More precisely, the set of admissible test functions is determined by the quality of the error terms in such counting functions. In addition to (1.1), we will consider exact estimates that include local conditions, which are of the form As part of this project, we produced numerical data suggesting that𝜃= 12 and any𝜔 >0 are admissible values in (1.2) (in particular indicating that the bound𝜔+𝜃≥ 12 in Theorem1.1 may be the best possible).
In the language of the Katz-Sarnak heuristic, the first and third terms on the right-hand side of (1.3) are a manifestation of the symplectic symmetry type of the family F±(𝑋). The first result in this direction is due to Yang [Ya], who showed that we have that under the condition 𝜎 < 501. Suppose conjecture 4.3 about the average of the shifts of the logarithmic derivative of 𝜁𝐾(𝑠)/𝜁(𝑠 ), as well as the Riemann hypothesis for 𝜁𝐾(𝑠), for.
Indeed, for𝜎 <1, the difference between these two evaluations of the one-level density is given by 𝐶±𝑋−13. We can now concentrate our efforts on the average of the third (and most crucial) period. To bound the integral involving the gamma function in (3.11), we note that Stirling's formula implies that we have the estimate for any fixed vertical strip minus slices centered at the poles of Γ±(𝑠).
However, since we have a different condition on𝜃 (that is, 𝜃+𝜔 < 12), there is an additional error term in the current estimate.
A refined Ratios Conjecture
A first step in understanding the 𝑅𝑗(𝛼, 𝛾;𝑋) will be achieved using the following exact evaluation of the expected value of𝜆𝐾(𝑚)𝜇𝐾(ℎ). However, our goal is to evaluate the one-level density by averaging 𝐿𝐿(12 +𝑟, 𝑓𝐾); therefore it is necessary to also calculate the partial derivative 𝜕𝛼𝜕 𝑅1(𝛼, 𝛾;𝑋)|𝛼=𝛾=𝑟. To do this, we need to make sure that the error term remains small after a differentiation. The resulting expression will converge in the said region. valid for all small enough𝜅 >0) and to bound the integrand using the approximation for𝑅1(𝛼, 𝛾;𝑋) above.
As for the key terms, you can differentiate them term by term and get the expected approximation. To account for this, we replace this step by using the following corollary of Lemma4.1. Traditionally, when applying the ratio estimation recipe, one has to limit the real part of the variable to fairly small positive values.
For example, in the family of quadratic DirichletL functions [CS,FPS3], one requires that log1𝑋 Re(𝑟) < 14. This ensures that one is far enough from a pole for the expression in the right-hand side. Under conjecture 4.3 and well-known arguments (see, e.g., [FPS3, Section 3.2]), the part of this sum involving the first integral is equal to.
We can also move the contour of integration to the line Re(𝑠)=𝑐with 0< 𝑐 < 16. The first integral on the right-hand side is identical to the integral just evaluated in the first part of this proof. Regarding the second one, moving the contour to the line Re(𝑠)=0, we find that it is equal to.
Applying Lemma 3.2 to the first term, we find the leading terms on the right-hand side of (4.15). The idea is to provide an analytic continuation on the Dirichlet series 𝐴3(−𝑠, 𝑠) and 𝐴4(−𝑠, 𝑠) in the strip 0 Therefore, the last step is to find a meromorphic continuation for the infinite product on the right-hand side of (5.5), which we will denote by 𝐷3(𝑠). Now that we have a meromorphic continuation of 𝐴4(−𝑠, 𝑠), we will calculate the principal Laurent coefficient at 𝑠= 16. Note that this is not contradictory, since in that theorem we assume such a bound uniformly for all partition types, and from the discussion above we expect that 𝐸+𝑝(𝑋, 𝑇1) 𝜀 𝑝𝜀𝑋12+𝜀 is essentially the best possible . We would require more data to guess as strongly as what we did for𝐸+𝑝(𝑋, 𝑇4). For the splitting type𝑇5, the error term appears to be even smaller (probably due to the fact that these fields are very rare). Here,𝑁all+(𝑋) counts all cubic fields of discriminant up to X, including Galois fields (of Cohn's work [C],𝑁all+(𝑋)−𝑁+(𝑋) ∼𝑐𝑋 program, in [BThe algorithm12,. ], can be found here: https://www.math.u-bordeaux.fr/~kbelabas/research/cubic.html It is also interesting that the graph is always positive, which is not without reminding us of Chebyshev's bias (see f .eg the graphs in the research paper [GM]) in the distribution of prime numbers. Given this numerical evidence, one can summarize this section by saying that in all cases it appears that we have square root cancellation. This is reminiscent of Montgomery's conjecture [Mo] for prime numbers in arithmetic progressions, which states that. The calculations in this paper were performed on a personal computer using pari/gp as well as Belabas's CUBIC program. Miller, 'Overcoming the ratio conjecture in the 1-level density of Dirichlet 𝐿 functions', Algebra Number Theory. Södergren, 'Low lying zeros of quadratic Dirichlet𝐿 functions: Lower order terms for extended support', Compos. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol 53 (American Mathematical Society, Providence, RI, 2004). Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy, American Mathematical Society Colloquium Publications, bind 45 (American Mathematical Society, Providence, RI, 1999). Snaith, Orthogonal and Symplectic𝑛-Level Densities, Memoirs of the American Mathematical Society, bind 251 (American Mathematical Society, Providence, RI, 2018). Miller, 'Et- og to-niveau tætheder for rationelle familier af elliptiske kurver: bevis for de underliggende gruppesymmetrier', Compos.Numerical investigations
