arXiv subjects
Publications and source records attributed to Roger Baker.
Let $1 < c < 24/19$. We show that the number of integers $n \le N$ that cannot be written as $[p_1^c] + [p_2^c]$ ($p_1$, $p_2$ primes) is $O(N^{1-\sigma+\varepsilon})$. Here $\sigma$ is a positive function of $c$ (given explicitly) and $\varepsilon$ is an arbitrary positive number.
Let $S(x,t)$ denote the Weyl sum with associated polynomial $xn + tn^2$. Suppose that $|S(x,t)|$ attains its maximum for given $x$ at $t = t(x)$. We give upper and lower bounds of the same order of magnitude for the $L^p$ norm of $S(x,t(x))$.
An asymptotic formula is given for the number of y-smooth numbers up to x in a Beatty sequence corresponding to an irrational number of finite type.
Let c > 0.55. Every large n can be written in the form p +ab, where p is prime, a and b are significantly smaller than x^1/2 and ab is less than n^c. This strengthens a result of Heath-Brown, which has the requirement c>3/4. We introduce the idea of 'intersecting two sieves' as a tool in the proof of this result.
The inequalities concern the sum of s powers of primes with non-integer exponent c>1. Here s =2,3,4,or 5. The equations are similar, taking integer part before summing; here s = 3 or 5. New ranges of c are found in all cases for which many solutions in primes exist.
We prove a theorem about approximation to an irrational number by rational numbers whose denominator n is free of prime factors bigger than a power of log n. We strengthen the result in version 1 by using an exponential sum over smooth numbers tailored to the application. The new exponent approaches 1/3 as the exponent of log n becomes large.
If a set S of pairwise coprime moduli q, less than x^(9/40), is considered, one obtains the expected behavior for primes up to x in arithmetic progressions mod q, except for a subset of S whose cardinality is bounded by a power of log x.
We obtain an analog of the Montgomery-Hooley asymptotic formula for the variance of the number of primes in arithmetic progressions. In the present paper the moduli are restricted to the sequences of integer parts $[F(n)]$, where $F(t) = t^c$ ($c > 1$, $c \not\in \mathbb{N}$) or $F(t) = \exp\big((\log t)^{\gamma}\big)$ ($1 < \gamma < 3/2$).
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.
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.
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.
We obtain the analog of the Bombieri-Vinogradov theorem for square moduli up to any power of x less than 1/2.
Let $p_n$ denote the $n$th smallest prime number, and let $\boldsymbol{L}$ denote the set of limit points of the sequence $\{(p_{n+1} - p_n)/\log p_n\}_{n = 1}^{\infty}$ of normalized differences between consecutive primes. We show that for $k = 9$ and for any sequence of $k$ nonnegative real numbers $\beta_1 \le \beta_2 \le ... \le \beta_k$, at least one of the numbers $\beta_j - \beta_i$ ($1 \le i < j \le k$) belongs to $\boldsymbol{L}$. It follows at least $12.5%$ of all nonnegative real numbers belong to $\boldsymbol{L}$.