Chebotarev's theorem and Littlewood complexity (Q2811188)

Let \(L/\mathbb{Q}\) be a finite Galois extension, with Galois group \(G\), and let \(M\) be the product of the primes dividing the discriminant. For any real valued function \(f\) on \(G\) which is constant on conjugacy classes, one writes NEWLINE\[NEWLINE\pi(x,f):=\sum_{p\leq x,\,p\nmid D}f(\mathrm{Frob}_p),NEWLINE\]NEWLINE and one takes \(\mu(f)\) to be the average of \(f\) over \(G\). Then, under the generalized Riemann hypothesis, the Chebotarev density may be given in the form NEWLINE\[NEWLINE\pi(x,f)-\mu(f)\pi(x)\ll x^{1/2}\lambda(f)\log (xM|G|).NEWLINE\]NEWLINE Here the implied constant is absolute, and \(\lambda(f)\) is the ``Littlewood complexity'', given by NEWLINE\[NEWLINE\lambda(f)= |G|^{-1}\sum_{\pi}\sum_{g\in G}\mathrm{Trace}(f(g))\pi(g^{-1})\mathrm{dim}(\pi),NEWLINE\]NEWLINE with \(\pi\) running over equivalence classes of irreducible complex representations of \(G\).NEWLINENEWLINEIt is not easy to give good estimates for \(\lambda(f)\) in general. The interested reader should look at the paper itself. However, one simple corollary is the following. Suppose \(P(X)\) is an irreducible monic polynomial of degree \(n\) over the integers, and let \(M\) be the product of the primes of bad reduction. Suppose one has Artin's conjecture and the generalized Riemann hypothesis for the \(L\)-functions attached to the splitting field of \(P\). Then there is a prime \(p\ll n^4(\log M+n\log n)^2\) such that \(P(X)\) has no root modulo \(p\). The corresponding bound obtained by previous methods would have been super-exponential in \(n\).NEWLINENEWLINEA second nice application concerns a conjecture of \textit{N. Koblitz} [Pac. J. Math. 131, No. 1, 157--165 (1988; Zbl 0608.10010)], according to which, if \(E\) is a non-CM elliptic curve over the rationals, there are asymptotically \(C_Ex(\log x)^{-2}\) primes \(p\leq x\) for which \(|E(\mathbb{F}_p)|\) is prime. Here \(C_E\) is an explicitly given positive constant. The present paper establishes an upper bound, 8 times the predicted bound, subject to Artin's conjecture and the generalized Riemann hypothesis. Under GRH alone one has a bound 20 times that which Koblitz conjectures.NEWLINENEWLINEA final application gives a new upper bound for the Lang-Trotter conjecture for abelian surfaces, \(\pi_A(0,x)\ll x^{9/10}(\log x)^{-3/5}\) subject again to Artin's conjecture and the generalized Riemann hypothesis.