A New Method in Arithmetical Functions and Contour Integration

1973 ◽  
Vol 16 (3) ◽  
pp. 381-387 ◽  
Author(s):  
Bruce C. Berndt
Keyword(s):  
Meromorphic Function ◽  
Closed Form ◽  
Meromorphic Functions ◽  
Arithmetical Function ◽  
Contour Integration ◽  
Partial Answer ◽  
Arithmetical Functions ◽  
Classical Technique ◽  
Limited Class ◽  
Image Position

If f is a suitable meromorphic function, then by a classical technique in the calculus of residues, one can evaluate in closed form series of the form,Suppose that a(n) is an arithmetical function. It is natural to ask whether or not one can evaluate by contour integration(1.1)where f belongs to a suitable class of meromorphic functions. We shall give here only a partial answer for a very limited class of arithmetical functions.

scholarly journals On Berndt's Method in Arithmetical Functions and Contour Integration

1979 ◽  
Vol 22 (2) ◽  
pp. 177-185 ◽  
Author(s):  
P. V. Krishnaiah ◽  
R. Sita Rama Chandra Rao
Keyword(s):  
Meromorphic Function ◽  
Closed Form ◽  
Contour Integration ◽  
Arithmetical Functions ◽  
Classical Technique ◽  
Image Position

If f is a suitable meromorphic function then, by a classical technique in the calculus of residues, one can evaluate in closed form series of the form1.1

Meromorphic functions with one deficient value

1969 ◽  
Vol 10 (3-4) ◽  
pp. 355-358 ◽  
Author(s):  
S. M. Shah
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Zero Order ◽  
Regularity Conditions ◽  
Deficient Value ◽  
Exceptional Value ◽  
Image Position

Let f(z) be a meromorphic function and write Here N(r, a) and T(r, f) have their usual meanings (see [4], [5]) and 0 ≧ |a| ≧ ∞. If δ(a, f) > 0 then a is said to be an exceptional (or deficient) value in the sense of Nevanlinna (N.e.v.), and if Δ(a, f) > 0 then a is said to be an exceptional value in the sense of Varliron (V.e.v.). The Weierstrass p(z) function has no exceptional value N or V. Functions of zero order can have atmost one N.e.v. [4, p. 114], but may have more than one V.e.v. (see [6], [8]). In this note we consider functions satisfying some regularity conditions and having one and only one exceptional value V.

On Ramanujan's Arithmetical Function Σr, s(n)

1929 ◽  
Vol 25 (3) ◽  
pp. 255-264 ◽  
Author(s):  
J. R. Wilton
Keyword(s):  
Error Term ◽  
Arithmetical Function ◽  
Arithmetical Functions ◽  
Image Position ◽  
The Right ◽  
Positive Integers

Let σs(n) denote the sum of the sth powers of the divisors of n,and let , where ζ(s) is the Riemann ζfunction. Ramanujan, in his paper “On certain arithmetical functions*”, proves that, ifandthen , whenever r and s are odd positive integers. He conjectures that the error term on the right of (1·1) is of the formfor every ε > 0, and that it is not of the form . He further conjectures thatfor all positive values of r and s; and this conjecture has recently been proved to be correct.

Sharing sets of q-difference of meromorphic functions

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
Xiaoguang Qi ◽  
Lianzhong Yang
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Zero Order ◽  
Partial Answer ◽  
Uniqueness Results

AbstractThis paper is devoted to proving some uniqueness results for meromorphic functions f(z) share sets with f(qz). We give a partial answer to a question of Gross concerning a zero-order meromorphic function f(z) and its q-difference f(qz).

On the order of magnitude of Ramanujan's arithmetical function τ(n)

1951 ◽  
Vol 47 (4) ◽  
pp. 668-678 ◽  
Author(s):  
W. B. Pennington
Keyword(s):  
Asymptotic Formula ◽  
Arithmetical Function ◽  
Modular Functions ◽  
Arithmetical Functions ◽  
Order Of Magnitude ◽  
Image Position ◽  
Ramanujan’S Formula

1. In his paper ‘On certain arithmetical functions' Ramanujan (23) considers the function τ(n) defined by the expansionThis function appears in the discussion of an asymptotic formula for the functionand also in Ramanujan's formula for the number of representations of an integer as the sum of 24 squares. It is also of interest as the coefficient in the expansion of g(z), which plays an important part in the theory of modular functions.

A Quantitative Estimate on Fixed-Points of Composite Meromorphic Functions

1995 ◽  
Vol 38 (4) ◽  
pp. 490-495 ◽  
Author(s):  
Jian-Hua Zheng
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Fixed Points ◽  
Lower Order ◽  
Quantitative Estimate ◽  
Meromorphic Functions ◽  
Transcendental Entire Function ◽  
Logarithmic Density ◽  
Image Position ◽  
Finite Lower Order

AbstractLet ƒ(z) be a transcendental meromorphic function of finite order, g(z) a transcendental entire function of finite lower order and let α(z) be a non-constant meromorphic function with T(r, α) = S(r,g). As an extension of the main result of [7], we prove thatwhere J has a positive lower logarithmic density.

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.

Note on Ramanujan's arithmetical function τ (n)

1927 ◽  
Vol 23 (6) ◽  
pp. 675-680 ◽  
Author(s):  
G. H. Hardy
Keyword(s):  
Arithmetical Function ◽  
Arithmetical Functions ◽  
Image Position

In his remarkable memoir ‘On certain arithmetical functions’* Ramanujan considers, among other functions of much interest, the. function τ(n) defined byThis function is important in the theory of the representation of a number as a sum of 24 squares. In factwhere r24 (n) is the number of representations;where σs(n) is the sum of the 8th powers of the divisors of n, and the sum of those of its odd divisors; and

scholarly journals Meromorphic or Holomorphic Completion of a Reinhardt Domain

1970 ◽  
Vol 38 ◽  
pp. 1-12 ◽  
Author(s):  
Eiichi Sakai
Keyword(s):  
Meromorphic Function ◽  
Power Series ◽  
Meromorphic Functions ◽  
Integral Formula ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Reinhardt Domain ◽  
Continuity Theorem ◽  
Cauchy’S Integral Formula ◽  
Long Time

In the theory of functions of several complex variables, the problem about the continuation of meromorphic functions has not been much investigated for a long time in spite of its importance except the deeper result of the continuity theorem due to E. E. Levi [4] and H. Kneser [3], The difficulty of its investigation is based on the following reasons: we can not use the tools of not only Cauchy’s integral formula but also the power series and there are indetermination points for the meromorphic function of many variables different from one variable. Therefore we shall also follow the Levi and Kneser’s method and seek for the aspect of meromorphic completion of a Reinhardt domain in Cn.

scholarly journals On Properties of Differences Polynomials about Meromorphic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Jianming Qi ◽  
Jie Ding ◽  
Wenjun Yuan
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Special Class ◽  
Meromorphic Functions ◽  
Value Distribution ◽  
Class Difference ◽  
Difference Polynomial

We study the value distribution of a special class difference polynomial about finite order meromorphic function. Our methods of the proof are also different from ones in the previous results by Chen (2011), Liu and Laine (2010), and Liu and Yang (2009).

Sign in / Sign up

Export Citation Format

Share Document