Some numerical results on Fekete polynomials
HTML articles powered by AMS MathViewer
- by Paul T. Bateman, George B. Purdy and Samuel S. Wagstaff PDF
- Math. Comp. 29 (1975), 7-23 Request permission
Abstract:
It is known that if $\chi$ is a real residue character modulo k with $\chi (p) = - 1$ for the first five primes p, then the corresponding Fekete polynomial $\Sigma _{n = 1}^k\;\chi (n){x^n}$ changes sign on (0, 1). In this paper it is shown that the condition that $\chi (p)$ be -1 for the first four primes p is not sufficient to guarantee such a sign change. More specifically, if $\chi$ is the real nonprincipal character modulo either 1277 or 1973, it is shown that the corresponding Fekete polynomial is positive throughout (0, 1) even though $\chi (2) = \chi (3) = \chi (5) = \chi (7) = - 1$.References
-
S. CHOWLA, "Note on Dirichlet’s L-functions," Acta Arith., v. 1, 1935, pp. 113-114.
M. FEKETE & G. PÓLYA, "Über ein Problem von Laguerre," Rend. Circ. Mat. Palermo, v. 34, 1912, pp. 89-120.
GOTTFRIED GRIMM, Über die reellen Nullstellen Dirichlet’scher L-Reihen, Dissertation, E. T. H., Zurich, 1932.
HWA S. HAHN, "On a conjecture of Fekete," J. Korean Math. Soc., v. 5, 1968, pp. 13-16.
H. HEILBRONN, "On real characters," Acta Arith., v. 2, 1937, pp. 212-213.
- M. E. Low, Real zeros of the Dedekind zeta function of an imaginary quadratic field, Acta Arith 14 (1967/1968), 117–140. MR 0236127, DOI 10.4064/aa-14-2-117-140 GEORG PÓLYA, "Verschiedene Bemerkungen zur Zahlentheorie," Jber. Deutsch. Math. Verein., v. 28, 1919, pp. 31-40. G. PÓLYA & G. SZEGÖ, Aufgaben und Lehrsätze aus der Analysis. Vol. 2, Springer, Berlin, 1925.
- J. Barkley Rosser, Real roots of real Dirichlet $L$-series, J. Research Nat. Bur. Standards 45 (1950), 505–514. MR 0041161
- J. H. Wilkinson, Rounding errors in algebraic processes, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0161456
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Math. Comp. 29 (1975), 7-23
- MSC: Primary 10-04; Secondary 10H10
- DOI: https://doi.org/10.1090/S0025-5718-1975-0480293-9
- MathSciNet review: 0480293