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prime number theorem is a key result in this subject. The Mathematics Subject Classification for multiplicative number theory is 11Nxx. Multiplicative number
Multiplicative_number_theory
Exploring properties of the integers with complex analysis
differences in technique. Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval
Analytic_number_theory
Study of subsets of integers and behavior under addition
Shapley–Folkman lemma Additive combinatorics Multiplicative combinatorics Multiplicative number theory Nathanson (1996) II:1 Henry Mann (1976). Addition
Additive_number_theory
Branch of pure mathematics
"Algebraic Number Theory". Retrieved 7 April 2020. Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative Number Theory: I, Classical Theory. Cambridge
Number_theory
American mathematician
and completed his Ph.D. in 1972. His dissertation, Topics in Multiplicative Number Theory, was supervised by Harold Davenport. He became an assistant professor
Hugh_Lowell_Montgomery
Function equal to the product of its values on coprime factors
In number theory, a multiplicative function is an arithmetic function f {\displaystyle f} of a positive integer n {\displaystyle n} with the property
Multiplicative_function
Type of quantum number
In quantum field theory, multiplicative quantum numbers are conserved quantum numbers of a special kind. A given quantum number q is said to be additive
Multiplicative_quantum_number
Concept in modular arithmetic
In number theory, given a positive integer n and an integer a coprime to n, the multiplicative order of a modulo n is the smallest positive integer k
Multiplicative_order
Characterization of how many integers are prime
Terence (10 December 2014). "254A, Notes 2: Complex-analytic multiplicative number theory". Terence Tao's blog. Edwards, Harold M. (2001). Riemann's zeta
Prime_number_theorem
Arithmetical operation
generalizations See Multiplication in group theory, above, and multiplicative group, which for example includes matrix multiplication. A very general, and
Multiplication
Mathematical operation on arithmetical functions
Dirichlet convolution of two multiplicative functions is again multiplicative, and every not constantly zero multiplicative function has a Dirichlet inverse
Dirichlet_convolution
Computation modulo a fixed integer
a modular multiplicative inverse of a modulo m. If a ≡ b (mod m) and a−1 exists, then a−1 ≡ b−1 (mod m) (compatibility with multiplicative inverse, and
Modular_arithmetic
Mathematical structure with multiplication as its operation
mathematics and group theory, the term multiplicative group refers to one of the following concepts: the group under multiplication of the invertible elements
Multiplicative_group
American mathematician and professor (born 1973)
interest is in analytic number theory, particularly in the subfields of automorphic L-functions, and multiplicative number theory. Soundararajan grew up
Kannan_Soundararajan
Arithmetic function
numbers, and such functions are called multiplicative functions. Outside of number theory, the term "multiplicative function" is often taken to be synonymous
Completely multiplicative function
Completely_multiplicative_function
Mathematical operation in linear algebra
algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the
Matrix_multiplication
Mathematical formula
In number theory, the multiplicative digital root of a natural number n {\displaystyle n} in a given number base b {\displaystyle b} is found by multiplying
Multiplicative_digital_root
Twelfth letter of the Greek alphabet
Retrieved 2025-01-24. "DLMF: §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory". dlmf.nist.gov. Retrieved 2025-01-31. Weisstein
Mu_(letter)
Analytic function in mathematics
Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative Number Theory. I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97
Riemann_zeta_function
Branch of elementary mathematics
New Approach to Multiplication and Exponential Functions". In Harel, Guershon; Confrey, Jere (eds.). The Development of Multiplicative Reasoning in the
Arithmetic
Concept in modular arithmetic
solution, i.e., when it exists, a modular multiplicative inverse is unique: If b and b' are both modular multiplicative inverses of a respect to the modulus
Modular multiplicative inverse
Modular_multiplicative_inverse
Type of mathematical function
Modern Number Theory (2nd ed.). Springer-Verlag. Montgomery, Hugh L.; Vaughan, Robert C. (2006). Multiplicative number theory. I. Classical theory. Cambridge
Dirichlet_L-function
Difference between logarithm and harmonic series
Sequences. OEIS Foundation. Ramaré, Olivier (2022). Excursions in Multiplicative Number Theory. Birkhäuser Advanced Texts: Basel Textbooks. Basel: Birkhäuser/Springer
Euler's_constant
Greek letter
number of primes less than or equal to... "DLMF: §27.12 Asymptotic Formulas: Primes ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory"
Pi_(letter)
Integer
1 is the multiplicative identity: x + (−1) ⋅ x = 1 ⋅ x + (−1) ⋅ x = (1 + (−1)) ⋅ x = 0 ⋅ x = 0. Here we have used the fact that any number x times 0
−1
Natural number
generally, in algebra, it denotes the multiplicative identity in any unital ring or field. An element with a multiplicative inverse is called a unit, generalizing
1
Way to write a number as a product of other numbers
In number theory, a multiplicative partition or unordered factorization of an integer n {\displaystyle n} is a way of writing n {\displaystyle n} as a
Multiplicative_partition
Group of units of the ring of integers modulo n
the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a, the multiplicative inverse
Multiplicative group of integers modulo n
Multiplicative_group_of_integers_modulo_n
Non-sinusoidal waveform
2021. Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97
Sawtooth_wave
Algebraic structure with addition and multiplication
distributive over the addition, and has a multiplicative identity. Some authors omit the requirement for a multiplicative identity in the definition of a ring
Ring_(mathematics)
Multiplication algorithm
Egyptian multiplication (also known as Egyptian multiplication, Ethiopian multiplication, Russian multiplication, or peasant multiplication), one of two
Ancient Egyptian multiplication
Ancient_Egyptian_multiplication
and multivectors with Grassmann algebra. Multiplicative number theory a subfield of analytic number theory that deals with prime numbers, factorization
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Algorithm to multiply two numbers
since antiquity as long multiplication or grade-school multiplication, consists of multiplying every digit in the first number by every digit in the second
Multiplication_algorithm
In mathematics, element with a multiplicative inverse
vu=uv=1,} where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u. The set of
Unit_(ring_theory)
Sum in algebraic number theory
Theorem 9.10 in H. L. Montgomery, R. C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, 97, (2006)
Gauss_sum
Algorithmic runtime requirements for matrix multiplication
multiplicative constant, the same computational complexity as matrix multiplication. The proof does not make any assumptions on matrix multiplication
Computational complexity of matrix multiplication
Computational_complexity_of_matrix_multiplication
Conjecture on zeros of the zeta function
MR 0820245 Montgomery, Hugh L.; Vaughan, Robert C. (2007), Multiplicative Number Theory I. Classical Theory, Cambridge studies in advanced mathematics, vol. 97
Riemann_hypothesis
Potential counterexample to the generalized Riemann hypothesis
ISBN 978-3-540-36364-4 Montgomery, H. L.; Vaughan, R. C. (2006). Multiplicative Number Theory I: Classical Theory. Cambridge Studies in Advanced Mathematics. Cambridge:
Siegel_zero
Formula in number theory
In number theory, the class number formula relates many important invariants of an algebraic number field to a special value of its Dedekind zeta function
Class_number_formula
Mathematical theory by Shinichi Mochizuki
symmetries of IUT: multiplicative arithmetic and additive geometric. On one hand, Hodge theaters generalize such classical objects in number theory as the adeles
Inter-universal Teichmüller theory
Inter-universal_Teichmüller_theory
root modulo n Multiplicative order Discrete logarithm Quadratic residue Euler's criterion Legendre symbol Gauss's lemma (number theory) Congruence of
List_of_number_theory_topics
Branch of mathematics that studies the properties of groups
groups. Group theory has three main historical sources: number theory, the theory of algebraic equations, geometry, and analysis. The number-theoretic strand
Group_theory
Mathematical subject
arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis. Arithmetic combinatorics is about
Arithmetic_combinatorics
Number used for counting
algorithms (such as the Euclidean algorithm), and ideas in number theory. The addition (+) and multiplication (×) operations on natural numbers as defined above
Natural_number
Mathematical function
In Multiplicative Number Theory. Springer. p. 104. ISBN 0-387-95097-4. Google Book Search. Apostol, Tom M. (1976), Introduction to analytic number theory
Chebyshev_function
British mathematician
ISBN 978-0-521-57347-4. Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97
Bob_Vaughan
Chinese mathematician and poet
full professor in Zhejiang University since 1998. Additive and multiplicative number theory, perfect numbers, congruence modulo integer power, Witten zeta
Tianxin_Cai
Integer that is a perfect square modulo some integer
zero modulo p has a multiplicative inverse. This is not true for composite moduli. Following this convention, the multiplicative inverse of a residue
Quadratic_residue
Result on density of prime numbers
492 Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97
Bertrand's_postulate
Whole number
for nonzero x. But x/0 is undefined, because 0 has no multiplicative inverse (no real number multiplied by 0 produces 1), a consequence of the previous
0
Theory of a class of elliptic curves
In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way
Complex_multiplication
English mathematician (1907–1969)
Introduction to analytic number theory, by K. Chandrasekharan; Arithmetical functions, by K. Chandrasekharan; Multiplicative number theory, by Harold Davenport;
Harold_Davenport
Number divisible only by 1 and itself
smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime
Prime_number
Mathematical conjecture about zeros of L-functions
Selberg class Grand Riemann hypothesis Davenport, Harold (2000). Multiplicative Number Theory. Graduate Texts in Mathematics. Vol. 74. Revised and with a preface
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Branch of mathematics
Abstract analytic number theory is a branch of mathematics which takes the ideas and techniques of classical analytic number theory and applies them to
Abstract analytic number theory
Abstract_analytic_number_theory
Symbol in number theory
761–784 Montgomery, Hugh L; Vaughan, Robert C. (2007). Multiplicative number theory. I. Classical theory. Cambridge Studies in Advanced Mathematics. Vol. 97
Kronecker_symbol
Set with associative invertible operation
abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation is used. Several other notations
Group_(mathematics)
British mathematician
Introduction to analytic number theory, by K. Chandrasekharan; Arithmetical functions, by K. Chandrasekharan; Multiplicative number theory, by Harold Davenport;
Heini_Halberstam
Property of a number
the number. Usually, this involves additive or multiplicative persistence of a non-negative integer, which is how often one has to replace the number by
Persistence_of_a_number
mathematics, a multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of
Multiplicative_character
Used in the summation of divergent series
40002. Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative number theory I. Classical theory. Cambridge Studies in Advanced Mathematics. Vol. 97
Abelian and Tauberian theorems
Abelian_and_Tauberian_theorems
Branch of mathematics that studies abstract algebraic structures
representation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication. Representation
Representation_theory
Number in {..., –2, –1, 0, 1, 2, ...}
integer has a multiplicative inverse (as is the case of the number 2), which means that Z {\displaystyle \mathbb {Z} } under multiplication is not a group
Integer
Number with a real and an imaginary part
particular, addition by a complex number corresponds to translation of the plane and multiplication by a complex number corresponds to scaling and rotation
Complex_number
Branch of algebraic number theory concerned with abelian extensions
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions
Class_field_theory
Number equal to the sum of its proper divisors
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number
Perfect_number
Mathematical table
columns for multiplication by 1, the multiplicative identity, which satisfies a × 1 = a. The traditional rote learning of multiplication was based on
Multiplication_table
Number used in combinatorial game theory
Nimber multiplication is associative and commutative, with the ordinal 1 as the multiplicative identity element. Moreover, nimber multiplication distributes
Nimber
Algebraic structure with addition, multiplication, and division
+ (−a) = 0. Multiplicative inverses: for every a ≠ 0 in F, there exists an element in F, denoted by a−1 or 1/a, called the multiplicative inverse of a
Field_(mathematics)
Size of a possibly infinite set
is a multiplicative absorbing element: κ·0 = 0·κ = 0. There are no nontrivial zero divisors: κ·μ = 0 → (κ = 0 or μ = 0). One is a multiplicative identity:
Cardinal_number
Multiplicative function in number theory
The Möbius function μ ( n ) {\displaystyle \mu (n)} is a multiplicative function in number theory introduced by the German mathematician August Ferdinand
Möbius_function
Branch of number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations
Algebraic_number_theory
Special function in mathematics
relationship to polygamma function.) Davenport, Harold (1967). Multiplicative number theory. Lectures in advanced mathematics. Vol. 1. Chicago: Markham.
Hurwitz_zeta_function
Condition under which an integer is a quadratic residue
Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical
Gauss's_lemma_(number_theory)
Debate on mathematics education
Another theory of learning multiplication derives from those studying embodied cognition, which examined the underlying metaphors for multiplication. Together
Multiplication and repeated addition
Multiplication_and_repeated_addition
Number system extending the rational numbers
In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers that is distinct from the real numbers, though
P-adic_number
Summation formula
S2CID 123419717. Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97
Euler–Maclaurin_formula
Mathematical conjecture
ISSN 1460-244X. Montgomery, Hugh L.; Vaughan, Robert C. (2006). Multiplicative Number Theory I: Classical Theory. Cambridge Studies in Advanced Mathematics. Cambridge:
Brocard's_conjecture
Mathematical group that can be generated as the set of powers of a single element
its inverse. Each element can be written as an integer power of g in multiplicative notation, or as an integer multiple of g in additive notation. This
Cyclic_group
Number representing a continuous quantity
number denoted 1 which is a multiplicative identity, which means that a × 1 = 1 × a = a {\displaystyle a\times 1=1\times a=a} for every real number a
Real_number
Tauberian theorem introduced by Shikao Ikehara (1931)
ISBN 3-540-04141-9. Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97
Wiener–Ikehara_theorem
Natural number
simplifies multiplication and division in base 60 because dividing a by b can be done by multiplying a by b's reciprocal when b is a regular number. For instance
54_(number)
Property of being an even or odd number
even number, and the Taylor series of an odd function contains only terms whose exponent is an odd number. In combinatorial game theory, an evil number is
Parity_(mathematics)
Number of integers coprime to and less than n
{\displaystyle \ln(x)} or log e ( x ) {\displaystyle \log _{e}(x)} . In number theory, Euler's totient function counts the positive integers up to a given
Euler's_totient_function
Identity obeyed by many special functions related to the gamma function
obeying the multiplication theorem from any totally multiplicative function. Let f ( n ) {\displaystyle f(n)} be totally multiplicative; that is, f (
Multiplication_theorem
In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any β , γ < α {\displaystyle
Additively indecomposable ordinal
Additively_indecomposable_ordinal
Modular arithmetic concept
classes modulo n. As explained in the article multiplicative group of integers modulo n, this multiplicative group Z n × {\displaystyle \mathbb {Z} _{n}^{\times
Primitive_root_modulo_n
Provides an asymptotic formula for counting the number of prime ideals of a number field
S2CID 119669682. Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97
Landau_prime_ideal_theorem
Mathematical construct
10036. Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97
Character_sum
Axioms for the natural numbers
nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and
Peano_axioms
in local class field theory establishes a one-to-one correspondence between open subgroups of finite index in the multiplicative group K× and finite abelian
Local_class_field_theory
Natural number
any number of up arrows. There are four dimensions in the theory of Minkowski space, three of space and the one being time. Four is the sacred number of
4
Commutative group (mathematics)
notational conventions for abelian groups – additive and multiplicative. Generally, the multiplicative notation is the usual notation for groups, while the
Abelian_group
Algebraic structure used in topology
these theories carry richer information than ordinary cohomology, but are harder to compute. A cohomology theory E is said to be multiplicative if E ∗
Cohomology
Generalization of vector spaces from fields to rings
vector spaces such as Lp spaces.) Suppose that R is a ring, and 1 is its multiplicative identity. A left R-module M consists of an abelian group (M, +) and
Module_(mathematics)
Generalization of the real numbers
closed under multiplication and forms a ring; and for birthday less than an (ordinal) epsilon number εα it is closed under multiplicative inverse and forms
Surreal_number
Complex-valued arithmetic function
N.G. "Theory of the characters of number semigroups". J. Indian Math. Soc. 20: 11–15. Davenport, Harold (1967). Multiplicative number theory. Lectures
Dirichlet_character
Number of prime factors of a natural number
the Theory of Numbers (6th ed.). Oxford University Press. H. L. Montgomery; R. C. Vaughan (2007). Multiplicative number theory I. Classical theory (1st ed
Prime_omega_function
Summatory function of the divisor-counting function
k\geq 2} . Montgomery, Hugh; R. C. Vaughan (2007). Multiplicative Number Theory I: Classical Theory. Cambridge: Cambridge University Press. ISBN 978-0-521-84903-6
Divisor_summatory_function
Generalization of "n-th" to infinite cases
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite
Ordinal_number
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MULTIPLICATIVE NUMBER-THEORY
MULTIPLICATIVE NUMBER-THEORY
MULTIPLICATIVE NUMBER-THEORY
MULTIPLICATIVE NUMBER-THEORY
MULTIPLICATIVE NUMBER-THEORY
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