Explicit bounds on exceptional zeroes of Dirichlet L-functions

Bordignon, Matteo

doi:10.1016/j.jnt.2019.02.003

Cited by 9 publications

(25 citation statements)

References 10 publications

(18 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By way of example, for 3 ≤ q ≤ 10, we have Here the constant 4.015 · 10 −4 sharpens the corresponding value 0.262 in Dusart [6] by a factor of roughly 650. We remark that x ≥ 7,932,309,757 is the best-possible range of validity for the error bound ( 1.14); indeed this is true for each x ψ (q), x θ (q), and x π (q), for 3 ≤ q ≤ 10 5 . For 3 ≤ q ≤ 10 5 , we observe that (as a consequence of our proofs), we have c ψ (q) ≤ c θ (q) ≤ c π (q) ≤ c 0 (q).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 63%

See 1 more Smart Citation

Explicit bounds for primes in arithmetic progressions

Bennett¹,

Martin²,

O’Bryant³

et al. 2018

Illinois J. Math.

View full text Add to dashboard Cite

We derive explicit upper bounds for various counting functions for primes in arithmetic progressions. By way of example, if q and a are integers with gcd(a, q) = 1 and 3 ≤ q ≤ 10 5 , and θ(x; q, a) denotes the sum of the logarithms of the primes p ≡ a (mod q) with p ≤ x, we show that θ(x; q, a) − x/ϕ(q) < 1 160x log x for all x ≥ 8 • 10 9 , with significantly sharper constants obtained for individual moduli q. We establish inequalities of the same shape for the other standard primecounting functions π(x; q, a) and ψ(x; q, a), as well as inequalities for the nth prime congruent to a (mod q) when q ≤ 1200. For moduli q > 10 5 , we find even stronger explicit inequalities, but only for much larger values of x. Along the way, we also derive an improved explicit lower bound for L(1, χ) for quadratic characters χ, and an improved explicit upper bound for exceptional zeros.

show abstract

Section: Introduction and Statement Of Resultsmentioning

confidence: 63%

“…Computations of the leading constants c ψ , c θ , and c π for q ≤ 10 5 88 A. 5. Dominant contributions to c ψ (q), c θ (q), and c π (q) for q ≤ 10 5 89 A.…”

Section: Explicit Upper Bound For Exceptional Zeros Of Quadratic DImentioning

confidence: 99%

Explicit bounds for primes in arithmetic progressions

Bennett¹,

Martin²,

O’Bryant³

et al. 2018

Illinois J. Math.

View full text Add to dashboard Cite

show abstract

“…Thus, remembering the lower bound for q, we obtain the desired result In this section we obtain a new explicit bound for exceptional zeros of Dirichlet L-functions to real characters and will be based on combining Theorem 5 with the argument of the first author [3]. Let χ a nonprincipal character mod q and suppose the L-function L(s, χ) := ∞ n=0 χ(n)n −s , has an exceptional zero β 0 satisfying…”

Section: Proof Of Theoremmentioning

confidence: 97%

“…A simple way to see this is via Fourier expansion into Gauss sums and hence transforms estimating sums of length N to sums of length q/N, an observation which first appears to be due to A. I. Vinogradov [35]. Making the above heuristics rigorous one obtains sums twisted by additive characters 1 λ q/N χ(λ)e q (aλ), (3) and the constant c in (1) which may be obtained by this method depends on how short sums of the form (3) may be estimated. For example, if for any integer a we have…”

Section: Introductionmentioning

confidence: 99%

An explicit Pólya-Vinogradov inequality via Partial Gaussian sums

Bordignon

Kerr

2020

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

In this paper we obtain a new fully explicit constant for the Pólya-Vinogradov inequality for squarefree modulus. Given a primitive character χ to squarefree modulus q, we prove the following upper boundwhere c = 1/(2π 2 ) + o(1) for even characters and c = 1/(4π) + o(1) for odd characters, with an explicit o(1) term. This improves a result of Frolenkov and Soundararajan for large q. We proceed via partial Gaussian sums rather than the usual Montgomery and Vaughan approach of exponential sums with multiplicative coefficients. This allows a power saving on the minor arcs rather than a factor of log q as in previous approaches and is an important factor for fully explicit bounds.

show abstract

“…1. Liu and Wang prove λ ≈ 6 for q > 987 in [5,Theorem 3] We can note, from [2,Theorem 1.3], that restricting the above results to odd characters we obtain a significantly better result, thus focusing on even characters will improve the overall result. The above results follow from the mean-value theorem, a lower bound for L (1, χ), obtained using the Class Number Formula, and an upper bound for |L ′ (σ, χ)|, with σ ∈ (β 0 , 1).…”

mentioning

confidence: 99%