A Cesàro Average of Hardy-Littlewood numbers

Abstract

Let $\Lambda$ be the von Mangoldt function and $r_{\text{HL}}(n)={\sum }_{m_1+m_2^2=n}\Lambda (m_1),$ be the counting function for the Hardy-Littlewood numbers. Let $N$ be a sufficiently large integer. We prove that $$\begin{array}{rl}\sum _{n\le N}{r}_{\text{HL}}(n)\frac{(1-n/N{)}^k}{\Gamma (k+1)}& =\frac{{\pi }^{1/2}}{2}\frac{N^{3/2}}{\Gamma (k+5/2)}-\frac{1}{2}\frac{N}{\Gamma (k+2)}-\frac{{\pi }^{1/2}}{2}\sum _{\rho }\frac{\Gamma (\rho )}{\Gamma (k+3/2+\rho )}{N}^{1/2+\rho }\\ & +1/2\sum _{\rho }\frac{\Gamma (\rho )}{\Gamma (k+1+\rho )}{N}^{\rho }+\frac{N^{3/4-k/2}}{{\pi }^{k+1}}\sum _{\ell \ge 1}\frac{J_{k+3/2}(2\pi \ell N^{1/2})}{{\ell }^{k+3/2}}\\ & -\frac{N^{1/4-k/2}}{{\pi }^k}\sum _{\rho }\Gamma (\rho )\frac{N^{\rho /2}}{{\pi }^{\rho }}\sum _{\ell \ge 1}\frac{J_{k+1/2+\rho }(2\pi \ell N^{1/2})}{{\ell }^{k+1/2+\rho }}+{\mathcal{O}}_k(1).\end{array}$$ for $k>1$, where $\rho$ runs over the non-trivial zeros of the Riemann zeta-function $\zeta (s)$ and $J_{\nu }(u)$ denotes the Bessel function of complex order $\nu$ and real argument $u$.

arXiv commentsubmitted

References in corpus (3)

• The number of Goldbach representations of an integer
• A Cesàro Average of Goldbach numbers
• Sums of many primes

Cited by in corpus (3)

• Sum of one prime and two squares of primes in short intervals
• A Cesàro average for an additive problem with an arbitrary number of prime powers and squares
• Explicit formula for the average of Goldbach and prime tuples representations