Moments and zero density estimates for Dirichlet L-functions

In this paper, we estimate the density of zeros of primitive Dirichlet L-functions in the half-plane ℜ(s) > 1/2, under the assumption of a plausible conjecture on high moments of Dirichlet L-functions on the critical line. We conditionally improve the results of Huxley [9], Jutila [13], Heath-Brown [5, 6] and Bourgain [1]

PDF

___

[1] Bourgain J, On large values estimates for Dirichlet polynomials and the density hypothesis for the Riemann zeta function, IMRN International Mathematics Research Notes 2000; No. 3
[2] Conrey B, Iwaniec H, Soundararajan K, The sixth power moment of Dirichlet L-functions, Geometric and Functional Analysis 2012; Vol. 22 : Issue 5: 1257-1288
[3] Heath-Brown DR, The twelfth power moment of the Riemann zeta-function, Quarterly Journal of Mathematics (Oxford) 1978; 29: 443-462
[4] Heath-Brown DR, Zero density estimates for the Riemann zeta-function and Dirichlet L-functions, Journal of the London Mathematical Society 1979; (2) : 19: No. 2: 221-232
[5] Heath-Brown DR, The density of zeros of Dirichlet’s L-functions, Canadian Journal of Mathematics 1979; 31: No. 2: 231-240
[6] Heath-Brown DR, A large values estimate for Dirichlet polynomials, Journal of the London Mathematical Society 1979; (2): 20: No.1: 8-18
[7] Harper AJ, Sharp conditional bounds for moments of the Riemann zeta function, preprint 2013; http://arxiv.org/abs/1305.4618
[8] Huxley MN, On the difference between consecutive primes, Inventiones mathematicae 1972; 15: 164-170
[9] Huxley MN, An imperfect hybrid zero-density theorem, Journal of the London Mathematical Society 1976; (2): 13: No.1: 53-56
[10] Ingham AE, Mean-value theorems in the theory of the Riemann zeta-function, Proceedings of the London Mathematical Society 1928; Volume s2-27: Issue 1: 273-300
[11] Ivić A, A note on the zero-density estimates for the zeta function, Archiv der Mathematik (Basel) 1979/80; 33: No. 2: 155-164
[12] Ivić A, The Riemann zeta-function, Theory and Applications, Dover Publications Mineola, New York 1985
[13] Jutila M, Zero density estimates for L-functions, Acta Arithmetica 1977; 32: 55-62
[14] Montgomery HL, Mean and large values of Dirichlet polynomials, Inventiones mathematicae 1969; 8: 334-345
[15] Montgomery HL, Topics in multiplicative number theory Springer, Berlin 1971
[16] Ramachandra K, Some new density estimates for the zeros of the Riemann zeta-function, Annales Academiæ Scientiarum Fennicæ 1975; Series A I Math. 1: No. 1: 177-182
[17] Soundararajan K, Moments of the Riemann zeta function, Annals of Mathematics (2) 2009; 170: (2): 981-993
[18] Zaccagnini A, A conditional density theorem for the zeros of the Riemann zeta-function, Acta Arithmetica 2000; 93: No. 3: 293-301