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Let be complex numbers, and consider the power sums , . Put , where the minimum is over all possible complex numbers satisfying the above. Turán conjectured that , for some positive absolute constant. Atkinson proved this conjecture by showing . It is now known that , for . Determining whether or approaches some other limiting value as is still an open problem. Our calculations show that an upper bound for decreases for , suggesting that decreases to a limiting value less than as .
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Wong KL Bitter M Hammett GW Heidbrink W Hendel H Kaita R Scott S Strachan JD Tait G Bell MG Budny R Bush C Chan A Coonrod J Efthimion PC England AC Eubank HP Fredrickson E Furth HP Goldston RJ Grek B Grisham L Hawryluk RJ Hill KW Johnson D Kamperschroer J Kugel H Ma C Mansfield D Manos D McCune DC McGuire K Medley SS Mueller D Nieschmidt E Owens DK Paré VK Park H Ramsey A Rasmussen D Roquemore AL Schivell J Sesnic S Taylor G Williams MD Zarnstorff MC 《Physical review letters》1985,55(23):2587-2590
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Strachan JD Bitter M Ramsey AT Zarnstorff MC Arunasalam V Bell MG Bretz NL Budny R Bush CE Davis SL Dylla HF Efthimion PC Fonck RJ Fredrickson E Furth HP Goldston RJ Grisham LR Grek B Hawryluk RJ Heidbrink WW Hendel HW Hill KW Hsuan H Jaehnig KP Jassby DL Jobes F Johnson DW Johnson LC Kaita R Kampershroer J Knize RJ Kozub T LeBlanc B Levinton F La Marche PH Manos DM Mansfield DK McGuire K McNeill DH Meade DM Medley SS Morris W Mueller D Nieschmidt EB Owens DK Park H Schivell J Schilling G 《Physical review letters》1987,58(10):1004-1007
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Sums of Three or More Primes 总被引：2，自引：0，他引：2
J. B. Friedlander D. A. Goldston 《Transactions of the American Mathematical Society》1997,349(1):287-310
It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error in the Prime Number Theorem, such bounds being within a factor of of each other and this fact being equivalent to the Riemann Hypothesis. In this paper we show that, provided ``Riemann Hypothesis' is replaced by ``Generalized Riemann Hypothesis', results of similar (often greater) precision hold in the case of the corresponding formula for the representation of an integer as the sum of primes for , and, in a mean square sense, for . We also sharpen, in most cases to best possible form, the original estimates of Hardy and Littlewood which were based on the assumption of a ``Quasi-Riemann Hypothesis'. We incidentally give a slight sharpening to a well-known exponential sum estimate of Vinogradov-Vaughan.