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.

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.

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 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.

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 Dirichlet series attached to cusp forms and the Siegel-zero

1984 ◽  
Vol 25 (1) ◽  
pp. 107-119 ◽  
Author(s):  
F. Grupp
Keyword(s):  
Dirichlet Series ◽  
Cusp Form ◽  
Fourier Expansion ◽  
Dirichlet Character ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Siegel Zero ◽  
Image Position

Let k be an even integer greater than or equal to 12 and f an nonzero cusp form of weight k on SL(2, Z). We assume, further, that f is an eigenfunction for all Hecke-Operators and has the Fourier expansionFor every Dirichlet character xmod Q we define

Heights and L-Series

1990 ◽  
Vol 42 (3) ◽  
pp. 533-560 ◽  
Author(s):  
Rhonda Lee Hatcher
Keyword(s):  
Dirichlet Series ◽  
Cusp Form ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Ideal Class ◽  
Quadratic Character ◽  
Trivial Character ◽  
Image Position

Let be a cusp form of weight 2k and trivial character for Γ0(N), where N is prime, which is orthogonal with respect to the Petersson product to all forms g(dz), where g is of level L < N, dL\N. Let K be an imaginary quadratic field of discriminant — D where the prime N is inert. Denote by ∊ the quadratic character of determined by ∊(p) = (—D/p) for primes p not dividing D. For A an ideal class in K, let rA(m) be the number of integral ideals of norm m in A. We will be interested in the Dirichlet series L(f,A,s) defined by

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

1939 ◽  
Vol 35 (3) ◽  
pp. 357-372 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Modular Form ◽  
Fundamental Region ◽  
Arithmetical Functions ◽  
Image Position

Suppose thatis an integral modular form of dimensions −κ, where κ > 0, and Stufe N, which vanishes at all the rational cusps of the fundamental region, and which is absolutely convergent for Thenwhere a, b, c, d are integers such that ad − bc = 1.

A further note on Ramanujan's arithmetical function τ(n)

1938 ◽  
Vol 34 (3) ◽  
pp. 309-315 ◽  
Author(s):  
G. H. Hardy
Keyword(s):  
Arithmetical Function ◽  
Image Position

This note is a sequel to two in earlier volumes of the Proceedings, the first by myself and the second by Wilton†.Suppose thatfor |z| <1; that x > 0; thatforr ≥ 0; and thatwhere τ(x) is to mean 0 if x is not an integer. Thuswhere the dash shows that the last term is to be halved when x is an integer;forr > 1; and

scholarly journals Subconvexity for a double Dirichlet series

2010 ◽  
Vol 147 (2) ◽  
pp. 355-374 ◽  
Author(s):  
Valentin Blomer
Keyword(s):  
Dirichlet Series ◽  
Functional Equations ◽  
Mean Square ◽  
Double Dirichlet Series ◽  
Image Position

AbstractFor two real characters ψ,ψ′ of conductor dividing 8 define where $\chi _d = (\frac {d}{.})$ and the subscript 2 denotes the fact that the Euler factor at 2 has been removed. These double Dirichlet series can be extended to $\Bbb {C}^2$ possessing a group of functional equations isomorphic to D12. The convexity bound for Z(s,w;ψ,ψ′) is |sw(s+w)|1/4+ε for ℜs=ℜw=1/2. It is proved that Moreover, the following mean square Lindelöf-type bound holds: for any Y1,Y2≥1.

scholarly journals Proximate order (R) of entire functions represented by Dirichlet series (IV)

1965 ◽  
Vol 7 (1) ◽  
pp. 15-18 ◽  
Author(s):  
Pawan Kumar Kamthan
Keyword(s):  
Dirichlet Series ◽  
Entire Functions ◽  
Proximate Order ◽  
Image Position

where be an entirefunction represented by a Dirichlet series whose order (R) and proximate order (R) are respectively ρ (0 < ρ < ∞) and ρ(σ). For proximate order (R) and its properties, see the paper of Balaguer [4, p. 28].

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