integral functions with a coefficient factor$$\exp (n^2 \alpha \pi i) \alpha = \raise.5ex\hbox{\scriptstyle 1}\kern-.1em/ \kern-.15em\lower.25ex\hbox{\scriptstyle 2} (\sqrt 5 - 1)$$ J. London Math. Soc.,43 (1968), 79–92.