Mean values of multiplicative functions on arithmetic progressions

1987 ◽  
Vol 42 (2) ◽  
pp. 674-684
Author(s):  
S. T. Tulyaganov
Keyword(s):  
Arithmetic Progressions ◽  
Multiplicative Functions ◽  
Mean Values

scholarly journals ON BINARY CORRELATIONS OF MULTIPLICATIVE FUNCTIONS

2018 ◽  
Vol 6 ◽  
Author(s):  
JONI TERÄVÄINEN
Keyword(s):  
Dirichlet Character ◽  
Arithmetic Progressions ◽  
Quadratic Polynomial ◽  
Multiplicative Functions ◽  
Real Character ◽  
Logarithmic Density ◽  
Mean Values ◽  
The Real ◽  
Smooth Numbers ◽  
Logarithmic Average

We study logarithmically averaged binary correlations of bounded multiplicative functions $g_{1}$ and $g_{2}$. A breakthrough on these correlations was made by Tao, who showed that the correlation average is negligibly small whenever $g_{1}$ or $g_{2}$ does not pretend to be any twisted Dirichlet character, in the sense of the pretentious distance for multiplicative functions. We consider a wider class of real-valued multiplicative functions $g_{j}$, namely those that are uniformly distributed in arithmetic progressions to fixed moduli. Under this assumption, we obtain a discorrelation estimate, showing that the correlation of $g_{1}$ and $g_{2}$ is asymptotic to the product of their mean values. We derive several applications, first showing that the numbers of large prime factors of $n$ and $n+1$ are independent of each other with respect to logarithmic density. Secondly, we prove a logarithmic version of the conjecture of Erdős and Pomerance on two consecutive smooth numbers. Thirdly, we show that if $Q$ is cube-free and belongs to the Burgess regime $Q\leqslant x^{4-\unicode[STIX]{x1D700}}$, the logarithmic average around $x$ of the real character $\unicode[STIX]{x1D712}\hspace{0.6em}({\rm mod}\hspace{0.2em}Q)$ over the values of a reducible quadratic polynomial is small.

Asymptotic Formulas for Mean Values of Multiplicative Functions

2020 ◽  
Author(s):  
Benjamin Logsdon
Keyword(s):  
Asymptotic Formulas ◽  
Multiplicative Functions ◽  
Mean Values

Multiplicative Functions on Arithmetic Progressions. VII: Large Moduli

2002 ◽  
Vol 66 (1) ◽  
pp. 14-28 ◽  
Author(s):  
P. D. T. A. Elliott
Keyword(s):  
Arithmetic Progressions ◽  
Multiplicative Functions

The mean values of multiplicative functions on shifted primes

1997 ◽  
Vol 37 (4) ◽  
pp. 443-451 ◽  
Author(s):  
G. Stepanauskas
Keyword(s):  
Multiplicative Functions ◽  
Mean Values ◽  
Shifted Primes ◽  
The Mean

Mean values of some multiplicative functions

2015 ◽  
Vol 97 (1-2) ◽  
pp. 111-123
Author(s):  
A. A. Sedunova
Keyword(s):  
Multiplicative Functions ◽  
Mean Values

scholarly journals SMOOTH‐SUPPORTED MULTIPLICATIVE FUNCTIONS IN ARITHMETIC PROGRESSIONS BEYOND THE ‐BARRIER

Mathematika ◽  
2017 ◽  
Vol 63 (3) ◽  
pp. 895-918 ◽  
Author(s):  
Sary Drappeau ◽  
Andrew Granville ◽  
Xuancheng Shao
Keyword(s):  
Arithmetic Progressions ◽  
Multiplicative Functions
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