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Extra info for Defects of properties in mathematics. Quantitative characterizations
We present an example. N. Baoulina and P. m 1/ 2/ 4 and k even. mod 4/. For a D 1 this was first established in Moree et al. [19, Lemma 12] using Bernoulli numbers. m/ D amk for integers a was given later by the second author . 1 makes use of the following lemma. 6 is even. mod 4/, otherwise. mod 8/. Note that k 3 C ord2 k. Indeed, for ord2 k D 1 and ord2 k D 2 it follows from the condition k 6 and for ord2 k 3 we have k 2ord2 k 3 C ord2 k. mod 8/. mod 4/. mod 8/. mod 4/. mod 8/. mod 8/. This completes the proof.
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0. qC1/ . 4, we conclude that this is impossible. Finally, assume that a D q2s for some s 1. mod 8/. Note that k has to be even. mod 16/. mod 16/. mod 16/. Acknowledgements This paper was begun during the stay of the first author in February–April 2014 at the Max Planck Institute for Mathematics. She likes to thank for the invitation and the pleasant research atmosphere. The second author was introduced to the subject around 1990 by the late Jerzy Urbanowicz. He will never forget his interest, help and kindness.
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