David T. Nguyen

American Institute of Mathematics | AIMATH

Let $\zeta^k(s) = \sum_{n=1}^\infty \tau_k(n) n^{-s}, \Re s > 1$. We present three conditional results on the additive correlation sum $$\sum_{n\le X} \tau_3(n) \tau_3(n+h)$$ and give numerical verifications of our method. The first is a conditional proof for the full main term of this correlation sum for the case $h=1$, on assuming an averaged lev...

In this paper, we confirm a smoothed version of a recent conjecture on the variance of the k-fold divisor function in arithmetic progressions to individual composite moduli, in a restricted range. In contrast to a previous result of Rodgers and Soundararajan, we do not require averaging over the moduli. Our proof adapts a technique of S. Lester who...

We prove some distribution results for the k-fold divisor function in arithmetic progressions to moduli that exceed the square-root of length X of the sum, with appropriate constrains and averaging on the moduli, saving a power of X from the trivial bound. On assuming the Generalized Riemann Hypothesis, we obtain uniform power saving error terms th...

This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude j and ending at a given altitude k, with additional constraints, for example, to never attain altitude 0 in-between. We first discuss the case of walks on the integers with steps \(-h, \dots , -1, +1, \dots ,...

This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude $j$ and ending at a given altitude $k$, with additional constraints such as, for example, to never attain altitude $0$ in-between. We first discuss the case of walks on the integers with steps $-h, \dots, -1,...