Analytic number theory studies integers, prime numbers, and arithmetic functions using tools from real analysis and complex analysis, especially Dirichlet series and their singularities.
If , infinitely many primes satisfy
Partial summation is the discrete analogue of integration by parts. If and is continuously differentiable, then
For a finite integer set and a set of primes , the sifting function counts elements of divisible by no prime in below :
An upper-bound sieve bounds the size of a sifted set from above. In a dimension-one problem with one forbidden class modulo each relevant prime , its main density factor is comparable to .
If three nonproportional linear forms exclude three distinct residue classes modulo every sufficiently large prime, the upper-bound sieve has dimension three and gives an upper bound of order .
A Dirichlet series is a series of the formMultiplication of absolutely convergent Dirichlet series corresponds to Dirichlet convolution of their coefficients.
An Euler product factors a Dirichlet series into local factors indexed by prime numbers. For a multiplicative arithmetic function and in a half-plane of absolute convergence,
For every complex number with ,Applying this to each prime-power term in logarithms of Euler products proves thatfor and every completely multiplicative bounded by one.
Perron's formula recovers a summatory arithmetic function from its Dirichlet series by the inverse Mellin integralwith the usual convergence and endpoint conventions.
The completed zeta functionis an entire function satisfying . Thus every nontrivial zero is accompanied by , , and .
A nontrivial zero of the Riemann zeta function lies in the critical strip . The Functional equation of the Riemann zeta function reflects such zeros across the critical line .
The Riemann hypothesis asserts that every Nontrivial zero of the Riemann zeta function lies on the critical line .
The classical zero-free region asserts that for some ,Together with bounds for the logarithmic derivative, it permits contour arguments with exponentially small errors in .
The logarithmic derivative of a nonzero differentiable function is . For , the Euler product for the Riemann zeta function gives
If, for some real ,then partial summation continues meromorphically to with no pole except at . The zeta function has no zero to the right of the critical line, and its functional equation then implies the Riemann hypothesis.
For , the alternating Dirichlet seriesconverges locally uniformly and defines a holomorphic function. For ,
The identity continues the Riemann zeta function meromorphically to . Apparent singularities at nonreal zeros of are removable, as one sees by replacing with an integer for which . At , and , so has a simple pole of residue one.
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Analytic number theory is a branch of mathematics that uses tools and techniques from mathematical analysis to solve problems about integers, particularly concerning the distribution of prime numbers. It is a rich field that combines elements of number theory with methods from analysis, particularly infinite series, functions, and complex analysis.