scholarly journals Mean values for a class of arithmetic functions in short intervals

2019 ◽  
Vol 293 (1) ◽  
pp. 178-202
Author(s):  
Jie Wu ◽  
Qiang Wu
Keyword(s):  
Arithmetic Functions ◽  
Mean Values ◽  
Short Intervals

scholarly journals Mean value theorems for a class of density-like arithmetic functions

2020 ◽  
pp. 1-15
Author(s):  
Lucas Reis
Keyword(s):  
Finite Field ◽  
Arithmetic Function ◽  
Field Extension ◽  
Mean Value ◽  
Arithmetic Functions ◽  
Mean Value Theorem ◽  
Primitive Elements ◽  
Mean Values ◽  
Mean Value Theorems ◽  
Nonzero Mean

This paper provides a mean value theorem for arithmetic functions [Formula: see text] defined by [Formula: see text] where [Formula: see text] is an arithmetic function taking values in [Formula: see text] and satisfying some generic conditions. As an application of our main result, we prove that the density [Formula: see text] (respectively, [Formula: see text]) of normal (respectively, primitive) elements in the finite field extension [Formula: see text] of [Formula: see text] are arithmetic functions of (nonzero) mean values.

VIII. On the corrections to be applied to the monthly means of meteorological observations taken at any hour, to convert them into mean monthly values

1848 ◽  
Vol 138 ◽  
pp. 125-139 ◽  
Keyword(s):  
Monthly Means ◽  
Mean Values ◽  
Short Intervals ◽  
Perfect Knowledge ◽  
Royal Observatory ◽  
The Mean ◽  
The Difference ◽  
Meteorological Observations ◽  
Essential Service

One of the most useful results of observations made at short intervals during the day and night, and continued for several years, is the knowledge we thus obtain of the diurnal ranges of the different subjects of investigation, and consequently the difference between the mean values of each element, as deduced from observations at one or more hours daily, and the true mean for the period over which the observations are spread. At the Royal Observatory at Greenwich magnetical and meteorological observations have been taken since the year 1840, as is familiar to the Fellows of this Society. These have been published to the end of the year 1845. The whole of these observations have been made under my immediate superintendence, under the direction of the Astronomer Royal, and I believe that no observations have been made and reduced with greater care or regularity. As the person entrusted with the superintendence of these operations, I have a more perfect knowledge of them than any other person can have; I feel it therefore a duty to communicate their results from time to time, when the doing so promises to be of essential service in promoting the advancement of the subjects of investigation.

scholarly journals On the mean values and distributions of arithmetic functions

1981 ◽  
Vol 40 (1) ◽  
pp. 63-77 ◽  
Author(s):  
Gutti Jogesh Babu
Keyword(s):  
Arithmetic Functions ◽  
Mean Values ◽  
The Mean

On the Changes of Sign of a Certain Error Function

1951 ◽  
Vol 3 ◽  
pp. 375-385 ◽  
Author(s):  
Paul Erdös ◽  
Harold N. Shapiro
Keyword(s):  
Error Function ◽  
Arithmetic Functions ◽  
Mean Values ◽  
Analytic Methods ◽  
The Mean

Though much effort has been expended in studying the mean values of arithmetic functions there is one case which has not yielded a great deal either to elementary or analytic methods. The case to which we refer is that of estimating

scholarly journals On the asymptotic formulae for some multiplicative functions in short intervals

2015 ◽  
Vol 11 (05) ◽  
pp. 1571-1587 ◽  
Author(s):  
Alisa Sedunova
Keyword(s):  
Short Interval ◽  
Multiplicative Functions ◽  
Density Estimates ◽  
Mean Values ◽  
Short Intervals ◽  
Asymptotic Formulae ◽  
Free Region ◽  
Complex Integration ◽  
The Mean ◽  
Riemann Ζ Function

We are going to study the mean values of some multiplicative functions connected with the divisor function in short interval of summation. The asymptotics for such mean values will be proved. Considering instead of well-known multiplicative functions, their inverses lead to very weak results of application of standard methods of complex integration. In order to get better estimations, we propose another method which uses as its main tools the density estimates and zero-free region for Riemann ζ-function and Dirichlet L-functions.

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