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In number theory, the gcd-sum function, also called Pillai's arithmetical function, is defined for every n {\displaystyle n} by P ( n ) = ∑ k = 1 n gcd
Pillai's arithmetical function
Pillai's_arithmetical_function
Mathematical operation on arithmetical functions
is Liouville's function. Id ∗ ϕ = P {\displaystyle {\text{Id}}*\phi =P} , where P {\displaystyle P} is Pillai's arithmetical function, also known as the
Dirichlet_convolution
Indian mathematician (1901–1950)
Subbayya Sivasankaranarayana Pillai (5 April 1901 – 31 August 1950) was an Indian mathematician specialising in number theory. His contribution to Waring's
Subbayya Sivasankaranarayana Pillai
Subbayya_Sivasankaranarayana_Pillai
Sequence of integers
sequence would require "hundreds of millions of digits". Pillai, S. S. (1930), "An arithmetical function concerning primes", Annamalai University Journal: 159–167
Pillai_sequence
Town in Tamilnadu, India
Sivasankaranarayana Pillai - Indian mathematician and inventor of Pillai's conjecture, Pillai's arithmetical function, Pillai prime, Pillai sequence. M.S.S
List_of_people_from_Nagercoil
Largest integer that divides given integers
_{k|a{\text{ and }}k|b}\varphi (k).} GCD Summatory function (Pillai's arithmetical function): ∑ k = 1 n gcd ( k , n ) = ∑ d | n d φ ( n d ) = n ∑ d | n
Greatest_common_divisor
Average value of a random variable
277. Billingsley 1995, Section 19. Edwards, A.W.F (2002). Pascal's arithmetical triangle: the story of a mathematical idea (2nd ed.). JHU Press. ISBN 0-8018-6946-3
Expected_value
Multiplicative function Additive function Dirichlet convolution Erdős–Kac theorem Möbius function Möbius inversion formula Divisor function Liouville function Partition
List_of_number_theory_topics
Indian mathematician (1918–1994)
India. He specialized in number theory, in particular the theory of arithmetic functions. Known to his friends as CSV, Venkataraman was born at Chelakkara
C._S._Venkataraman
Sequence of natural numbers
68 (6): 557–560. doi:10.2307/2311150. JSTOR 2311150. S. S. Pillai, "An arithmetical function concerning primes", Annamalai University Journal (1930), pp
Complete_sequence
Probability distribution
distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution f ( x ; x 0 , γ
Cauchy_distribution
Indian educator
of Travancore. His father was Kunchukkali Pillai. He received elementary education in Malayalam and arithmetic as a child. At the age of twelve, he obtained
C._Krishna_Pillai
Characterization of how many integers are prime
particular, the Riemann zeta function). The first such distribution found is π(N) ~ N/log(N), where π(N) is the prime-counting function (the number of primes
Prime_number_theorem
Indian mathematician and astronomer (1114–1185)
below.) Bhaskara's arithmetic text Līlāvatī covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions
Bhāskara_II
(1996). "The Development of Arithmetical Thinking: On the Role of Calculating Aids in Ancient Egyptian & Babylonian Arithmetic". Abstraction & Representation:
History_of_mathematics
Mathematical treatise by Bhāskara II
contains thirteen chapters, mainly definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid
Līlāvatī
Procedure for comparing multivariate sample means
Sreedharan Pillai–M. S. Bartlett trace, Λ Pillai = ∑ 1 , … , p ( λ p / ( 1 + λ p ) ) = tr ( A ( I + A ) − 1 ) {\displaystyle \Lambda _{\text{Pillai}}=\sum
Multivariate analysis of variance
Multivariate_analysis_of_variance
Indian mathematician (1887–1920)
certain arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called τ(n) (the Ramanujan tau function). He proved
Srinivasa_Ramanujan
fact examples had been found earlier of functions that were nowhere differentiable (see Weierstrass function). According to Weierstrass in his paper,
List_of_conjectures
Indian mathematician (1920–2017)
reprinting 2012 Arithmetical Functions. Grundlehren der Mathematischen Wissenschaften. Springer. 1970. LCCN 49010722. Elliptic Functions. Springer. 1985
K._S._Chandrasekharan
normalized pair correlation function between pairs of zeros of the Riemann zeta function is the same as the pair correlation function of random Hermitian matrices
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Indian actress and former Chief Minister of Tamil Nadu (1948–2016)
the Cradle Baby Scheme (CBS) Mother Teresa was also present at the State function on the International Women's Day on 9 March 1994 at Chennai, which Jayalalithaa
J._Jayalalithaa
Integer of the form 3 × 2^n – 1 for non-negative n
Rashed, Roshdi (1994). The development of Arabic mathematics: between arithmetic and algebra. Vol. 156. Dordrecht, Boston, London: Kluwer Academic Publishers
Thabit_number
Product of prime numbers, plus one
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Euclid_number
Number of form 2^(2^p-1)-1 with prime exponent
number is known as a "Martian prime". Cunningham chain Double exponential function Fermat number Perfect number Wieferich prime Chris Caldwell, Mersenne Primes:
Double_Mersenne_number
Indian mathematician (1901-1954)
dealt with non-differentiable functions (an example of an everywhere non-differentiable function is the Weierstrass function). Singh published about a dozen
Avadhesh_Narayan_Singh
Positive integer that is an integer power of another positive integer
(k))\approx 0.874464368\dots } where μ(k) is the Möbius function and ζ(k) is the Riemann zeta function. According to Euler, Goldbach showed (in a now-lost
Perfect_power
were not stored. There are known formulae to evaluate the prime-counting function (the number of primes smaller than a given value) faster than computing
List_of_prime_numbers
Indian mathematician and astronomer (1019–1066)
major work on astronomy in 19 chapters; and Gaṇita-tilaka, an incomplete arithmetical treatise in 125 verses based on a work by Shridhara. Śrīpati was born
Śrīpati
Statistical test
also be considered. SPSS provides an F-ratio from four different methods: Pillai's trace, Wilks’ lambda, Hotelling's trace, and Roy's largest root. In general
Mauchly's_sphericity_test
Conjecture in number theory
Ivan (September 2015). "Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi
Abc_conjecture
Mathematical concept
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Cullen_number
Type of numeral systems
_{j=1}^{i}p_{j}} , and pj = jth prime, p0# = p0 = 1. S. S. Pillai, "An arithmetical function concerning primes", Annamalai University Journal (1930), pp
Mixed_radix
Natural number whose divisor sum is greater than that of any smaller number
{\displaystyle \sigma (n)>\sigma (m)} where σ denotes the sum-of-divisors function. The first few highly abundant numbers are 1, 2, 3, 4, 6, 8, 10, 12, 16
Highly_abundant_number
Number of the form x^y + y^x
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Leyland_number
Mathematical Institute located in Mumbai, India
esteemed mathematician known for his contributions to number theory and arithmetic geometry. Dr. Murty holds a Ph.D. from Harvard and has served as the Director
Lodha Mathematical Sciences Institute
Lodha_Mathematical_Sciences_Institute
Measure of the joint variability
{\displaystyle X} and Y {\displaystyle Y} have the following joint probability mass function, in which the six central cells give the discrete joint probabilities f
Covariance
Number of the form (n * 2^n) - 1
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Woodall_number
8th century Indian mathematician
century) was an Indian mathematician, known for two extant treatises about arithmetic and practical mathematics, Pātīgaṇita and Pāṭīgaṇita-sāra, and a now-lost
Sridhara
Canadian-American mathematician (born 1974)
Jonathan Hanke) of the 290 theorem. A novel generalization of the factorial function, Bhargava factorial, providing an answer to a decades-old question of George
Manjul_Bhargava
1356 mathematical treatise by Narayana Pandita
written by Indian mathematician Narayana Pandita in 1356. It was an arithmetical treatise alongside the other algebraic treatise called "Bijganita Vatamsa"
Ganita_Kaumudi
Type of positive integer
coprime to both endpoints. They observed that earlier results of S. S. Pillai and George Szekeres implied that such an element exists for every interval
Erdős–Woods_number
Prime number of the form 2^n – 1
representation of Mn equals ⌊n log102⌋ + 1, where ⌊x⌋ denotes the floor function (or equivalently ⌊log10Mn⌋ + 1). In September 2008, mathematicians at UCLA
Mersenne_prime
mathematical literature to denote the area of mathematics dealing with arithmetic and mensuration. The term is a compound word formed by combining the words
Pātīgaṇita
Indian mathematician and astronomer (598–668)
new values of the sine function from other values already tabulated. The formula gives an estimate for the value of a function f at a value a + xh of
Brahmagupta
Most common system for writing numbers
medieval Arabs and Persians, they called it al-ḥisāb al-hindī ("Indian arithmetic"). These numerals were gradually adopted in Europe starting around the
Hindu–Arabic_numeral_system
Even integers as sums of two primes
believed to be of roughly comparable difficulty. The Goldbach partition function associates to each even integer the number of ways it can be decomposed
Goldbach's_conjecture
Type of prime number
to do with the role cubes (third powers) play in the equations. Cubic function List of prime numbers Prime number Allan Joseph Champneys Cunningham, On
Cuban_prime
Mathematical problem in number theory
Mat. (in Russian). 23 (5): 637–642. Karatsuba, A. A. (1985). "On the function G(n) in Waring's problem". Izv. Akad. Nauk SSSR Ser. Mat. 27 (4): 935–947
Waring's_problem
Mathematical work of Brahmagupta
(Princeton University Press, 2012). Henry Thomas Colebrooke. Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara, London
Brāhmasphuṭasiddhānta
Ratio of competing statistical models
PMC 2796821. PMID 19880371. Robert, C.P.; J. Cornuet; J. Marin & N.S. Pillai (2011). "Lack of confidence in approximate Bayesian computation model choice"
Bayes_factor
Indian mathematician (about 1325–1400)
worked in the northern half of India. Narayana Pandit wrote two works, an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijaganita
Narayana Pandita (mathematician)
Narayana_Pandita_(mathematician)
Indian mathematician-astronomer (476–550)
sin(3π/2+x)=sinπ/2-versinπ/2-sinπ/2+versinx These formulae are no longer used as the function versin has become obsolete.[citation needed] A problem of great interest
Aryabhata
Result on density of prime numbers
relationship with π ( x ) {\displaystyle \pi (x)} , the prime-counting function (number of primes less than or equal to x {\displaystyle x} ): π ( x )
Bertrand's_postulate
Indian historian of mathematics (1888-1958)
History of Hindu Mathematics – A Source Book. Part I. Numeral Notation and Arithmetic". The American Mathematical Monthly. 43 (6): 367–368. doi:10.2307/2301803
Bibhutibhushan_Datta
Tamil language book about Arithmetic
authored by Paṇṭala Rāmasvāmi Nāykkar and published in 1825 dealing with arithmetic. It is the first Tamil book on mathematics ever to be printed and it is
Kaṇita_Tīpikai
Positive integer of the form (2^(2^n))+1
are constructible partially depends on Fermat primes. Double exponential function Lucas' theorem Mersenne prime Pierpont prime Primality test Proth's theorem
Fermat_number
Book by Bhaskara II
Siddhānta Śiromaṇi. The book contains thirteen chapters, 278 verses, mainly arithmetic and measurement. It is the second volume of Siddhānta Śiromaṇi. It is
Siddhānta_Shiromani
Indian Hindu Brahmin subcaste
Deshastha families, the name that the child inevitably uses in secular functioning is the one decided by his parents. If a name is chosen on the basis of
Deshastha_Brahmin
Statistical method that summarizes and/or integrates data from multiple sources
bias in research synthesis: sensitivity analysis using a priori weight functions". Psychological Methods. 10 (4): 428–443. doi:10.1037/1082-989X.10.4.428
Meta-analysis
9th-century Indian mathematician
He asserted that the square root of a negative number does not exist. Arithmetic operations utilized in his works like Gaṇita-sāra-saṅgraha(Ganita Sara
Mahāvīra_(mathematician)
the 16th-17th century CE, dealing with elementary arithmetic and methods for solving arithmetical problems arising in the everyday life of members of
Kanakkusaram
Ancient Indian alphasyllabic numeral system
1989, Lilavati, gracious lady of arithmetic – India – A Mathematical Mystery Tour "Lilavati, gracious lady of arithmetic - India - A Mathematical Mystery
Katapayadi_system
Indian-born Canadian mathematician (1921–2006)
1960s Subbarao began to study the congruence properties of the partition function, p(n), which became one of his favourite problems. For example, he conjectured
Mathukumalli_V._Subbarao
Jain cosmological text
in the world that clearly uses three principles of positional decimal arithmetic system together – graphical signs and terms as numerals, assigning a value
Lokavibhaga
Treatise on algebra by Bhāskara II
Bijaganita into English include: 1817. Henry Thomas Colebrooke, Algebra, with Arithmetic and mensuration, from the Sanscrit of Brahmegupta and Bháscara 1813. Ata
Bijaganita
Satellite navigation system
uplinks navigation data and is used for tracking, telemetry and command functions. Seven 7.2-metre (24 ft) FCA and two 11-metre (36 ft) FMA of IRSCF are
Indian Regional Navigation Satellite System
Indian_Regional_Navigation_Satellite_System
Prime such that p^2 divides 2^(p-1)-1
the congruence 2φ(n) ≡ 1 (mod n2), where φ denotes the Euler's totient function (according to Euler's theorem, 2φ(n) ≡ 1 (mod n) for every odd natural
Wieferich_prime
Two different Tamil mathematics books
invocation stanza. These stanzas specify rules for carrying out elementary arithmetical operations. One interesting feature of this work is that bulk of the
Asthana_Kolahalam
Indian scholar in Sanskrit and mathematics and translator (1821-1900)
Hindu Brahmin family of Maharashtra. He received his early education in arithmetic and algebra at the Marathi school in Nagpur. He also studied the Lilāvati
Bapudeva_Sastri
Type of prime number
{\displaystyle (p-1)!+1} , where " ! {\displaystyle !} " denotes the factorial function; compare this with Wilson's theorem, which states that every prime p {\displaystyle
Wilson_prime
Sanskrit astronomical treatise by Aryabhata
and vowels denoting place value. This innovation allows for advanced arithmetical computations which would have been considerably more difficult without
Aryabhatiya
Tamil mathematics book
version of the book is Kaṇakkadhikāram. Kaṇakkatikāram deals with the arithmetical calculations that are likely to be required in the daily life of ordinary
Kaṇakkatikāram
Treatise on mathematics and astronomy
Some of its important topics include the infinite series expansions of functions; power series, including of π and π/4; trigonometric series of sine, cosine
Yuktibhāṣā
Ancient mathematical text
problems involve arithmetic, algebra and geometry, including mensuration. The topics covered include fractions, square roots, arithmetic and geometric progressions
Bakhshali_manuscript
Texts belonging to the Śrauta ritual
supervene the conviction that it is impossible to arrive at an accurate arithmetical expression of the value. And lastly (3) the impossibility must be proved
Shulba_Sutras
Sanskrit word for "rule of three"
p. 204. Retrieved 21 June 2024. H. T. Colebrooke (1817). Algebra with Arithmetic and Mensuration from the Sanscrit of Brhmagupta and Bhaskara. London:
Trairāśika
Astronomy Book by Sankara Variar
a circle. This yields the infinite series expansion of the arctangent function. This result is also ascribed to Madhava. "Now, by just the same argument
Kriyakramakari
Institute. Retrieved 22 June 2024. H. T. Colebrooke (1817). Algebra with Arithmetic and Mensuration from the Sanscrit of Brhmagupta and Bhaskara. London:
Bījapallava
PILLAIS ARITHMETICAL-FUNCTION
PILLAIS ARITHMETICAL-FUNCTION
PILLAIS ARITHMETICAL-FUNCTION
PILLAIS ARITHMETICAL-FUNCTION
PILLAIS ARITHMETICAL-FUNCTION
PILLAIS ARITHMETICAL-FUNCTION
PILLAIS ARITHMETICAL-FUNCTION
PILLAIS ARITHMETICAL-FUNCTION
PILLAIS ARITHMETICAL-FUNCTION