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Let $x$ be a large positive real number and let $q\leq 2$ be a positive integer. It is known that for a positive integer $n,$ there exists a unique sequence $\left\{0\leq n_k\leq q-1\right\}_{k\geq 0}$ such that $$n=\sum_{k\geq 0} n_k q^k.$$ Define the sum of digits of $n$ in basis $q$ by $$S_q(n):=\sum_{k\geq 0} n_k.$$ Is there an asymptotic formula for the following sum $$\sum_{a+b\leq x} \sigma_q(a,b),$$ where $\sigma_q(a,b)$ is the number of carry operations needed when adding $a$ and $b$ in basis $q.$

Many thanks.

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    $\begingroup$ Khadija, why do you define $S_q(n)$ if you don't use it in the rest of the question? Or am I missing something? $\endgroup$
    – Joël
    Commented Dec 10, 2016 at 0:53

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