In this work we continue the investigation, started in [8], about the interplay between hypergeometric functions and Fourier-Legendre (FL) series expansions. In the section "Hypergeometric series related to π, π 2 and the lemniscate constant", through the FL-expansion of [x(1 − x)] µ (with µ + 1 ∈ 1 4 N) we prove that all the hypergeometric series n≥0 (−1) n (4n + 1) p(n)
Let Λ be the von Mangoldt function andbe the counting function for the numbers that can be written as sum of a prime and two squares (that we will call Linnik numbers, for brevity). Let N a sufficiently large integer. We prove that for k > 3/2 we havewhere M (N, k) is essentially a weighted sum, over non-trivial zeros of the Riemann zeta function, of Bessel functions of complex order and real argument. We also prove that with this technique the bound k > 3/2 is optimal.
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