Highly biased prime number races (Q466560)

After a classical observation of Chebyshev concerning the bias for the primes of the form \(4k+3\) and \(4k+1\), the investigation of prime number races became an interesting and important issue. Many great mathematicians have considered the problem and obtained interesting results.NEWLINENEWLINELet \(q\) be a positive integer \(>1\), and consider the set NEWLINE\[NEWLINE P_{q;a_1,\dots,a_r}:=\{n\;:\;\pi(n;q,a_1)>\dots>\pi(n;q,a_r)\}, NEWLINE\]NEWLINE where the \(a_i\) are representatives of distinct coprime residue classes modulo \(q\), and as usual, \(\pi(n;q,a_i)\) stands for the number of primes \(p\) with \(p\leq n\) such that \(p\equiv a_i\pmod{q}\). By a result of \textit{J. Kaczorowski} [Analysis 15, No. 2, 159--171 (1995; Zbl 0826.11042)], it is known that the natural density of \(P_{q;a_1,\dots,a_r}\) does not exist in general. Hence, instead of the natural density, the logarithmic density is studied, defined for \(P\subset{\mathbb N}\) as NEWLINE\[NEWLINE \delta(P)=\lim\limits_{N\to\infty} \frac{1}{\log N}\underset{n\in P}{\sum\limits_{n\leq N}}\frac{1}{n}. NEWLINE\]NEWLINE The author deduces several theorems concerning two types of prime number races, where different residue classes mod \(q\) are combined, under the generalized Riemann hypothesis (GRH) and the linear independence hypothesis (LI).NEWLINENEWLINEThe first type of results concern races between quadratic residues and nonresidues modulo \(q\). Note that such races were previously studied among others by \textit{M. Rubinstein} and \textit{P. Sarnak} [Exp. Math. 3, No. 3, 173--197 (1994; Zbl 0823.11050)]. Put NEWLINE\[NEWLINE \pi(x;q,NR)=\#\{p\leq x\:\;p\;\text{is not a quadratic residue mod}\;q\} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \pi(x;q,R)=\#\{p\leq x\:\;p\;\text{is a quadratic residue mod}\;q\}, NEWLINE\]NEWLINE for moduli \(q\) having a primitive root. Among other (even more general) results the author proves that under the GRH and LI, for any \(\varepsilon>0\) there exists a \(q\) such that NEWLINE\[NEWLINE 1-\varepsilon < \delta(q;NR,R) < 1, NEWLINE\]NEWLINE where the set \((q;NR,R)\) is defined in the natural way.NEWLINENEWLINEThe second type of results concern linear combinations of the \(\pi\) function corresponding to different residue classes, more precisely, logarithmic densities of the form NEWLINE\[NEWLINE \delta(q;\bar{a},\bar{\alpha}):= \delta(\{n\;:\;\alpha_1\pi(n;q,a_1)+\dots+\alpha_k\pi(n;q,a_k)>0\}). NEWLINE\]NEWLINE Here \(\bar{a}=(a_1,\dots,a_k)\) is a vector of reduced residues modulo \(q\), and \(\bar{\alpha}=(\alpha_1,\dots,\alpha_k)\) is a nonzero vector of real numbers such that \(\sum_{i=1}^k\alpha_i=0\).NEWLINENEWLINEBeside these, one can find many interesting remarks and comments, revealing properties and objects hidden in the background (such as effects of low-lying zeros of certain \(L\)-functions on decreasing the bias).