Roger Baker

Roger Baker
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Roger Baker

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Mathematics - Number Theory (10)

Publications Authored By Roger Baker

We consider the distance to the nearest integer of f(p), where f is a quadratic polynomial with irrational leading coefficient. This distance is very small as a function of p, for infinitely many primes p. We give a 14% improvement in the exponent that measures the distance, compared with the most recent result in the literature. Read More

Let f be a polynomial with irrational leading coefficient. We obtain inequalities for the distance from the nearest integer of f(p) that hold for infinitely many primes p. These results improve work of Harman in 1981 and 1983 and Wong in 1997. Read More

Using the recent result of Bourgain, Demeter and Guth on Vinogradov's mean value, a number of new results about small fractional parts of polynomials and fractional parts of additive forms are obtained. These improve work of Baker, Cook, Danicic, Vaughan and Wooley. Read More

We obtain the analog of the Bombieri-Vinogradov theorem for square moduli up to any power of x less than 1/2. Read More

Let $d_n = p_{n+1} - p_n$, where $p_n$ denotes the $n$th smallest prime, and let $R(T) = \log T \log_2 T\log_4 T/(\log_3 T)^2$ (the "Erd{\H o}s--Rankin" function). We consider the sequence $(d_n/R(p_n))$ of normalized prime gaps, and show that its limit point set contains at least $25\%$ of nonnegative real numbers. We also show that the same result holds if $R(T)$ is replaced by any "reasonable" function that tends to infinity more slowly than $R(T)\log_3 T$. Read More

Let $\mathcal{R}$ be a finite set of integers satisfying appropriate local conditions. We show the existence of long clusters of primes $p$ in bounded length intervals with $p-b$ squarefree for all $b \in \mathcal{R}$. Moreover, we can enforce that the primes $p$ in our cluster satisfy any one of the following conditions: (1) $p$ lies in a short interval $[N, N+N^{\frac{7}{12}+\epsilon}]$, (2) $p$ belongs to a given inhomogeneous Beatty sequence, (3) with $c \in (\frac{8}{9},1)$ fixed, $p^c$ lies in a prescribed interval mod $1$ of length $p^{-1+c+\epsilon}$. Read More

Let $t \in \mathbb{N}$, $\eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \leq x^{5/12-\eta}$, $q$ not a multiple of the conductor of the exceptional character $\chi^*$ (if it exists). Suppose further that, \[ \max \{p : p | q \} < \exp (\frac{\log x}{C \log \log x}) \; \; {and} \; \; \prod_{p | q} p < x^{\delta}, \] where $C$ and $\delta$ are suitable positive constants depending on $t$ and $\eta$. Read More

In this paper, we study the gaps between primes in Beatty sequences following the methods in the recent breakthrough of J. Maynard. Read More

Let $m$ be a natural number, and let $\mathcal{Q}$ be a set containing at least $\exp(C m)$ primes. We show that one can find infinitely many strings of $m$ consecutive primes each of which has some $q\in\mathcal{Q}$ as a primitive root, all lying in an interval of length $O_{\mathcal{Q}}(\exp(C'm))$. This is a bounded gaps variant of a theorem of Gupta and Ram Murty. Read More

We consider various arithmetic questions for the Piatetski-Shapiro sequences $\fl{n^c}$ ($n=1,2,3,... Read More