Stepanov's Method Applied to Binomial Exponential Sums

Stepanov’s Method Applied to Binomial Exponential Sums

For a prime p and binomial axk+bxl with 1 ≤ l < k < 1 32 (p−1) 2 3 , we use Stepanov’s method to obtain the bound ∣∣∣∣∣ p−1 ∑ x=1 ep(ax k + bx) ∣∣∣∣∣ max { 1, l∆− 1 3 } 1 4 k 1 4 p 3 4 , where ∆ = k−l (k,l,p−1) .

Explicit Bounds on Monomial and Binomial Exponential Sums

Let p be a prime and ep(·) = e2πi·/p. First, we make explicit the monomial sum bounds of Heath-Brown and Konyagin: ̨̨̨Pp−1 x=1 ep(ax d) ̨̨̨ ≤ min{λ d5/8p5/8, λ d3/8p3/4}, where λ = 2/ 4 √ 3 = 1.51967 . . .. Second, letting d = (k, l, p− 1), we obtain the explicit binomial sum bound ̨̨̨Pp−1 x=1 ep(ax k + bxl) ̨̨̨ ≤ (k − l, p− 1) + 2.292 d13/46p89/92, for any nonconstant binomial axk + bxl on Zp, by sharpeni...

Binomial Sums and Identities

Generalising Tuenter’s binomial sums

Tuenter [Fibonacci Quarterly 40 (2002), 175-180] and other authors have considered centred binomial sums of the form Sr(n) = ∑

Multiple binomial sums

Multiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of products of binomial coefficients and also all the sequences with algebraic generating function. We study the representation of the generating functions of binomial sums by integrals of rational functions. The outcome is two...