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Infinite sets of b-additive and b-multiplicative Ramanujan-Hardy numbers
[article]

2018
*
arXiv
*
pre-print

We show that for any

arXiv:1811.10035v1
fatcat:efh36hqogfh4jecyxpqldut5ri
*b*there exists an infinity*of**b*-*additive**Ramanujan*-*Hardy**numbers**and*an infinity*of**additive*multipliers. ... We show that for an even*b*,*b*≡ 1 3,*and*for*b*=2, there exists an infinity*of**b*-*multiplicative**Ramanujan*-*Hardy**numbers**and*an infinity*of**multiplicative*multipliers. ... In Niţicȃ [7] , motivated by some properties*of*the taxicab*number*, 1729, we introduce the classes*of**b*-*additive**Ramanujan*-*Hardy*(or*b*-ARH)*numbers**and**b*-*multiplicative**Ramanujan*-*Hardy*(or*b*-MRH)*numbers*...##
###
Srinivasa Ramanujan: Going Strong at 125, Part II

2013
*
Notices of the American Mathematical Society
*

*Ramanujan*was fascinated by the coefficients

*of*the function where q := e 2π iz

*and*Im(z) > 0. This function is a weight 12 modular form. ... In other words, ∆(z) is a function on the upper half

*of*the complex plane such that for every matrix a

*b*c d ∈ SL 2 (Z). He conjectured (see p. 153

*of*[12]) that τ(nm) = τ(n)τ(m), ... In

*addition*to the partition function,

*Hardy*

*and*

*Ramanujan*also observe that the same ideas can be applied to the

*number*

*of*representations

*of*a

*number*as a sum

*of*a fixed

*number*

*of*squares. ...

##
###
Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood

2014
*
Czechoslovak Mathematical Journal
*

Ram Murty for encouragement given

doi:10.1007/s10587-014-0098-5
fatcat:ixl55igqtvfpnlkevmcyyoeys4
*and*also for drawing our attention to the interesting reference*of*Golomb [8] . ... We are grateful to the referee for suggesting several changes in the paper that have improved the presentation*of*the paper. ... In [10] ,*Hardy*proved that the*Ramanujan*sum is a*multiplicative*function*of*q, that is (2.20) c qq ′ (n) = c q (n)c q ′ (n) if (q, q ′ ) = 1. ...##
###
Page 175 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 43, Issue 3
[page]

1937
*
American Mathematical Society. Bulletin of the American Mathematical Society
*

Consider a Boolean ring, K,

*of*point*sets*, a,*b*, c,- + - , where a—), the ring*addition*, represents all points in either a or 6 but not in both,*and*ab, the ring*multiplication*, represents all points in ... In 1917*Hardy**and**Ramanujan*gave a remarkable asymptotic series for p(n), the*number**of*partitions*of*m, whose kth coefficient A:(m) was given as a complicated sum*of*24th roots*of*unity associated with ...##
###
Linking the Circle and the Sieve: Ramanujan - Fourier Series
[article]

2006
*
arXiv
*
pre-print

Currently the circle

arXiv:math/0601574v1
fatcat:fjglapsrcrgrpocpjcw425qque
*and*the sieve methods are the key tools in analytic*number*theory. In this paper the unifying theme*of*the two methods is shown to be the*Ramanujan*- Fourier series. ... It is immediately obvious that the dens(N )=1*and*dens(A p ) = 1 p where A p is the*set**of**multiples**of*p. ... The two well known methods in*additive**number*theory are the circle method*and*the sieve method. ...##
###
Page 1836 of Mathematical Reviews Vol. 53, Issue 6
[page]

1977
*
Mathematical Reviews
*

In the present paper the problem

*of**Hardy**and**Ramanujan*is completely solved for*additive*arithmetic functions. {The same problem for*multiplicative*arithmetic functions was solved by*B*. J. Birch [J. ...*Hardy**and*S.*Ramanujan*[Quart. J. Pure Appl. ...##
###
Multiplicative Ramanujan coefficients of null-function
[article]

2020
*
arXiv
*
pre-print

A G:N →C is called a

arXiv:2005.14666v2
fatcat:cavx46lcw5fzrj5eql6uiito5a
*Ramanujan*coefficient, abbrev. R.c., iff (if*and*only if) ∑_q=1^∞G(q)c_q(a) converges in all a∈N; given F:N →C, we call , the*set**of*its R.c.s, the*Ramanujan*cloud*of*F. ... The null-function 0(a):=0, ∀ a∈N, has*Ramanujan*expansions: 0(a)=∑_q=1^∞(1/q)c_q(a) (where c_q(a):=*Ramanujan*sum), given by*Ramanujan*,*and*0(a)=∑_q=1^∞(1/φ(q))c_q(a), given by*Hardy*(φ:= Euler's totient ... We denote by P the*set**of*prime*numbers**and*by N 0 def = N ∪ {0} the*set**of*non-negative integers. ...##
###
Page 1187 of Mathematical Reviews Vol. , Issue 87c
[page]

1987
*
Mathematical Reviews
*

*Hardy*-

*Ramanujan*J. 7 (1984), 17-20. For A > 0,

*and*an even integer N (> 1),

*set*> 977(a)u2(9)C,(-N). q<(log N)4 87c:11094 E,(N) = Here, C,(n) is the

*Ramanujan*sum. R. Balasubramanian

*and*C. J. ...

*B*. 87c:11090 The

*number*

*of*prime divisors

*of*Gaussian

*numbers*in a sectorial layer. (Russian) Vestnik Akad. Nauk Kazakh. SSR 1985, no. 4, 79-81. ...

##
###
An Invitation to the Rogers-Ramanujan Identities

2020
*
Notices of the American Mathematical Society
*

Srinivasa

doi:10.1090/noti2013
fatcat:x3abi4rrzzaa5dxypn7oy4ogga
*Ramanujan*, FRS (1887-1920). identified with the*set**of*mappings 𝜙(𝜏) = 𝑎𝜏 +*𝑏*𝑐𝜏 + 𝑑 where Im(𝜏) > 0*and*𝑎,*𝑏*, 𝑐, 𝑑 are integers satisfying 𝑎𝑑 − 𝑏𝑐 = 1. ... A Rogers-*Ramanujan*-type (R-R type) identity is an*infinite*𝑞-series =*infinite*𝑞-product identity, where the series is the generating function*of*partitions whose parts satisfy gap conditions,*and*the ... Credits Photos*of*Srinivasa*Ramanujan**and*G. H.*Hardy*are courtesy*of*the Archives*of*the Mathematisches Forschungsinstitut, Oberwolfach. ...##
###
The Development of "Partitio Numerorum," With Particular Reference to the Work of Messrs. Hardy, Littlewood and Ramanujan

1923
*
The American mathematical monthly
*

How must an

doi:10.1080/00029890.1923.11986272
fatcat:yjmokpuwvfdqpmni3oszvnw6ru
*infinite**set**of**numbers*be chosen in order that every n can be represented as a sum*of*a fixed*number**of*elements*of*the*set*? ... “Outlines*of*seven lectures on the partition*of**numbers*.” (Delivered 1859.) (*b*) Papers by G. H.*Hardy*, J. E. Littlewood*and*8. ...##
###
Ramanujan in Computing Technology
[article]

2021
*
arXiv
*
pre-print

In his short life period he made substantial contributions to mathematical analysis,

arXiv:2103.09654v1
fatcat:t2hkkyaivzhexdkxurrmuhfqk4
*number*theory,*infinite*series,*and*continued fractions, including solutions to mathematical problems then considered ... We shall discuss the application*of*certain*infinite*series discovered by*Ramanujan*in computing the value*of*the mathematical constant π. ... We denote by Z q the field whose*set**of*elements is the*set*{0, 1, 2, . . . , q − 1}, the*addition*operation is*addition*modulo q*and**multiplication*operation is*multiplication*modulo q. ...##
###
Ramanujan series for arithmetical functions

2013
*
Hardy-Ramanujan Journal
*

International audience We give a short survey

doi:10.46298/hrj.2013.180
fatcat:q3e7pwyt2jbufowifzk4xrpcaa
*of*old*and*new results in the theory*of**Ramanujan*expansions for arithmetical functions. ... Gadiyar*and*R. Padma for their comments on an earlier version*of*this article. ...*of**Hardy**and*Littlewood on the*number**of*twin primes up to x. ...##
###
Arithmetical Functions : Infinite Products
[article]

2012
*
arXiv
*
pre-print

The notion

arXiv:1212.1644v1
fatcat:4ddy5cuukrcvfjv3prxfwv5lj4
*of**additively*decomposable*and**multiplicatively*decomposable arithmetical functions is proposed. The concepts*of*arithmetical polynomials*and*arithmetical power series are introduced. ... Using these concepts, an interesting Theorem relating arithmetical power series*and**infinite*products has been proved. Also arithmetical polynomials are related to probabilistic*number*theory. ... For instance*Ramanujan*studied this function in his famous paper with*Hardy*[Har] (the normal*number**of*prime factors*of*a*number*n ). Let us define the following arithmetical polynomial i.e. J(!) ...##
###
A Hardy-Ramanujan type inequality for shifted primes and sifted sets
[article]

2021
*
arXiv
*
pre-print

We establish an analog

arXiv:2101.03440v3
fatcat:e2txcwpvhfcfrlwpcwgl72iycu
*of*the*Hardy*-*Ramanujan*inequality for counting members*of*sifted*sets*with a given*number**of*distinct prime factors. ... In particular, we establish a bound for the*number**of*shifted primes p+a below x with k distinct prime factors, uniformly for all positive integers k. ... To show this, Erdős proved an upper bound*of**Hardy*-*Ramanujan*type for the*number**of*primes p x with ω(p−1) = k in a restricted range*of*k. ...##
###
Page 583 of Mathematical Reviews Vol. 25, Issue 4
[page]

1963
*
Mathematical Reviews
*

*of*G

*and*such that o(a)=a for all ae G, a unique element

*of*G takes the role

*of*zero

*and*¢ is not defined for sequences having an

*infinite*

*number*

*of*non-zero elements; it is not assumed that the sum

*of*... In the study

*of*Waring’s problem, Siegel has extended the so-called circle method

*of*

*Hardy*

*and*Littlewood so that it can be applied to Waring’s problem in algebraic

*number*- fields. ...

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