Functiones et Approximatio Commentarii Mathematici

Notes on low discriminants and the generalized Newman conjecture

Jeffrey Stopple

Full-text: Open access


Generalizing work of Polya, de Bruijn and Newman, we allow the backward heat equation to deform the zeros of quadratic Dirichlet $L$-functions. There is a real constant $\Lambda_{Kr}$ (generalizing the de Bruijn-Newman constant $\Lambda$) such that for time $t\ge\Lambda_{Kr}$ all such $L$-functions have all their zeros on the critical line; for time $t<\Lambda_{Kr}$ there exist zeros off the line. Under GRH, $\Lambda_{Kr}\le 0$; we make the complementary conjecture $0\le \Lambda_{Kr}$. Following the work of Csordas \emph{et. al}. on Lehmer pairs of Riemann zeros, we use low-lying zeros of quadratic Dirichlet $L$-functions to show that $-1.13\cdot 10^{-7}<\Lambda_{Kr}$. In the last section we develop a precise definition of a Low discriminant which is motivated by considerations of random matrix theory. The existence of infinitely many Low discriminants would imply $0\le \Lambda_{Kr}$.

Article information

Funct. Approx. Comment. Math., Volume 51, Number 1 (2014), 23-41.

First available in Project Euclid: 24 September 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M20: Real zeros of $L(s, \chi)$; results on $L(1, \chi)$
Secondary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 11M50: Relations with random matrices 11Y35: Analytic computations 11Y60: Evaluation of constants

generalized Riemann hypothesis de Bruijn-Newman constant backward heat equation Lehmer pair Low discriminant random matrix theory


Stopple, Jeffrey. Notes on low discriminants and the generalized Newman conjecture. Funct. Approx. Comment. Math. 51 (2014), no. 1, 23--41. doi:10.7169/facm/2014.51.1.2.

Export citation


  • N.G. de Bruijn, The roots of trigonometric integrals, Duke J. Math. 17 (1950), 197–226.
  • G. Csordas, W. Smith, R. Varga, Lehmer pairs of zeros, the de Bruijn-Newman constant, and the Riemann Hypothesis, Constructive Approximation 10 (1994), 107–129.
  • G. Csordas, A. Odlyzko, W. Smith, R. Varga, A new Lehmer pair of zeros and a new lower bound for the de Bruijn-Newman constant $\Lambda$, Electron. Trans. Numer. Anal. 1 (1993), 104–111.
  • N. Katz, P. Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy, AMS Colloquium Publications 45, 1999.
  • J. Keating, N. Snaith, Random matrix theory and $L$-functions at $s=1/2$, Comm. Math. Phys. 214 (2000), 91–110.
  • M.E. Low, Real zeros of the Dedekind zeta function of an imaginary quadratic field, Ph.D. thesis, University of Illinois Urbana-Champaign (1965)
  • H. Montgomery, P. Weinberger, Notes on small class numbers, Acta Arith. XXIV (1974), 529–542.
  • C.M. Newman, Fourier transforms with only real zeros, Proc. AMS, 61 (1976), 245–251.
  • A.M. Odlyzko, The $10^{20}$-th zero of the Riemann zeta function and $175$ million of its neighbors,
  • A.M. Odlyzko, An improved bound for the de Bruijn-Newman constant, Numerical Algorithms 25 (2000), 293–303.
  • G. Polya, Über trigonometrische Integrale mit nur reelen Nullstellen, J. für die reine und angewandte Mathematik 158 (1927), 6–18.
  • C.L. Siegel, On the zeros of Dirichlet $L$-functions, Annals of Mathematics 46 (1945) no. 3, 409–422.
  • J. Stopple, Computing $L$-functions with large conductor, Mathematics of Computation 76 (2007), 2051–2062.
  • –––– The quadratic character experiment, Experimental Mathematics, 18 (2009), 193–200.
  • P. van der Steen, On differential operators of infinite order, Ph.D. thesis, TU Delft, 1968.
  • M. Watkins, Class numbers of imaginary quadratic fields, Math. Comp. 73 (2003), 907–938.
  • ––––, Real zeros of real odd Dirichlet $L$-functions, Math. Comp. 73 (2003), 415–423.
  • P. Weinberger, On small zeros of Dirichlet L-functions, Math. Comp. 29 (1975), 319–328.