Dirichlet $L$-functions of quadratic characters of prime conductor at the central point

@article{Baluyot2021DirichletO,
  title={Dirichlet \$L\$-functions of quadratic characters of prime conductor at the central point},
  author={Siegfred Alan C. Baluyot and Kyle Pratt},
  journal={Journal of the European Mathematical Society},
  year={2021}
}
  • S. Baluyot, Kyle Pratt
  • Published 26 September 2018
  • Mathematics
  • Journal of the European Mathematical Society
We prove that more than nine percent of the central values $L(\frac{1}{2},\chi_p)$ are non-zero, where $p\equiv 1 \pmod{8}$ ranges over primes and $\chi_p$ is the real primitive Dirichlet character of conductor $p$. Previously, it was not known whether a positive proportion of these central values are non-zero. As a by-product, we obtain the order of magnitude of the second moment of $L(\frac{1}{2},\chi_p)$, and conditionally we obtain the order of magnitude of the third moment. Assuming the… Expand
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References

SHOWING 1-10 OF 40 REFERENCES
Mean Value Theorems for L-functions over Prime Polynomials for the Rational Function Field
The first and second moments are established for the family of quadratic Dirichlet $L$--functions over the rational function field at the central point $s=\tfrac{1}{2}$ where the character $\chi$ isExpand
CENTRAL VALUES OF DERIVATIVES OF DIRICHLET L-FUNCTIONS
Let $\mathscr{C}_{q}^{+}$ be the set of even, primitive Dirichlet characters (mod q). Using the mollifier method, we show that L(k)(½, χ) ≠ 0 for almost all the charactersExpand
Non-vanishing of Dirichlet L-functions in Galois orbits
A well known result of Iwaniec and Sarnak states that for at least one third of the primitive Dirichlet characters to a large modulus q, the associated L-functions do not vanish at the central point.Expand
Moments and distribution of central $$L$$L-values of quadratic twists of elliptic curves
We show that if one can compute a little more than a particular moment for some family of L-functions, then one has upper bounds of the conjectured order of magnitude for all smaller (positive, real)Expand
Simple zeros of the zeta function of a quadratic number field. I
Let K be a fixed quadratic extension of Q and write ζK(s) for the Dedekind zeta-function of K, where s = σ + it. It is wellknown, and easy to prove, that the number NK(T) of zeros of ζK(s) in theExpand
The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros
We describe a number of results and techniques concerning the non-vanishing of automorphic L-functions at s = ½. In particular we show that as N → ∞ at least 50% of the values L(½, f), with f varyingExpand
Explicit upper bound for the (analytic) rank of J0(q)
We refine the techniques of our previous paper [KM1] to prove that the average order of vanishing of L-functions of primitive automorphic forms of weight 2 and prime level q satisfiesExpand
On the Mean Value of L(1/2, χ ) FW Real Characters
Asymptotic formulae are derived for the sums ZL''(i,x), where k=l or 2 and x runs over all Dirichlet characters defined by Kronecker's symbol ^ with d being restricted to fundamental discriminants inExpand
Zeroes of zeta functions and symmetry
Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidenceExpand
Long mollifiers of the Riemann Zeta-function
The best current bounds for the proportion of zeros of ζ( s ) on the critical line are due to Conrey [C], using Levinson's method [Lev]. This method can also be used to detect simple zeros on theExpand
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