A further note on Ramanujan's arithmetical function τ(n)

1938 ◽  
Vol 34 (3) ◽  
pp. 309-315 ◽  
Author(s):  
G. H. Hardy
Keyword(s):  
Arithmetical Function ◽  
Image Position

This note is a sequel to two in earlier volumes of the Proceedings, the first by myself and the second by Wilton†.Suppose thatfor |z| <1; that x > 0; thatforr ≥ 0; and thatwhere τ(x) is to mean 0 if x is not an integer. Thuswhere the dash shows that the last term is to be halved when x is an integer;forr > 1; and

On Ramanujan's Arithmetical Function Σr, s(n)

1929 ◽  
Vol 25 (3) ◽  
pp. 255-264 ◽  
Author(s):  
J. R. Wilton
Keyword(s):  
Error Term ◽  
Arithmetical Function ◽  
Arithmetical Functions ◽  
Image Position ◽  
The Right ◽  
Positive Integers

Let σs(n) denote the sum of the sth powers of the divisors of n,and let , where ζ(s) is the Riemann ζfunction. Ramanujan, in his paper “On certain arithmetical functions*”, proves that, ifandthen , whenever r and s are odd positive integers. He conjectures that the error term on the right of (1·1) is of the formfor every ε > 0, and that it is not of the form . He further conjectures thatfor all positive values of r and s; and this conjecture has recently been proved to be correct.

On the order of magnitude of Ramanujan's arithmetical function τ(n)

1951 ◽  
Vol 47 (4) ◽  
pp. 668-678 ◽  
Author(s):  
W. B. Pennington
Keyword(s):  
Asymptotic Formula ◽  
Arithmetical Function ◽  
Modular Functions ◽  
Arithmetical Functions ◽  
Order Of Magnitude ◽  
Image Position ◽  
Ramanujan’S Formula

1. In his paper ‘On certain arithmetical functions' Ramanujan (23) considers the function τ(n) defined by the expansionThis function appears in the discussion of an asymptotic formula for the functionand also in Ramanujan's formula for the number of representations of an integer as the sum of 24 squares. It is also of interest as the coefficient in the expansion of g(z), which plays an important part in the theory of modular functions.

A Divisor Problem for Values of Polynomials

1992 ◽  
Vol 35 (1) ◽  
pp. 108-115
Author(s):  
Armel Mercier ◽  
Werner Georg Nowak
Keyword(s):  
Exponential Sums ◽  
Modern Method ◽  
Divisor Problem ◽  
Arithmetical Function ◽  
Asymptotic Result ◽  
Average Order ◽  
Equal Degree ◽  
Image Position ◽  
Classical Divisor ◽  
Hardy Littlewood Method

AbstractIn this article we investigate the average order of the arithmetical functionwhere p1(t), p2(t) are polynomials in Z [t], of equal degree, positive and increasing for t ≥ 1. Using the modern method for the estimation of exponential sums ("Discrete Hardy-Littlewood Method"), we establish an asymptotic result which is as sharp as the best one known for the classical divisor problem.

A New Method in Arithmetical Functions and Contour Integration

1973 ◽  
Vol 16 (3) ◽  
pp. 381-387 ◽  
Author(s):  
Bruce C. Berndt
Keyword(s):  
Meromorphic Function ◽  
Closed Form ◽  
Meromorphic Functions ◽  
Arithmetical Function ◽  
Contour Integration ◽  
Partial Answer ◽  
Arithmetical Functions ◽  
Classical Technique ◽  
Limited Class ◽  
Image Position

If f is a suitable meromorphic function, then by a classical technique in the calculus of residues, one can evaluate in closed form series of the form,Suppose that a(n) is an arithmetical function. It is natural to ask whether or not one can evaluate by contour integration(1.1)where f belongs to a suitable class of meromorphic functions. We shall give here only a partial answer for a very limited class of arithmetical functions.

A note on Ramanujan's arithmetical function τ(n)

1929 ◽  
Vol 25 (2) ◽  
pp. 121-129 ◽  
Author(s):  
J. R. Wilton
Keyword(s):  
Dirichlet Series ◽  
Arithmetical Function ◽  
Arithmetical Functions ◽  
Image Position

The function τ (n), defined by the equationis considered by Ramanujan* in his memoir “On certain arithmetical functions”The associated Diriċhlet seriesconverges when σ = > σ0, for a sufficiently large positive σ0.

Note on Ramanujan's arithmetical function τ (n)

1927 ◽  
Vol 23 (6) ◽  
pp. 675-680 ◽  
Author(s):  
G. H. Hardy
Keyword(s):  
Arithmetical Function ◽  
Arithmetical Functions ◽  
Image Position

In his remarkable memoir ‘On certain arithmetical functions’* Ramanujan considers, among other functions of much interest, the. function τ(n) defined byThis function is important in the theory of the representation of a number as a sum of 24 squares. In factwhere r24 (n) is the number of representations;where σs(n) is the sum of the 8th powers of the divisors of n, and the sum of those of its odd divisors; and

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

A Magnetic Spectrometer for a Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 436-437
Author(s):  
A. Kosiara ◽  
J. W. Wiggins ◽  
M. Beer
Keyword(s):  
Scanning Transmission Electron Microscope ◽  
Magnetic Spectrometer ◽  
Fringe Field ◽  
Curvature Radius ◽  
Magnetic Pole ◽  
Quadrupole Field ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Under Construction ◽  
Image Position

A magnetic spectrometer to be attached to the Johns Hopkins S. T. E. M. is under construction. Its main purpose will be to investigate electron interactions with biological molecules in the energy range of 40 KeV to 100 KeV. The spectrometer is of the type described by Kerwin and by Crewe Its magnetic pole boundary is given by the equationwhere R is the electron curvature radius. In our case, R = 15 cm. The electron beam will be deflected by an angle of 90°. The distance between the electron source and the pole boundary will be 30 cm. A linear fringe field will be generated by a quadrupole field arrangement. This is accomplished by a grounded mirror plate and a 45° taper of the magnetic pole.

Optimizing conditions for x-ray microchemical analysis in analytical electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 548-549 ◽  
Author(s):  
N. J. Zaluzec
Keyword(s):  
Solid Angle ◽  
Analytical Electron Microscopy ◽  
Incident Beam ◽  
Thin Foil ◽  
Experimental Conditions ◽  
X Ray ◽  
Microchemical Analysis ◽  
Analytical Electron ◽  
Image Position ◽  
Incident Beam Energy

The ultimate sensitivity of microchemical analysis using x-ray emission rests in selecting those experimental conditions which will maximize the measured peak-to-background (P/B) ratio. This paper presents the results of calculations aimed at determining the influence of incident beam energy, detector/specimen geometry and specimen composition on the P/B ratio for ideally thin samples (i.e., the effects of scattering and absorption are considered negligible). As such it is assumed that the complications resulting from system peaks, bremsstrahlung fluorescence, electron tails and specimen contamination have been eliminated and that one needs only to consider the physics of the generation/emission process.The number of characteristic x-ray photons (Ip) emitted from a thin foil of thickness dt into the solid angle dΩ is given by the well-known equation

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