The number of Goldbach representations of an integer (Q2880639)

Let the weighted number of representations, of the integer \(n\geq 1\) as a sum of two primes, be \(G(n)\) defined as NEWLINE\[NEWLINE G(n):=\sum_{k_1+k_2=n}\Lambda(k_1)\Lambda(k_2) NEWLINE\]NEWLINE with the familiar von Mangoldt function, compare my review [\textit{G. Bhowmik} and \textit{J. Schlage-Puchta}, Nagoya Math. J. 200, 27--33 (2010; Zbl 1217.11089)], to which the notations are consistent (in the present paper it is called \(R(n)\) instead).NEWLINENEWLINEThe authors give conditional results under the Riemann Hypothesis (RH) of the following kind: defining NEWLINE\[NEWLINE H(N):=-2\sum_{\rho}{{N^{1+\rho}}\over {\rho(1+\rho)}} \qquad \forall N\geq 2 NEWLINE\]NEWLINE where the sum is over all non-trivial (i.e., \(\text{Re}\,\rho>0\)) zeros \(\rho\) of the Riemann \(\zeta\)-function and \(E(N)\) inside NEWLINE\[NEWLINE \sum_{n\leq N}G(n)=N^2/2+H(N)+E(N), \leqno{(*)} NEWLINE\]NEWLINE writing (RH) to mean: assuming Riemann Hypothesis (i.e., all \(\rho\), see above, have real part \(1/2\)), NEWLINE\[NEWLINE \text{(RH)} \enspace \Rightarrow \enspace E(N)=O(N\log^{3} N). NEWLINE\]NEWLINE The authors improve previous bounds of Bhowmik and Schlage-Puchta for \((*)\) error term (see the quoted review, for details).NEWLINENEWLINEThey feed the circle method with smoothed exponential sums over primes, of the kind introduced by Yu. V. Linnik, applying a previous result of Languasco and Perelli. This, in turn, rests on a conditional estimate, under (RH), due to Selberg (yes, the 1943 milestone). Apart from this technical smoothing (from which it steams their gain over log-powers), such bound for the ``Selberg integral'', of the primes, is the core of the arithmetic information needed; this, even if they don't use Gallagher's Lemma, that gives upper bounds for exponential sums (needed in the circle method) in terms of the Selberg integral, hidden in the RHS!NEWLINENEWLINEThey also give, under RH, other results, like a more general formula for \(G(n)\) sum in short intervals.