Selberg's upper bound method provides rather good results in certain circumstances. We wish to apply ideas from this upper bound method to that of the lower bound sifting problem. The sum G(x) arises in Selberg's method and in this account we study the related sum Hz(x). We provide an asymptotic estimate for the sum Hz(x) by investigating the residual sum Iz(x) = Hz(oo) Hz(x) and transferring back to Hz(x). We obtain a lower bound for the sum which counts the number of a G A with the logarithmic weight log pj log z attached to the smallest prime factor of the number a subject to the condition v(D, A) < R combining ideas from Selberg's A2A" method with Richert's weights. v(D, A) counts the number of prime factors p of a number a according to multiplicity when p > D but counting each p at most once when p < D.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:583958 |
| Date | January 2006 |
| Creators | Howie, Moira |
| Publisher | Cardiff University |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://orca.cf.ac.uk/56127/ |
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