TME-EMT Wiki

Fully explicit results in multiplicative number theory are often scattered through the litterature. The aim of this site is to present annoted bibliographies in order to keep track of the current knowledge. By the way, the acronym TME-EMT stands for

Théorie Multiplicative Explicite des nombres / Explicit Multiplicative number Theory

Managing editors: André de Camargo.

Note

Modifications are to be sent to: tmeemt.wiki@gmail.com

Programming and Python side: Dominique Benielli (Aix Marseille Université, Cellule de développement Institut Archimède)

Here are the contributors:

Olivier Bordellès, André de Camargo, Pierre Dusart, Charles Greathouse, Harald Helfgott, Kiran S. Kedlaya, Ethan Lee, Pieter Moree, Akhilesh P, Olivier Ramaré, Enrique Treviño.

• Explicit bounds on primes
  • 1. Bounds on primes, in special ranges
  • 2. Bounds on primes, without any congruence condition
  • 3. Bounds containing the \(n\)-th prime
  • 4. Bounds on primes in arithmetic progressions
  • 5. Least prime verifying a condition
• Explicit bounds on the Moebius function
  • 1. Bounds on \(M(D)=\sum\limits_{d\le D}\mu(d)\)
  • 2. Bounds on \(m(D)=\sum\limits_{d\le D}\frac{\mu(d)}{d}\)
  • 3. Bounds on \(\check{m}(D)=\sum\limits_{d\le D}\mu(d)\log(D/d)/d\)
  • 4. Miscellanae
  • 5. The Moebius function and arithmetic progressions
• Exact computations of the number of primes
• Computations of arithmetical constants
  • 1. Euler products of rational functions
  • 2. Some special sums over prime values that are derivatives
• Explicit results on exponential sums
  • 1. Bounds with the first derivative
  • 2. Bounds with the second derivative
  • 3. Bounds with the third derivative
  • 4. Bounds with higher derivatives
  • 5. Iterated Van der Corput Inequality
  • 6. Explicit Poisson Formula
• Bounds for the Gamma function
  • 1. Bounds for \(\Gamma(x)\) for real \(x\)
  • 2. Bounds for the Digamma function \(\psi(x) = \Gamma'(x)/\Gamma(x)\) for real \(x\)
  • 3. Bounds for the Polygamma functions \(\psi(x)', \psi(x)'', \psi(x)''' ...\) for real \(x\)
  • 4. Complex Stirling formula
  • 4. Complex Stirling formula
• Bounds for \(|\zeta(s)|\), \(|L(s,\chi)|\) and related questions
  • 1. Approximating \(|\zeta(s)|\) or \(L\)-series in the critical strip
  • 2. Size of \(|\zeta(s)|\) and of \(L\)-series
  • 3. On the total number of zeroes
  • 4. \(L^2\)-averages
  • 5. Bounds on the real line
• Size of \(L(1,\chi)\)
  • 1. Upper bounds for \(|L(1,\chi)|\)
  • 2. Lower bounds for \(|L(1,\chi)|\)
• Explicit zero-free regions for the \(\zeta\) and \(L\) functions
  • 1. Numerical verifications of the Generalized Riemann Hypothesis
  • 2. Asymptotical zero-free regions
  • 3. Real zeros
  • 4. Density estimates
  • 5. Miscellanae
• Short intervals containing primes
  • 1. Interval with primes, without congruence condition, saving a constant
  • 2. Interval with primes, no congruence condition, saving more than a constant
  • 3. Interval with primes under RH, without any congruence condition
  • 4. Interval with primes, with congruence condition
• Averages of non-negative multiplicative functions
  • 1. Asymptotic estimates
  • 2. Upper bounds
  • 3. Estimates of some special functions
  • 4. Euler products
• Integer Points near Smooth Plane Curves
  • 1. Bounds using elementary methods
  • 2. Bounds using exponential sums techniques
  • 3. Integer points on curves
• Explicit pointwise upper bounds for some arithmetic functions
• Explicit bounds for class numbers
  • 1. Majorising \(h_K\mathcal{R}_K\)
  • 2. Majorising \(h_K\)
  • 3. Using the influence of the small primes
  • 4. The \(h^-_K\) of CM-fields
• Sieve bounds
  • 1. Some upper bounds
  • 2. Density estimates
  • 3. Combinatorial sieve estimates
  • 4. Integers free of small prime factors
• Character sums
  • 1. Explicit Polya-Vinogradov inequalities
  • 2. Burgess type estimates
• Tools on Fourier transforms
  • 1. The large sieve inequality
• Tools on Mellin transforms
  • 1. Explicit truncated Perron formula
  • 2. \(L{}^2\)-means
• Bounds on the Dedekind zeta-function
  • 1. Size
  • 2. Number of ideals
  • 3. Bounds for the residue of the Dedekind zeta-function
  • 4. Zeroes and zero-free regions
  • 5. Real Zeroes.
• Primitive Roots
• Primes in number fields
  • 1. On the number of primes in a number field
  • 2. On the Chebotarev Density Theorem
  • 3. On the least prime ideal
• Bibliography

Note

PDF version of TME-EMT-Wiki