Let Λ be the von Mangoldt function and r G (n) = m1+m2=n Λ(m 1 )Λ(m 2 ) be the counting function for the Goldbach numbers. Let N ≥ 2 be an integer. We prove thatfor k > 1, where ρ, with or without subscripts, runs over the non-trivial zeros of the Riemann zeta-function ζ(s).version of such a result with an error term o(N 2 ). The technique here is completely different and RH has no consequences on the lower bound for the size of k. Similar averages of arithmetical functions are common in the literature, see, e.g., Chandrasekharan-Narasimhan [2] and Berndt [1] who built on earlier classical works. In their setting the generalized Dirichlet series associated to the arithmetical function satisfies a suitable functional equation and this leads to an asymptotic formula containing Bessel functions of real order. In our case we have no functional equation, and Bessel functions are naturally replaced by Gamma functions; in fact we plan to develop further the present technique to deal with the cases p ℓ 1 1 +p ℓ 2 2 and p+m 2 , where Bessel functions with complex order arise; we expect many technical complications.The most interesting explicit formula in Goldbach's problem has been recently given by Pintz [12]. It is too complicated to be reproduced here, but we remark that in his formula, which deals with individual values of r G (n), the summation is over zeros of suitable Dirichlet L-functions, whereas in an average problem like the present one, only the zeros of the Riemann ζ-function are relevant. The same phenomenon occurs in our papers [7] and [8].The method we will use is based on a formula due to Laplace [9], namely

Let $\Lambda$ be the von\ud Mangoldt function and \(R(n) = \sum_{h+k=n} \Lambda(h)\Lambda(k) \) be the\ud counting function for the Goldbach numbers. Let $N \geq 2$ and assume that\ud the Riemann Hypothdfesis holds. We prove that \[ \sum_{n=1}^{N} R(n) =\ud \frac{N^{2}}{2} -2 \sum_{\rho} \frac{N^{\rho + 1}}{\rho (\rho + 1)} + \Odi{N\ud \log^{3}N }, \] where $\rho=1/2+i\gamma$ runs over the non-trivial zeros of\ud the Riemann Zeta-function $\zeta(s)$. This improves a recent result by\ud Bhowmik and Schlage-Puchta

We introduce a new algorithm, which is faster and requires less computing resources than the ones previously known, to compute the Euler–Kronecker constants $${\mathfrak {G}}_q$$ G q for the prime cyclotomic fields $$ {\mathbb {Q}}(\zeta _q)$$ Q ( ζ q ) , where q is an odd prime and $$\zeta _q$$ ζ q is a primitive q-root of unity. With such a new algorithm we evaluated $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}_q^+$$ G q + , where $${\mathfrak {G}}_q^+$$ G q + is the Euler–Kronecker constant of the maximal real subfield of $${\mathbb {Q}}(\zeta _q)$$ Q ( ζ q ) , for some very large primes q thus obtaining two new negative values of $${\mathfrak {G}}_q$$ G q : $${\mathfrak {G}}_{9109334831}= -0.248739\dotsc $$ G 9109334831 = - 0.248739 ⋯ and $${\mathfrak {G}}_{9854964401}= -0.096465\dotsc $$ G 9854964401 = - 0.096465 ⋯ We also evaluated $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}^+_q$$ G q + for every odd prime $$q\le 10^6$$ q ≤ 10 6 , thus enlarging the size of the previously known range for $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}^+_q$$ G q + . Our method also reveals that the difference $${\mathfrak {G}}_q - {\mathfrak {G}}^+_q$$ G q - G q + can be computed in a much simpler way than both its summands, see Sect. 3.4. Moreover, as a by-product, we also computed $$M_q=\max _{\chi \ne \chi _0} \vert L^\prime /L(1,\chi ) \vert $$ M q = max χ ≠ χ 0 | L ′ / L ( 1 , χ ) | for every odd prime $$q\le 10^6$$ q ≤ 10 6 , where $$L(s,\chi )$$ L ( s , χ ) are the Dirichlet L-functions, $$\chi $$ χ run over the non trivial Dirichlet characters mod q and $$\chi _0$$ χ 0 is the trivial Dirichlet character mod q. As another by-product of our computations, we will provide more data on the generalised Euler constants in arithmetic progressions.

We study the Mertens product over primes in arithmetic progressions, and find a uniform version of previous results on the asymptotic formula, improving at the same time the size of the error term, and giving an alternative, simpler value for the constant appearing in the main term

Assuming the Riemann Hypothesis we prove that the interval \ud $[N,N+H]$\ud contains an integer which is a sum of a prime and two squares of primes provided that $H \ge C (\log N)^{4}$, \ud where $C >0$ is an effective constant