Beatty sequences and prime numbers with restrictions on strongly q-additive functions
Tóm tắt
We estimate exponential sums over a non-homogenous Beatty sequence with restriction on strongly q-additive functions. We then apply our result in a few special cases to obtain an asymptotic formula for the number of primes
$$p=\lfloor \alpha n +\beta \rfloor $$
and
$$f(p)\equiv a (\mathrm{mod\,}b)$$
, with
$$n\ge N $$
, where
$$\alpha $$
,
$$\beta $$
are real numbers and f is a strongly q-additive function (for example, the sum of digits function in base q is a strongly q-additive function). We also prove that for any fixed integer
$$k\ge 3 $$
, all sufficiently large
$$N\equiv k (\mathrm{mod\,}2) $$
could be represented as a sum of k prime numbers from a Beatty sequence with restriction on strongly q-additive functions.
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