Moments of zeta and correlations of divisor-sums: I

We examine the calculation of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. Previously, this approach has proved unsuccessful in computing moments beyond the eighth, even heuristically. A careful analysis of the second and fourth moments illustrates the nature of the problem and enables us to identify the terms that are missed in the standard application of these methods.

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On the Size of an Expression in the Nyman–Beurling-Báez–Duarte Criterion for the Riemann Hypothesis

Canadian Mathematical Bulletin ◽

10.4153/cmb-2017-070-3 ◽

2018 ◽

Vol 61 (3) ◽

pp. 622-627

Author(s):

Helmut Maier ◽

Michael Th. Rassias

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Crucial Role ◽

Riemann Hypothesis ◽

Critical Line ◽

Nontrivial Zeros ◽

The Riemann Zeta Function ◽

Dirichlet Polynomials ◽

Riemann Zeta

AbstractA crucial role in the Nyman–Beurling–Báez-Duarte approach to the Riemann Hypothesis is played by the distancewhere the infimum is over all Dirichlet polynomialsof length N. In this paper we investigate under the assumption that the Riemann zeta function has four nontrivial zeros off the critical line.

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Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line

Journal de Théorie des Nombres de Bordeaux ◽

10.5802/jtnb.836 ◽

2013 ◽

Vol 25 (2) ◽

pp. 285-305 ◽

Cited By ~ 1

Author(s):

Thomas Christ ◽

Justas Kalpokas

Keyword(s):

Zeta Function ◽

Lower Bounds ◽

Riemann Zeta Function ◽

Critical Line ◽

The Riemann Zeta Function ◽

Riemann Zeta ◽

Discrete Moments ◽

Derivatives Of

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Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line

Communications on Pure and Applied Mathematics ◽

10.1002/cpa.21791 ◽

2018 ◽

Vol 72 (3) ◽

pp. 500-535 ◽

Cited By ~ 7

Author(s):

Louis-Pierre Arguin ◽

David Belius ◽

Paul Bourgade ◽

Maksym Radziwiłł ◽

Kannan Soundararajan

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Critical Line ◽

Short Interval ◽

The Riemann Zeta Function ◽

Riemann Zeta

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Small values of the Riemann zeta function on the critical line

Acta Arithmetica ◽

10.4064/aa169-3-1 ◽

2015 ◽

Vol 169 (3) ◽

pp. 201-220 ◽

Cited By ~ 2

Author(s):

Justas Kalpokas ◽

Paulius Šarka

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Critical Line ◽

The Riemann Zeta Function ◽

Riemann Zeta

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Null trajectories for the symmetrized Hurwitz zeta function

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2006.1763 ◽

2006 ◽

Vol 463 (2077) ◽

pp. 303-319 ◽

Cited By ~ 1

Author(s):

Ross C McPhedran ◽

Lindsay C Botten ◽

Nicolae-Alexandru P Nicorovici

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Critical Line ◽

Hurwitz Zeta Function ◽

Trigonometric Functions ◽

Asymptotic Results ◽

End Points ◽

The Riemann Zeta Function ◽

Riemann Zeta ◽

Pass Through

We consider the Hurwitz zeta function ζ ( s , a ) and develop asymptotic results for a = p / q , with q large, and, in particular, for p / q tending to 1/2. We also study the properties of lines along which the symmetrized parts of ζ ( s , a ), ζ + ( s , a ) and ζ − ( s , a ) are zero. We find that these lines may be grouped into four families, with the start and end points for each family being simply characterized. At values of a =1/2, 2/3 and 3/4, the curves pass through points which may also be characterized, in terms of zeros of the Riemann zeta function, or the Dirichlet functions L −3 ( s ) and L −4 ( s ), or of simple trigonometric functions. Consideration of these trajectories enables us to relate the densities of zeros of L −3 ( s ) and L −4 ( s ) to that of ζ ( s ) on the critical line.

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