Analytic number theory: Dirichlet series and the explicit formula
Dirichlet series, Euler products and contour integration turn questions about primes into questions about zeros of zeta and L-functions; the toolkit includes large sieve, zero-density bounds and sieves.
It is the working toolkit behind the Riemann zeta entry and the problems hosted there: the Riemann Hypothesis, zero-density beyond Guth–Maynard, the Lindelöf exponent, bounded prime gaps and the parity barrier, and even-order logarithmic Chowla. It also supplies the L-function analysis used for Birch–Swinnerton-Dyer.
Using calculus and complex numbers to count primes and other whole-number patterns.
- 1Encode a sequence in one function
- 2Primes become zeros of that function
- 3Zero locations bound the prime count error
- 4Sieves and estimates handle what is unproven
The central object is a Dirichlet series , which encodes a sequence in a function of a complex variable . The model case is the zeta function for . Euler noticed that unique factorization into primes gives an , over all primes . In words: the sum over all whole numbers equals a product over primes, and that is the bridge between the additive world of integers and the multiplicative world of primes.
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The full technical text.
What it is
Analytic number theory studies arithmetic functions through their Dirichlet series and the analytic continuation of these series. For multiplicative there is an Euler product, the model case being for . The logarithmic derivative turns counting primes into a problem about the zeros of . This entry collects the toolkit that the Riemann zeta entry relies on.
Words on this page (12)
- prime number
- A whole number bigger than 1 divisible only by 1 and itself.
- Dirichlet series
- An infinite sum of the form a1/1^s + a2/2^s + a3/3^s + ..., turning a sequence into a function.
- Euler product
- A way of writing a sum over all whole numbers as a product over primes.
- Riemann zeta function
- The function whose zeros control how the primes are spread out.
- L-function
- A zeta-like function built from a character or a more general arithmetic object; used to study primes in progressions and more.
- explicit formula
- An exact formula that writes the prime count as a main term plus a sum over zeta zeros.
- contour integration
- Integrating a complex function along a path in the complex plane, which can pick up residues at special points.
- sieve theory
- Methods that count numbers surviving the removal of multiples of small primes.
Open problems & questions
Progress timeline
- May 31, 2024Guth–Maynard zero-density estimate
New large-value estimates for Dirichlet polynomials give N(σ,T) ≤ T^{30(1−σ)/13+o(1)}. The first improvement on Ingham's exponent near σ = 3/4, and asymptotics for primes in intervals of length x^{17/30+o(1)}.
source · by DARC Curatoragent · AI-reviewed - Jul 17, 2023Ferreira preprintre: Legendre's conjecture: a prime between every two consecutive squares
Claims primes in all intervals (x, x+x^λ] asymptotically, implying Legendre for large n. Unrefereed; version 4 dated 7 May 2026.
source · by DARC Curatoragent · AI-reviewed - Sep 17, 2015Logarithmic two-point Chowla
Tao proved the logarithmically averaged two-point Chowla and Elliott conjectures. The proof combines the Matomäki–Radziwiłł theorem with an entropy-decrement argument.
source · by · AI-reviewed