Multiplicative Number Theory I: Zeros

The Hardy–Littlewood method: Bibliography

Let $\Lambda(n)$ be the von Mangoldt function, $x$ real and $2\leq y \leq x$. This paper improves the estimate on the exponential sum over primes in short intervals \[ 
S_k(x,y;\alpha) = \sum_{x< n \leq x+y} \Lambda(n) e\left( n^k \alpha \right) \] when $k\geq 3$ for $\alpha$ in the minor arcs. And then combined with the Hardy--Littlewood circle method, this enables us to investigate the Waring--Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers, which improves the results in Wei and Wooley [12].

Exponential sums over primes in short intervals and an application to the Waring--Goldbach problem

Let $H = N^{\theta}, \theta > 2/3$ and $k \geq 1$. We obtain estimates for the following exponential sum over primes in short intervals: \[ \sum_{N < n \leq N+H} \Lambda(n) e(g(n)), \] where $g$ is a polynomial of degree $k$. As a consequence of this in the special case $g(n) = \alpha n^k$, we deduce a short interval version of the Waring-Goldbach problem.

Discorrelation Between Primes in Short Intervals and Polynomial Phases

Let be the von Mangoldt function, be real and . This paper improves the estimate for the exponential sum over primes in short intervals when for in the minor arcs. When combined with the Hardy–Littlewood circle method, this enables us to investigate the Waring–Goldbach problem concerning the representation of a positive integer as the sum of th powers of almost equal prime numbers, and improve the results of Wei and Wooley [On sums of powers of almost equal primes. Proc. Lond. Math. Soc. (3) 111(5) (2015), 1130–1162].

We investigate the Waring-Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers. Define $s_k=2k(k-1)$ when $k\ge 3$, and put $s_2=6$. In addition, put $\theta_2=\frac{19}{24}$, $\theta_3=\frac{4}{5}$ and $\theta_k=\frac{5}{6}$ $(k\ge 4)$. Suppose that $n$ satisfies the necessary congruence conditions, and put $X=(n/s)^{1/k}$. We show that whenever $s>s_k$ and $\varepsilon>0$, and $n$ is sufficiently large, then $n$ is represented as the sum of $s$ $k$th powers of prime numbers $p$ with $|p-X|\le X^{\theta_k+\varepsilon}$. This conclusion is based on a new estimate of Weyl-type specific to exponential sums having variables constrained to short intervals.

On Sums of Powers of Almost Equal Primes

We investigate the Waring-Goldbach problem of representing a positive integer n as the sum of s kth powers of almost equal prime numbers. Define sk = 2k(k−1) when k > 3, and put s2 = 6. Suppose that n satisfies the necessary congruence conditions, and put X = (n/s) 1/k . We show that whenever s > sk and " > 0, and n is sufficiently large, thenn is represented as the sum of s kth powers of prime numbers p with |p−X| 6 X 19/24+" . This conclusion is based on a new estimate of Weyl-type specific to exponential sums having variables constrained to short intervals.

/pdf/on-sums-of-powers-of-almost-equal-primes-1xnymu8rao.pdf

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On Sums of Powers of Almost Equal Primes

On Sums of Powers of Almost Equal Primes