# What Do You Know About Analytical Number Theory

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In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progression.

• 1.

### Analytic number theory is a subject in ......?

• A.

Mathematics

• B.

English Language

• C.

History

• D.

Philosophy

A. Mathematics
Explanation
Analytic number theory is a branch of mathematics that deals with the study of properties of numbers using techniques from analysis. It involves the application of analytical methods to understand the distribution of prime numbers, the behavior of arithmetic functions, and the properties of number-theoretic functions. Therefore, the correct answer is Mathematics.

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• 2.

### Analytic number theory can be split up into how many major parts?

• A.

Two

• B.

Three

• C.

Four

• D.

Five

A. Two
Explanation
Analytic number theory can be split up into two major parts.

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• 3.

### Much of analytic number theory was inspired by the ........?

• A.

Linear algebra

• B.

Prime number theorem

• C.

Lattices

• D.

Geometry

B. Prime number theorem
Explanation
The correct answer is the Prime number theorem. Analytic number theory, which deals with the properties of integers and prime numbers using analytic methods, was greatly influenced by the Prime number theorem. This theorem provides an estimate for the distribution of prime numbers and has been a fundamental result in number theory. Its discovery and proof have led to significant advancements in the field of analytic number theory.

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• 4.

### In two papers from years ......., the Russian mathematician Pafnuty L'vovich Chebyshev attempted to prove the asymptotic law of distribution of prime numbers.

• A.

1564 and 1566

• B.

1888 and 1890

• C.

1788 and 1791

• D.

1848 and 1850

D. 1848 and 1850
Explanation
In 1848 and 1850, the Russian mathematician Pafnuty L'vovich Chebyshev attempted to prove the asymptotic law of distribution of prime numbers.

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• 5.

### In two papers from 1848 and 1850, the Russian mathematician ........... attempted to prove the asymptotic law of distribution of prime numbers.

• A.

Johann Peter Gustav Lejeune Dirichlet

• B.

Pafnuty L'vovich Chebyshev

• C.

Bernhard Riemann

• D.

B. Pafnuty L'vovich Chebyshev
Explanation
Pafnuty L'vovich Chebyshev, a Russian mathematician, attempted to prove the asymptotic law of distribution of prime numbers in two papers from 1848 and 1850.

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• 6.

### Pafnuty L'vovich Chebyshev is a mathematician from which country?

• A.

Russia

• B.

British

• C.

Austria

• D.

China

A. Russia
Explanation
Pafnuty L'vovich Chebyshev is a mathematician from Russia.

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• 7.

### ........ is concerned with the additive structure of the integers?

• A.

• B.

Multiplicative number theory

• C.

Divisive number theory

• D.

Subtractive number theory

Explanation
Additive number theory is concerned with the additive structure of the integers. It focuses on the properties and relationships of addition in the context of number theory. This branch of mathematics explores topics such as partitions, arithmetic progressions, and additive bases. It aims to understand how integers can be combined together through addition and uncover patterns and properties related to this operation.

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• 8.

### ......... deals with the distribution of the prime numbers?

• A.

Subtractive number theory

• B.

Divisive number theory

• C.

Multiplicative number theory

• D.

C. Multiplicative number theory
Explanation
Multiplicative number theory deals with the distribution of prime numbers. This branch of number theory focuses on studying the properties and patterns of prime numbers, including their distribution, relationships, and behavior under multiplication. It involves concepts such as the prime number theorem, prime factorization, and the study of multiplicative functions. By analyzing the multiplicative properties of numbers, this theory provides insights into the distribution of primes and their connections to other mathematical concepts.

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• 9.

### In what year did Bernhard Riemann used complex analysis and a special meromorphic function now known as the Riemann zeta function to derive an analytic expression for the number of primes less than or equal to a real number x.

• A.

1859

• B.

1865

• C.

1888

• D.

1824

A. 1859
Explanation
In 1859, Bernhard Riemann used complex analysis and a special meromorphic function, now known as the Riemann zeta function, to derive an analytic expression for the number of primes less than or equal to a real number x.

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• 10.

### ........ are concerned with integer solutions to polynomial equations?

• A.

• B.

Diophantine problem

• C.

Multiplicative number theory

• D.

Divisive number theory

B. Diophantine problem
Explanation
The Diophantine problem is concerned with finding integer solutions to polynomial equations. This problem is named after the ancient Greek mathematician Diophantus, who studied and wrote about these types of equations. The other options, additive number theory, multiplicative number theory, and divisive number theory, are not specifically focused on finding integer solutions to polynomial equations.

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