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Auxiliary Polynomials in Number Theory by David Masser

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10

Integer-valued entire functions II – Gramain

With regard to the previous chapter, it is even arguable that the analogous problem for entire functions mapping Z + iZ into Z + iZ is more natural, and the only extremal example that easily springs to mind is the Weierstrass sigma function

image (10.1)

with the product taken over all γ ≠ 0 in G = Z + iZ. This vanishes on G and grows as fast as image with C = exp(π/2) (see Exercise 10.4 and Exercise 20.98(b) with μ = 0). It could be considered an analogue of

and so we might expect that such a growth with ...

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