January 14, 2019
We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we assume the Riemann Hypothesis.
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May 23, 2018
We investigate the average number of representations of a positive integer as the sum of $k + 1$ perfect $k$-th powers of primes. We extend recent results of Languasco and the last Author, which dealt with the case $k = 2$ [6] and $k = 3$ [5] respectively. We use the same technique to study the corresponding problem for sums of just $k$ perfect $k$-th powers of primes.
June 13, 2018
In this paper we extend and improve our results on weighted averages for the number of representations of an integer as a sum of two powers of primes. Let $1\le \ell_1 \le \ell_2$ be two integers, $\Lambda$ be the von Mangoldt function and % \(r_{\ell_1,\ell_2}(n) = \sum_{m_1^{\ell_1} + m_2^{\ell_2}= n} \Lambda(m_1) \Lambda(m_2) \) % be the weighted counting function for the number of representation of an integer as a sum of two prime powers. Let $N \geq 2$ be an integer. We ...
October 26, 2018
We improve some results about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell_1}+p_{2}^{\ell_2}$ and $n=p^{\ell_1} + m^{\ell_2}$, where $\ell_1, \ell_2\ge 2$ are fixed integers, $p,p_1,p_2$ are prime numbers and $m$ is an integer.
February 13, 2019
We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.
June 13, 2018
Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{k}+p_{2}^{2}+p_{3}^{2}$, where $p_1,p_2,p_3$ are prime numbers, holds in intervals shorter than the ones previously known.
April 9, 2015
We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case.
June 20, 2008
Assuming the Riemann Hypothesis we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Omega-term, we prove that our result is essentially the best possible.
November 18, 2022
Let $k$ be a natural number and let $c=2.134693\ldots$ be the unique real solution of the equation $2c=2+\log (5c-1)$ in $[1,\infty)$. Then, when $s\ge ck+4$, we establish an asymptotic lower bound of the expected order of magnitude for the number of representations of a large positive integer as the sum of one prime and $s$ positive integral $k$-th powers.
April 18, 2015
We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime square and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case.
July 26, 2018
Let $R_{k,\ell}(N)$ be the representation function for the sum of the $k$-th power of a prime and the $\ell$-th power of a positive integer. Languasco and Zaccagnini (2017) proved an asymptotic formula for the average of $R_{1,2}(N)$ over short intervals $(X,X+H]$ of the length $H$ slightly shorter than $X^{\frac{1}{2}}$, which is shorter than the length $H=X^{\frac{1}{2}+\epsilon}$ in the exceptional set estimates of Mikawa (1993) and of Perelli and Pintz (1995). In this pap...