Joint value-distribution of Dirichlet series associated with periodic arithmetical functions

2020 ◽  
Vol 62 (1) ◽  
pp. 63-79
Author(s):  
Philipp Muth ◽  
Jörn Steuding
Keyword(s):  
Dirichlet Series ◽  
Value Distribution ◽  
Arithmetical Functions

On the Value Distribution of Shifts of Universal Dirichlet Series

2006 ◽  
Vol 147 (4) ◽  
pp. 309-317 ◽  
Author(s):  
Jerzy Kaczorowski ◽  
Antanas Laurinčikas ◽  
Jörn Steuding
Keyword(s):  
Dirichlet Series ◽  
Value Distribution

scholarly journals On Ramanujan and Dirichlet series with Euler products

1984 ◽  
Vol 25 (2) ◽  
pp. 203-206 ◽  
Author(s):  
S. Raghavan
Keyword(s):  
Modular Form ◽  
Dirichlet Series ◽  
Modular Forms ◽  
Hecke Operators ◽  
Euler Product ◽  
Arithmetical Functions ◽  
In Trans ◽  
Euler Products ◽  
Unpublished Manuscripts ◽  
Profound Insight

In his unpublished manuscripts (referred to by Birch [1] as Fragment V, pp. 247–249), Ramanujan [3] gave a whole list of assertions about various (transforms of) modular forms possessing naturally associated Euler products, in more or less the spirit of his extremely beautiful paper entitled “On certain arithmetical functions” (in Trans. Camb. Phil. Soc. 22 (1916)). It is simply amazing how Ramanujan could write down (with an ostensibly profound insight) a basis of eigenfunctions (of Hecke operators) whose associated Dirichlet series have Euler products, anticipating by two decades the famous work of Hecke and Petersson. That he had further realized, in the event of a modular form f not corresponding to an Euler product, the possibility of restoring the Euler product property to a suitable linear combination of modular forms of the same type as f, is evidently fantastic.

Value distribution of general Dirichlet series. VII

2006 ◽  
Vol 46 (2) ◽  
pp. 155-162 ◽  
Author(s):  
J. Genys ◽  
A. Laurinčikas ◽  
D. Šiaučiūnas
Keyword(s):  
Dirichlet Series ◽  
Value Distribution ◽  
General Dirichlet Series

Value Distribution of General Dirichlet series. VI

2005 ◽  
Vol 10 (3) ◽  
pp. 235-246
Author(s):  
A. Laurinčikas
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Dirichlet Series ◽  
Complex Plane ◽  
Value Distribution ◽  
Probability Measures ◽  
General Dirichlet Series ◽  
New Class ◽  
Convergence Of Probability Measures

In the paper a limit theorem in the sense of weak convergence of probability measures on the complex plane for a new class of general Dirichlet series is obtained.

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.

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