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- W1978333622 abstract "Let $$qge 3$$ and $$2le rle phi (q)$$ be positive integers, and $$a_1,ldots ,a_r$$ be distinct reduced residue classes modulo $$q$$ . Rubinstein and Sarnak defined $$delta (q;a_1,ldots ,a_r)$$ to be the logarithmic density of the set of real numbers $$x$$ such that $$pi (x;q,a_1)>pi (x;q,a_2)>cdots >pi (x;q,a_r)$$ . In this paper, we establish an asymptotic formula for $$delta (q;a_1,ldots ,a_r)$$ when $$rge 3$$ is fixed and $$q$$ is large. Several applications concerning these prime number races are then deduced. First, comparing with a recent work of Fiorilli and Martin on the case $$r=2$$ , we show that these densities behave differently when $$rge 3$$ . Another surprising consequence of our results is that, unlike two-way races, biases do appear in races involving three or more squares (or non-squares) to large moduli. Furthermore, we establish a partial result towards a conjecture of Rubinstein and Sarnak on biased races, and disprove a recent conjecture of Feuerverger and Martin concerning bias factors. Lastly, we use our method to derive the Fiorilli and Martin asymptotic formula for the densities when $$r=2$$ ." @default.
- W1978333622 created "2016-06-24" @default.
- W1978333622 creator A5086450144 @default.
- W1978333622 date "2012-11-25" @default.
- W1978333622 modified "2023-10-15" @default.
- W1978333622 title "Prime number races with three or more competitors" @default.
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- W1978333622 doi "https://doi.org/10.1007/s00208-012-0874-1" @default.
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