https://nova.newcastle.edu.au/vital/access/ /manager/Index en-au 5 https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:12961 Wed 11 Apr 2018 10:00:33 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13081 k(n) and σk/2(n) in the terms of Riemann Zeta function ζ(s) only. In this paper, we explore other arithmetical functions enjoying this remarkable property. In Theorem 2.1 below, we are able to generalize the above result and prove that if f_iand g_i are completely multiplicative, then we have [formula could not be replicated] where L_f(s):= Σ∞/n=1 f(n)n_-s is the Dirichlet series corresponding to f. Let r_N(n) be the number of solutions of x²/1 +...+ x²/N = n and r₂,_P(n) be the number of solutions of x² + Py² = n. One of the applications of Theorem 2.1 is to obtain closed forms, in terms of ζ(s) and Dirichlet L-functions, for the generating functions of rN(n), r²/N(n),r2,P(n) and r2,P(n)² for certain N and P. We also use these generating functions to obtain asymptotic estimates of the average values for each function for which we obtain a Dirichlet series.]]> Sat 24 Mar 2018 08:15:38 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13064 11 and cannot occur if we assume the Generalized Riemann Hypothesis.]]> Sat 24 Mar 2018 08:15:37 AEDT ]]>