@Article{LanguascoZaccagnini2013a,
  Author    = "A. Languasco and A. Zaccagnini",
  Title     = "A {Ces\`aro} average of {Hardy}-{Littlewood} numbers",
  Journal   = "J. Math. Anal. Appl.",
  Volume    = 401,
  Pages     = "568--577",
  Url       = "\url{http://dx.doi.org/10.1016/j.jmaa.2012.12.046}",
  Year      = 2013,
  Abstract  = "We study averages of the quantity
$R_{\textit{HL}}(n)  =  \sum_{m_1 + m_2^2 = n} \Lambda(m_1)$.
In particular, for any $k > 1$ we give an ``explicit formula'' for
\[
  \sum_{n \le N} R_{\textit{HL}}(n) \frac{(1 - n / N)^k}{\Gamma(k + 1)}
\]
in terms of the Gamma function evaluated at suitable combinations of
the Riemann zeta-function, and of Bessel functions of complex order."
}