Recent Progress on Topics of Ramanujan Sums and Cotangent Sums Associated with the Riemann Hypothesis

RAMANUJAN SUMS AND THE BURGESS ZETA FUNCTION

International Journal of Number Theory ◽

10.1142/s1793042112501187 ◽

2012 ◽

Vol 08 (08) ◽

pp. 2069-2092 ◽

Cited By ~ 7

Author(s):

EMRE ALKAN

Keyword(s):

Zeta Function ◽

Infinite Series ◽

Riemann Hypothesis ◽

Mellin Transform ◽

Bernoulli Numbers ◽

Weighted Averages ◽

Summatory Function ◽

Ramanujan Sums

The Mellin transform of a summatory function involving weighted averages of Ramanujan sums is obtained in terms of Bernoulli numbers and values of the Burgess zeta function. The possible singularity of the Burgess zeta function at s = 1 is then shown to be equivalent to the evaluation of a certain infinite series involving such weighted averages. Bounds on the size of the tail of these series are given and specific bounds are shown to be equivalent to the Riemann hypothesis.

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Electron-Mirror Microscopic Aspects of Ferroelectric Domains of BaTiO3 And Ca2Sr (C2H5CO2)6

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100069387 ◽

1970 ◽

Vol 28 ◽

pp. 478-479

Author(s):

Teruo Someya ◽

Jinzo Kobayashi

Keyword(s):

Surface Layer ◽

Electric Fields ◽

Quantitative Study ◽

Recent Progress ◽

Ferroelectric Domain ◽

Resolving Power ◽

Surface Pattern ◽

Crystal Surfaces ◽

One Dimensional ◽

Ferroelectric Domains

Recent progress in the electron-mirror microscopy (EMM), e.g., an improvement of its resolving power together with an increase of the magnification makes it useful for investigating the ferroelectric domain physics. English has recently observed the domain texture in the surface layer of BaTiO3. The present authors ) have developed a theory by which one can evaluate small one-dimensional electric fields and/or topographic step heights in the crystal surfaces from their EMM pictures. This theory was applied to a quantitative study of the surface pattern of BaTiO3).

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The Nature of Oxide Surfaces: Topographic and Electronic Structure

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100150666 ◽

1993 ◽

Vol 51 ◽

pp. 966-967

Author(s):

Dawn A. Bonnell ◽

Yong Liang

Keyword(s):

Transition Metal Oxides ◽

Electronic Structures ◽

Recent Progress ◽

Spatial Localization ◽

Level Of Detail ◽

Scanning Tunneling ◽

Oxide Surfaces ◽

Tunneling Microscopy ◽

Considerable Effort ◽

Complete Understanding

Recent progress in the application of scanning tunneling microscopy (STM) and tunneling spectroscopy (STS) to oxide surfaces has allowed issues of image formation mechanism and spatial resolution limitations to be addressed. As the STM analyses of oxide surfaces continues, it is becoming clear that the geometric and electronic structures of these surfaces are intrinsically complex. Since STM requires conductivity, the oxides in question are transition metal oxides that accommodate aliovalent dopants or nonstoichiometry to produce mobile carriers. To date, considerable effort has been directed toward probing the structures and reactivities of ZnO polar and nonpolar surfaces, TiO2 (110) and (001) surfaces and the SrTiO3 (001) surface, with a view towards integrating these results with the vast amount of previous surface analysis (LEED and photoemission) to build a more complete understanding of these surfaces. However, the spatial localization of the STM/STS provides a level of detail that leads to conclusions somewhat different from those made earlier.

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