https://nova.newcastle.edu.au/vital/access/ /manager/Index en-au 5 https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:12923 b,c = Σ_n≥1 1/(cⁿb^cn), for coprime integers b ≥ 2 and c ≥ 2. These are of interest because, according to previous studies, α_b,c is known to be b-normal, meaning that every m-long string of base-b digits appears in the base-b expansion of the constant with precisely the limiting frequency b^-m. So, for example, the constant α_2,3 = Σn≥1 1/(3ⁿ2³ⁿ) is 2-normal. More recently it was established that α_b,c is not bc-normal, so, for example,α_2,3 is provably not 6-normal. In this paper, we extend these findings by showing that α_b,c is not B-normal, where B = b^pc^q r, for integers b and c as above, p, q, r ≥ 1, neither b nor c divide r, and the condition D=c^q/pr^1/p/b^c-1 < 1 is satisfied. It is not known whether or not this is a complete catalog of bases to which α_b,c is nonnormal. We also show that the sum of two B-nonnormal Stoneham constants as defined above, subject to some restrictions, is B-nonnormal.]]> Wed 11 Apr 2018 10:56:19 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:12331 Wed 11 Apr 2018 10:18:35 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:12247 Wed 11 Apr 2018 09:16:08 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:11935 Sat 24 Mar 2018 11:09:02 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:10331 Sat 24 Mar 2018 11:07:38 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:12922 n = ∫₀¹logⁿΓ(x)dx for 1≤n≤4 and make some comments regarding the general case. The subsidiary computational challenges are substantial, interesting and significant in their own right]]> Sat 24 Mar 2018 08:18:13 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:12941 Sat 24 Mar 2018 08:17:40 AEDT ]]> 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:13072 ∞₀ ⁿ∏_k=0 ^sin(akx)―_akxdx. We then give quite striking closed form evaluations of such integrals and finish by discussing various extensions and applications.]]> Sat 24 Mar 2018 08:15:38 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13069 Sat 24 Mar 2018 08:15:36 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:10334 Sat 24 Mar 2018 08:07:00 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:10335 Sat 24 Mar 2018 08:07:00 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:17309 Sat 24 Mar 2018 08:01:48 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:23396 Sat 24 Mar 2018 07:13:56 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:22779 Sat 24 Mar 2018 07:12:15 AEDT ]]>