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

Viorel Nitica
2018 arXiv   pre-print
We show that for any 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  ... 
arXiv:1811.10035v1 fatcat:efh36hqogfh4jecyxpqldut5ri

Srinivasa Ramanujan: Going Strong at 125, Part II

Krishnaswami Alladi, Ken Ono, K. Soundararajan, R. C. Vaughan, S. Ole Warnaar
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.  ... 
doi:10.1090/noti926 fatcat:mextxcfvcfgxxcgxwr7omac6vi

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

H. Gopalakrishna Gadiyar, Ramanathan Padma
2014 Czechoslovak Mathematical Journal  
Ram Murty for encouragement given 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.  ... 
doi:10.1007/s10587-014-0098-5 fatcat:ixl55igqtvfpnlkevmcyyoeys4

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]

H. Gopalkrishna Gadiyar, R. Padma
2006 arXiv   pre-print
Currently the circle 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.  ... 
arXiv:math/0601574v1 fatcat:fjglapsrcrgrpocpjcw425qque

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]

Giovanni Coppola, Luca Ghidelli
2020 arXiv   pre-print
A G:N →C is called a 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.  ... 
arXiv:2005.14666v2 fatcat:cavx46lcw5fzrj5eql6uiito5a

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

Krishnaswami Alladi
2020 Notices of the American Mathematical Society  
Srinivasa 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.  ... 
doi:10.1090/noti2013 fatcat:x3abi4rrzzaa5dxypn7oy4ogga

The Development of "Partitio Numerorum," With Particular Reference to the Work of Messrs. Hardy, Littlewood and Ramanujan

Aubrey J. Kempner
1923 The American mathematical monthly  
How must an 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.  ... 
doi:10.1080/00029890.1923.11986272 fatcat:yjmokpuwvfdqpmni3oszvnw6ru

Ramanujan in Computing Technology [article]

V. N. Krishnachandran
2021 arXiv   pre-print
In his short life period he made substantial contributions to mathematical analysis, 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.  ... 
arXiv:2103.09654v1 fatcat:t2hkkyaivzhexdkxurrmuhfqk4

Ramanujan series for arithmetical functions

M. Ram Murty
2013 Hardy-Ramanujan Journal  
International audience We give a short survey 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.  ... 
doi:10.46298/hrj.2013.180 fatcat:q3e7pwyt2jbufowifzk4xrpcaa

Arithmetical Functions : Infinite Products [article]

Garimella Rama Murthy (International Institute of Information Technology, Hyderabad, AP, India)
2012 arXiv   pre-print
The notion 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(!)  ... 
arXiv:1212.1644v1 fatcat:4ddy5cuukrcvfjv3prxfwv5lj4

A Hardy-Ramanujan type inequality for shifted primes and sifted sets [article]

Kevin Ford
2021 arXiv   pre-print
We establish an analog 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.  ... 
arXiv:2101.03440v3 fatcat:e2txcwpvhfcfrlwpcwgl72iycu

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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