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:

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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 \(x.\)