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.

scholarly journals On the Mean Values of Certain Character Sums

2013 ◽  
Vol 2013 ◽  
pp. 1-19
Author(s):  
Zhefeng Xu ◽  
Huaning Liu
Keyword(s):  
Character Sums ◽  
Fourth Power ◽  
Power Mean ◽  
Mean Values ◽  
Asymptotic Formulae ◽  
Fourth Power Mean ◽  
The Mean

Letq≥5be an odd number. In this paper, we study the fourth power mean of certain character sums∑χmodq,χ-1=-1*∑1≤a≤q/4aχa4and∑χmodq,χ-1=1*∑1≤a≤q/4aχa4, where∑‍*denotes the summation over primitive characters moduloq, and give some asymptotic formulae.

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.

Multiplicative Functions in Short Intervals

1987 ◽  
Vol 39 (3) ◽  
pp. 646-672 ◽  
Author(s):  
Adolf Hildebrand
Keyword(s):  
Number Theory ◽  
Mean Value ◽  
Considerable Progress ◽  
Central Problem ◽  
Partial Sums ◽  
Multiplicative Functions ◽  
Short Intervals ◽  
Probabilistic Number ◽  
The Mean ◽  
Image Position

A central problem in probabilistic number theory is to evaluate asymptotically the partial sumsof multiplicative functions f and, in particular, to find conditions for the existence of the “mean value”1.1In the last two decades considerable progress has been made on this problem, and the results obtained are very satisfactory.

scholarly journals On mean value results for the Riemann zeta-function in short intervals.

2009 ◽  
Vol Volume 32 ◽  
Author(s):  
Aleksandar Ivić
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Mean Value ◽  
Upper And Lower Bounds ◽  
Mean Values ◽  
Short Intervals ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta ◽  
The Mean

International audience We discuss the mean values of the Riemann zeta-function $\zeta(s)$, and analyze upper and lower bounds for $$\int_T^{T+H} \vert\zeta(\frac{1}{2}+it)\vert^{2k}\,dt~~~~~~(k\in\mathbb{N}~{\rm fixed,}~1<\!\!< H \leq T).$$ In particular, the author's new upper bound for the above integral under the Riemann hypothesis is presented.

scholarly journals Mean Values of Certain Multiplicative Functions and Artin's Conjecture on Primitive Roots

2014 ◽  
Vol Volume 37 ◽  
Author(s):  
Sankar Sitaraman
Keyword(s):  
Dirichlet Series ◽  
Mean Value ◽  
Multiplicative Functions ◽  
New Approach ◽  
Mean Values ◽  
Artin's Conjecture ◽  
International Audience ◽  
The Mean ◽  
Primitive Roots

International audience We discuss how one could study asymptotics of cyclotomic quantities via the mean values of certain multiplicative functions and their Dirichlet series using a theorem of Delange. We show how this could provide a new approach to Artin's conjecture on primitive roots. We focus on whether a fixed prime has a certain order modulo infinitely many other primes. We also give an estimate for the mean value of one such Dirichlet series.

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