A Mean-Value Theorem for Multiplicative Functions on the Set of Shifted Primes

1998 ◽  
pp. 153-165
Author(s):  
Karl-Heinz Indlekofer ◽  
Nikolai M. Timofeev
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Multiplicative Functions ◽  
Shifted Primes

Integral Limit Laws for Additive Functions

1973 ◽  
Vol 25 (1) ◽  
pp. 194-203
Author(s):  
J. Galambos
Keyword(s):  
Characteristic Function ◽  
Fourier Transforms ◽  
Mean Value ◽  
Characteristic Functions ◽  
Asymptotic Formulas ◽  
Mean Value Theorem ◽  
Multiplicative Functions ◽  
Additive Functions ◽  
Limit Laws ◽  
Integral Limit

In the present paper a general form of integral limit laws for additive functions is obtained. Our limit law contains Kubilius’ results [5] on his class H. In the proof we make use of characteristic functions (Fourier transforms), which reduces our problem to finding asymptotic formulas for sums of multiplicative functions. This requires an extension of previous results in order to enable us to take into consideration the parameter of the characteristic function in question. We call this extension a parametric mean value theorem for multiplicative functions and its proof is analytic on the line of [4].

On multiplicative functions with mean-value one on the set of shifted primes

2002 ◽  
Vol 78 (6) ◽  
pp. 454-458
Author(s):  
K.-H. Indlekofer ◽  
N. M. Timofeev
Keyword(s):  
Mean Value ◽  
Multiplicative Functions ◽  
Shifted Primes

scholarly journals Asymptotics for multilinear averages of multiplicative functions

2016 ◽  
Vol 161 (1) ◽  
pp. 87-101 ◽  
Author(s):  
NIKOS FRANTZIKINAKIS ◽  
BERNARD HOST
Keyword(s):  
Unit Disc ◽  
Convergence Result ◽  
Mean Value ◽  
Arithmetic Mean ◽  
Mean Value Theorem ◽  
Two Dimensional ◽  
Multiplicative Functions ◽  
Structural Result ◽  
Convergence Results ◽  
The Mean

AbstractA celebrated result of Halász describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions providing similar asymptotics, thus verifying a two dimensional variant of a conjecture of Elliott. As a consequence, we get several convergence results for such multilinear expressions, one of which generalises a well known convergence result of Wirsing. The key ingredients are a recent structural result for multiplicative functions with values on the unit disc proved by the authors and the mean value theorem of Halász.

A mean-value theorem for multiplicative functions

1980 ◽  
Vol 172 (3) ◽  
pp. 255-271 ◽  
Author(s):  
Karl-Heinz Indlekofer
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Multiplicative Functions

NONNEGATIVE MULTIPLICATIVE FUNCTIONS ON SIFTED SETS, AND THE SQUARE ROOTS OF −1 MODULO SHIFTED PRIMES

2019 ◽  
Vol 62 (1) ◽  
pp. 187-199
Author(s):  
PAUL POLLACK
Keyword(s):  
Arithmetic Progression ◽  
Multiplicative Function ◽  
Mean Value ◽  
Multiplicative Functions ◽  
Square Roots ◽  
Shifted Primes ◽  
The Mean

AbstractAn oft-cited result of Peter Shiu bounds the mean value of a nonnegative multiplicative function over a coprime arithmetic progression. We prove a variant where the arithmetic progression is replaced by a sifted set. As an application, we show that the normalized square roots of −1 (mod m) are equidistributed (mod 1) as m runs through the shifted primes q − 1.

scholarly journals On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1303
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Faraidun Kadir Hamasalh
Keyword(s):  
Time Scale ◽  
Initial Value Problems ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Discrete Fractional Calculus ◽  
Initial Value ◽  
Forward Difference ◽  
The Mean ◽  
Monotonicity Analysis ◽  
Monotonicity Results

Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) for each z∈Ma+h,h. Conversely, if y(a+υh) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) on Ma,h. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale hZ utilizing the monotonicity results.

scholarly journals On a hybrid version of the Vinogradov mean value theorem

2021 ◽  
Vol 163 (1) ◽  
pp. 1-17
Author(s):  
C. Chen ◽  
I. E. Shparlinski
Keyword(s):  
Mean Value ◽  
Mean Value Theorem
Sign in / Sign up

Export Citation Format

Share Document