An application of generalized mollifiers to the Riemann zeta-function. (English) Zbl 1429.11158
Summary: In this paper, we establish the asymptotic formula for the second moment of the Riemann zeta-function twisted by a \((\Lambda+1)\)-piece mollifier which is a generalization of the two-piece mollifier considered by
H. M. Bui et al. [Acta Arith. 150, No. 1, 35–64 (2011; Zbl 1250.11083)]. As an application, we obtain a lower bound for the proportion of critical zeros of the Riemann zeta-function.
MSC:
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
Citations:
Zbl 1250.11083References:
| [1] | H. M. Bui, J. B. Conrey and M. P. Young. More than 41 · Zbl 1250.11083 |
| [2] | J. B. Conrey. More than two fifths of the zeros of the Riemann zeta function are on the critical line. J. Reine Angew. Math. 399 (1989), 1-26. · Zbl 0668.10044 |
| [3] | J. M. Deshouillers and H. Iwaniec. Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70 (1982), 219-288. · Zbl 0502.10021 |
| [4] | J. M. Deshouillers and H. Iwaniec. Power mean values of the Riemann zeta function II. Acta Arith. 48 (1984), 305-312. · Zbl 0531.10041 |
| [5] | S. Feng. Zeros of the Riemann zeta function on the critical line. J. Number Theory 132(4), 511-542. · Zbl 1333.11086 |
| [6] | H. Iwaniec. Lectures on the Riemann Zeta Function (University Lecture Series, 62). American Mathematical Society, 2014. |
| [7] | . |
| [8] | . |
| [9] | N. Levinson. More than one third of the zeros of Riemann’s zeta function are on σ = 1/2. Adv. Math. 13 (1974), 383-436. · Zbl 0281.10017 |
| [10] | N. Robles, A. Roy and A. Zaharescu. Twisted second moments of the Riemann zeta-function and applications. J. Math. Anal. Appl. 434 (2016), 271-314. · Zbl 1383.11103 |
| [11] | A. Selberg. On the zeros of Riemann’s zeta-function. Skr. Norske Vid Akad. Oslo I. (1942), No. 10, 1-59. |
| [12] | E. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
