AbstractWe discuss the generalizations of the concept of Chebyshev’s bias from two perspectives. First, we give a general framework for the study of prime number races and Chebyshev’s bias attached to general L-functions satisfying natural analytic hypotheses. This extends the cases previously considered by several authors and involving, among others, Dirichlet L-functions and Hasse–Weil L-functions of elliptic curves over Q. This also applies to new Chebyshev’s bias phenomena that were beyond the reach of the previously known cases. In addition, we weaken the required hypotheses such as GRH or linear independence properties of zeros of L-functions. In particular, we establish the existence of the logarithmic density of the set $ \{x \ge 2:\sum\nolimits_{p \le x} {\lambda _f}(p) \ge 0\}$ for coefficients (λf(p)) of general L-functions conditionally on a much weaker hypothesis than was previously known.

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Abstract. Let X be a scheme of finite type over Z. For p ∈ P the set of prime numbers, let N X (p) be the number of Fp-points of X/Fp. For fixed n ≥ 1 and a 1 , . . . , an ∈ Z, we study the set n i=1 {p ∈ P − Σ X , N X (p) = a i [mod p]} where Σ X is the finite set of primes of bad reduction for X. In case dim X ≤ 3, we show the set is either empty or has positive lower-density. We also address the question of the size of the smallest prime in that set. Using sieve methods, we obtain for example an upper bound for the size of the least prime of {p ∈ P, p ∤ N X (p)} on average in particular families of hyperelliptic curves.

We generalize current known distribution results on Shanks-Rényi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let π(x; q, a) be the number of primes up to x that are congruent to a modulo q. For a fixed integer q and distinct invertible congruence classes a 0 , a 1 , . . . , a D , assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of real x for which the inequalities π(x; q, a 0 ) > π(x; q, a 1 ) > . . . > π(x; q, a D ) are simultaneously satisfied admits a logarithmic density.

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We study low-lying zeros of L-functions attached to holomorphic cusp forms of level 1 and large even weight. In this family, the Katz–Sarnak heuristic with orthogonal symmetry type was established in the work of Iwaniec, Luo and Sarnak for test functions ϕ satisfying the condition supp$(\widehat \phi) \subset(-2,2)$. We refine their density result by uncovering lower-order terms that exhibit a sharp transition when the support of $\widehat \phi$ reaches the point 1. In particular, the first of these terms involves the quantity $\widehat \phi(1)$ which appeared in the previous work of Fouvry–Iwaniec and Rudnick in symplectic families. Our approach involves a careful analysis of the Petersson formula and circumvents the assumption of the Generalized Riemann Hypothesis (GRH) for higher-degree automorphic L-functions. Finally, when supp$(\widehat \phi)\subset (-1,1)$ we obtain an unconditional estimate which is significantly more precise than the prediction of the L-functions ratios conjecture.

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