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.

Fourier coefficients of cusp forms

1986 ◽  
Vol 100 (1) ◽  
pp. 5-29 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Modular Functions ◽  
Field Of Study ◽  
Important Paper ◽  
Present Survey ◽  
Arithmetical Functions ◽  
Survey Article ◽  
Order Of Magnitude

The object of this survey article is to trace the influence on the theory of modular forms of the ideas contained in L. J. Mordell's important paper ‘On Mr Ramanujan's empirical expansions of modular functions’, which appeared in October 1917 in this Society's Proceedings [32]. The equally important paper [42] by S. Ramanujan, ‘On certain arithmetical functions’, referred to in Mordell's title, was published in May 1916 in the same Society's older journal, the Transactions, which was regrettably suppressed in 1928, 107 years after its foundation. Ramanujan's paper was concerned not only with multiplicative properties of Fourier coefficients of modular forms, but also with their order of magnitude. Since subsequent papers on the latter subject have also appeared in the Proceedings, it seems appropriate to include further developments in this field of study in the present survey.

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.

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.

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 On the coefficients in the expansions of certain modular functions

1918 ◽  
Vol 95 (667) ◽  
pp. 144-155 ◽  
Keyword(s):  
Unit Circle ◽  
New Method ◽  
Sums Of Squares ◽  
Additional Property ◽  
Modular Functions ◽  
Natural Boundary ◽  
Arithmetical Functions ◽  
Order Of Magnitude ◽  
Restricted Region

1. A very large proportion of the most interesting arithmetical functions —of the functions, for example, which occur in the theory of partitions, the theory of the divisors of numbers, or the theory of the representation of numbers by sums of squares—occur as the coefficients in the expansions of elliptic modular functions in powers of the variable q = e π i τ . All of these functions have a restricted region of. existence, the unit circle | q | = 1 being a “ natural boundary” or line of essential singularities. The most important of them, such as the functions (ω 1 /π) 12 ∆ = q 2 {(1- q 2 ) (1- q 4 )...} 24 , (1, 1) ϑ 3 (0) = 1 + 2 q + 2 q 4 + 2 q 9 + ....., (1. 2) 12 (ω 1 /π) 4 g 2 = 1 + 240 (1 3 q 2 /1- q 2 + 2 3 q 4 /1- q 4 + ...), (1, 3) 216 (ω 1 /π) 6 g 3 = 1 - 504 (1 5 q 2 /1- q 2 + 2 5 q 4 /1- q 4 + ...), (1, 4) are regular inside the unit circle ; and many, such as the functions (1, 1) and (1, 2), have the additional property of having no zeros inside the circle, so that their reciprocals are also regular. In a series of recent papers we have applied a new method to the study of these arithmetical functions. Our aim has been to express them as series which exhibit explicitly their order of magnitude, and the genesis of their irregular variations as n increases. We find, for example, for p ( n ) the number of unrestricted partitions of n ,and for r s ( n ), the number of repre­sentations of n as the sum of an even number s of squares, the series

Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions

1939 ◽  
Vol 35 (3) ◽  
pp. 351-356 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Asymptotic Formula ◽  
Modular Forms ◽  
Arithmetical Functions ◽  
Image Position

In this and the succeeding paper I solve two problems suggested by Prof. Hardy, namely (1) that of proving that Ramanujan's functionhas no zeros on the line and (2) that of finding an asymptotic formulawhere A is a constant. I also prove similar results concerning the coefficients of general modular forms. I am indebted to Prof. Hardy and Mr Ingham for various suggestions, and in particular to Mr Ingham's paper, “A note on Riemann's ζ-function and Dirichlet's L-functions”.

scholarly journals The Distribution of Heights of Binary Trees and Other Simple Trees

1993 ◽  
Vol 2 (2) ◽  
pp. 145-156 ◽  
Author(s):  
Philippe Flajolet ◽  
Zhicheng Gao ◽  
Andrew Odlyzko ◽  
Bruce Richmond
Keyword(s):  
Asymptotic Formula ◽  
Positive Constant ◽  
Binary Trees ◽  
Asymptotic Results ◽  
Image Position

The number, , of rooted plane binary trees of height ≤ h with n internal nodes is shown to satisfyuniformly for δ−1(log n)−1/2 ≤ β ≤ δ(log n)1/2, where and δ is a positive constant. An asymptotic formula for is derived for h = cn, where 0 < c < 1. Bounds for are also derived for large and small heights. The methods apply to any simple family of trees, and the general asymptotic results are stated.

ON A NEW CIRCLE PROBLEM

2016 ◽  
Vol 103 (2) ◽  
pp. 231-249
Author(s):  
JUN FURUYA ◽  
MAKOTO MINAMIDE ◽  
YOSHIO TANIGAWA
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Dirichlet Character ◽  
Arithmetical Function ◽  
Circle Problem ◽  
The Riemann Zeta Function ◽  
Primitive Dirichlet Character ◽  
Riemann Zeta ◽  
The Mean ◽  
Voronoi Formula

We attempt to discuss a new circle problem. Let $\unicode[STIX]{x1D701}(s)$ denote the Riemann zeta-function $\sum _{n=1}^{\infty }n^{-s}$ ($\text{Re}\,s>1$) and $L(s,\unicode[STIX]{x1D712}_{4})$ the Dirichlet $L$-function $\sum _{n=1}^{\infty }\unicode[STIX]{x1D712}_{4}(n)n^{-s}$ ($\text{Re}\,s>1$) with the primitive Dirichlet character mod 4. We shall define an arithmetical function $R_{(1,1)}(n)$ by the coefficient of the Dirichlet series $\unicode[STIX]{x1D701}^{\prime }(s)L^{\prime }(s,\unicode[STIX]{x1D712}_{4})=\sum _{n=1}^{\infty }R_{(1,1)}(n)n^{-s}$$(\text{Re}\,s>1)$. This is an analogue of $r(n)/4=\sum _{d|n}\unicode[STIX]{x1D712}_{4}(d)$. In the circle problem, there are many researches of estimations and related topics on the error term in the asymptotic formula for $\sum _{n\leq x}r(n)$. As a new problem, we deduce a ‘truncated Voronoï formula’ for the error term in the asymptotic formula for $\sum _{n\leq x}R_{(1,1)}(n)$. As a direct application, we show the mean square for the error term in our new problem.

scholarly journals The order of magnitude of the mth coefficients of cyclotomic polynomials

1985 ◽  
Vol 27 ◽  
pp. 143-159 ◽  
Author(s):  
H. L. Montgomery ◽  
R. C. Vaughan
Keyword(s):  
Cyclotomic Polynomial ◽  
Cyclotomic Polynomials ◽  
Order Of Magnitude ◽  
Euler’S Function ◽  
Image Position

We define the nth cyclotomic polynomial Φn(z) by the equationand we writewhere ϕ is Euler's function.Erdös and Vaughan [3] have shown thatuniformly in n as m-→∞, whereand that for every large m

Sign in / Sign up

Export Citation Format

Share Document