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Small zeros of quadratic L-functions

  • Ali E. Özlük (a1) and C. Snyder (a1)
Abstract

We study the distribution of the imaginary parts of zeros near the real axis of quadratic L-functions. More precisely, let K(s) be chosen so that |K(1/2 ± it)| is rapidly decreasing as t increases. We investigate the asymptotic behaviour of

as D → ∞. Here denotes the sum over the non-trivial zeros p = 1/2 + of the Dirichlet L-function L(s, χd), and χd = () is the Kronecker symbol. The outer sum is over all fundamental discriminants d that are in absolute value ≤ D. Assuming the Generalized Riemann Hypothesis, we show that for

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References
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[1]Ayoub R.G., An introduction to the analytic theory of numbers 10 (Math. Surveys, Providence, R.I., 1963).
[2]Bellman R., Analytic number theory, an introduction (Benjamin Cummings Pub. Co., 1980).
[3]Bentz H.-J. and Pintz J., ‘Quadratic residues and the distribution of prime numbers’, Monatshefte für Mathematik 90 (1982), 91100.
[4]Bentz H.-J., ‘Discrepancies in the distribution of prime numbers’, J. Number Theory 15 (1982), 252274.
[5]Chebysev P.L., ‘Lettres de M. le professeur Tchebychev á M. Fuss sur un nouveau théorème relative aux nombres premiers contenus dans les formes 4n + 1 et 4n + 3’, Bull. Classe Phys. de l'Acad. Imp. Sciences St. Petersburg 11 (1853), 208.
[6]Davenport H., Multiplicative number theory, 2nd ed. (Springer-Verlag, Berlin, Heidelberg, New York, 1980).
[7]Ellison F. and Ellison W., Prime numbers (John Wiley, New York, 1985).
[8]Montgomery H.L. and Weinberger P.J., ‘Notes on small class numbers’, Acta Arith. 24 (1974), 529542.
[9]Shanks D., ‘Quadratic residues and the distribution of primes’, Math. Tables and other Aids to Comp. 13 (1959), 272284.
[10]Turán P., On a new method of analysis and its applications (John Wiley and Sons, Inc., New York, 1984).
[11]Weinberger P.J., ‘On small zeros of Dirichlet L-functions’, Math. Comp. 29 (1975), 319328.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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