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On SA, CA, and GA numbers. (English) Zbl 1280.11049
Summary: Gronwall’s function \(G\) is defined for \(n>1\) by \(G(n)=\frac{\sigma(n)}{n \log\log n}\) where \(\sigma (n)\) is the sum of the divisors of \(n\). We call an integer \(N>1\) a GA1 number if \(N\) is composite and \(G(N)\geq G(N/p)\) for all prime factors \(p\) of \(N\). We say that \(N\) is a GA2 number if \(G(N)\geq G(aN)\) for all multiples \(aN\) of \(N\). In [Integers 11, No. 6, 753–763, A33 (2011; Zbl 1235.11082)], we used Robin’s and Gronwall’s theorems on \(G\) to prove that the Riemann Hypothesis (RH) is true if and only if 4 is the only number that is both GA1 and GA2. In the present paper, we study GA1 numbers and GA2 numbers separately. We compare them with superabundant (SA) and colossally abundant (CA) numbers (first studied by Ramanujan). We give algorithms for computing GA1 numbers; the smallest one with more than two prime factors is 183783600, while the smallest odd one is 1058462574572984015114271643676625. We find nineteen GA2 numbers \(\leq 5040\), and prove that a GA2 number \(N>5040\) exists if and only if RH is false, in which case \(N\) is even and \(>10^{8576}\).
MSC:
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11A41 Primes
11Y55 Calculation of integer sequences
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