# Research

My research is in analytic and elementary number theory, with a focus on additive prime number theory and the distribution of prime numbers.

Additive prime number theory studies additive representations of integers as sums of powers of primes or other sequences closely related to the primes. The most famous problem in this field is, of course, Goldbach’s conjecture that every even integer $$n \ge 4$$ is the sum of two primes. This conjecture has been extended and is now a special case of the Waring–Goldbach problem which studies the additive representations of sufficiently large integers $$n$$ in the form $n = p_1^k + p_2^k + \dots + p_s^k, \tag{WG}$ where $$p_1, p_2, \dots, p_s$$ are prime variables and $$s > k \ge 1$$ are fixed positive integers.
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To avoid degeneracy, we study the Diophantine equation (WG) only for sufficiently large “admissible” integers $$n$$ (where “admissibility” restricts $$n$$ to certain arithmetic progressions). For each pair $$k,s$$, let $$\mathcal E_{k,s}$$ denote the set of admissible $$n$$ for which (WG) has no solutions. Ideally, we want to show that $$\mathcal E_{k,s}$$ is finite whenever $$s > k$$, so it is convenient to let $$H(k)$$ denote the least $$s$$ for which the set $$\mathcal E_{k,s}$$ is indeed finite. Short of proving that $$H(k) \le s$$, we sometimes settle for a quantitative measure of the “sparsity” of the set $$\mathcal E_{k,s}.$$ Such results are commonly stated in terms of the quantity $$E_{k,s}(N)$$, defined as $E_{k,s}(N) = \#\{ n \in \mathcal E_{k,s} : 1 \le n \le N \}.$ My work in additive prime number theory falls into several categories:

Three of my papers [2005b, 2016b, 2017a] establish improved upper bounds for the function $$H(k)$$. For example, in recent joint work with Trevor Wooley [2017a], we proved the sharpest known upper bounds for $$H(k)$$ when $$k \ge 7$$. In particular, for large values of $$k$$, we showed that  $H(k)< 4k\log k \; – (2\log 2 \; – 1)k.$ Hide

Several of my papers [2005a, 2006a, 2009, 2010a, 2016a] establish quantitative bounds for the numbers of “exceptional integers” counted by $$E_{k,s} (N)$$ when $$N$$ is large. For example, Lilu Zhao and I [2016a] showed that $E_{2,4}(N) = O( N^{11/32} ) \qquad \text{as } N \to \infty.$Hide

Sometimes, we ask whether we can solve (WG) not merely in the primes but in primes subject to additional requirements. I have a couple of papers [2012b, 2017c] on one such hybrid question that seeks solutions of the Waring–Goldbach problem in “almost equal” primes $$p_1, \dots, p_s$$: i.e., we want to have $$|p_i – p_j| \le p_i^{\theta}$$ for some fixed $$\theta < 1$$ and for all pairs of indices $$i,j$$. For example, H.F. Liu and I [2017c] proved such a result when $$\theta > 31/40$$, $$k \ge 2$$ and $$s \ge k^2+k+1$$. I have written also some papers on the solubility of the ternary Goldbach problem (the case $$k=1$$ and $$s=3$$ of (WG)) in primes from special sequences, such as Beatty primes (see [2008]) and Piatetski-Shapiro primes (see [1997b]).
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Early in my career, I worked on several additive problems with prime variables that are variants of (WG). Some of my earliest papers [1996, 1998, 1999b, 2002b] focused on analogues of (WG) for fractional exponents. For example, an old result [1999b] from my Ph.D. thesis establishes that when $$1 < c < 61/55$$ and $$\varepsilon > 0$$ is fixed, the Diophantine inequality $\big| p_1^c + p_2^c + p_3^c – x \big| < \varepsilon$ can be solved in primes $$p_1, p_2, p_3$$ for all sufficiently large $$x$$. Later, I worked also on Diophantine inequalities involving cubic forms with prime and “almost-prime” variables (see [2000b, 2001]) and on more general additive equations with integer exponents (see [2004b, 2006b]).
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The proofs of virtually all of the theorems mentioned above use some version of the Hardy–Littlewood circle method, and therefore, rely on various bounds for exponential sums. Consequently, I have done a fair amount of work on such exponential sums. Two of my papers [2006c, 2013c] focus mainly on such estimates for the so-called “Weyl sums” over primes, which are the main generating functions used to study the Waring–Goldbach problem. A third [2006b] establishes a mean-value estimate for Dirichlet polynomials, which has proven to be a useful tool for estimating certain averages of exponential sums over primes. Finally, several other papers contain estimates for exponential sums over primes whose applicability seems limited to the specific problem studied in the paper (see [1997a, 1997b, 1999b, 2009a, 2016a]).
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I have studied also the distribution of primes among the elements of sequences of arithmetic interest. The most fundamental result in this subject is the Prime Number Theorem: If $$\pi(x)$$ denotes the number of primes $$p \le x$$, one has $\pi(x) \sim \int_2^x \frac{dt}{\log t} \qquad \text{as } x \to \infty.$ In fact, we know quantitative bounds for the error in this approximation, and the conjecture that $\pi(x) {}-{} \int_2^x \frac{dt}{\log t} = O\big( x^{1/2}\log x \big)$ is one way to state the Riemann Hypothesis—arguably the most famous open problem in mathematics.
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Most problems dealing with the distribution of primes can be seen as refinements of the PNT. For example, if $$\pi(x; q, a)$$ denotes the number of primes $$p \le x$$ such that $$p \equiv a \pmod q$$, then a strong form of Dirichlet’s theorem on primes in arithmetic progressions states that when $$\gcd(a,q) = 1$$, one has $\pi(x; q, a) \sim \frac {\pi(x)}{\phi(q)} \qquad \text{as } x \to \infty; \tag{SW}$ $$\phi(q)$$ being Euler’s totient function. In other words, the primes are uniformly distributed among the $$\phi(q)$$ arithmetic progressions modulo $$a \bmod q$$, with $$\gcd(a,q) = 1$$. Another problem asks how small can one choose $$y$$ relative to $$x$$ and preserve the asymptotic relation $\pi(x+y) – \pi(x) \sim \frac {y}{x} \pi(x) \qquad \text{as } x \to \infty,$ which claims that primes are uniformly distributed in “short intervals” near $$x.$$