The summatory function of the sum-of-digits function on polynomial sequences

The summatory function of q-additive functions on pseudo-polynomial sequences

“…Now we draw our attention towards a recent result by Peter [13] who considered polynomial sequences of the form αn k n≥1 where α is one or an irrational of finite type. In particular, he could show the following.…”

“…The goal of this intermediate section is the transformation of the integral I r,a to give the result the shape as in the results of Delange [1] and Peter [13]. Since we will often use the functions J f,β and F f,β defined in (1.4), we will write J := J f,β and F := F f,β for short.…”

“…I would like to thank Jörg Thuswaldner for drawing my attention to this problem, in general, and, in particular, to the paper of Manfred Peter [13].…”

“…Thus we will apply our estimates from Section 2 and 3. (2) Then we will give the integral resulting from the transformation in Section 3 the shape of a periodic function such as in the results of Delange [1] and Peter [13]. (3) Finally we will put all together in order to gain the desired result.…”

The Sum-of-Digits Function of Squares

“…There are some known facts, for example, Peter [17] has proved that 1 N n<N s(n 2 ) = log 2 N + γ(log 2 N ) + O(N −η ) (1.1)…”