An arithmetic formula is an expression involving only the constant 1, and the binary operations of addition and multiplication, with multiplication by 1 not allowed. We obtain an asymptotic formulaâ€¦ (More)

Let b â‰¥ 2 be a fixed integer. Let sb(n) denote the sum of digits of the nonnegative integer n in the base-b representation. Further let q be a positive integer. In this paper we study the length k ofâ€¦ (More)

Let b 2 be an integer and wb(n) be the sum of digits of the nonnegative integer n written in hereditary base b notation. We give optimal upper bounds for the exponential sum PN 1 n=0 exp(2â‡¡iwb(n)t),â€¦ (More)

A natural number is a binary kâ€™th power if its binary representation consists of k consecutive identical blocks. We prove an analogue of Waringâ€™s theorem for sums of binary kâ€™th powers. Moreâ€¦ (More)