In his paper "Primes with restricted digits," James Maynard explores the existence of infinitely many prime numbers that do not contain a specific digit $a_0$ in their decimal expansion. The proof utilizes the Hardy-Littlewood circle method applied to a binary problem. To control the minor arcs, Maynard obtains suitable arithmetic information of both "Type I" and "Type II" using Harman's sieve. This involves decorrelating Diophantine conditions that determine when the Fourier transform of primes is large from digital conditions that determine when the Fourier transform of numbers with restricted digits is large. The estimates used in this study rely on a combination of techniques including the geometry of numbers, the large sieve, and moment estimates obtained through comparison with a Markov process. Overall, this research successfully demonstrates the existence of infinitely many prime numbers without a specific digit in their decimal expansion by employing advanced mathematical methods and analyzing various types of arithmetic information.
James Maynard's paper explores the idea that there are always more prime numbers with certain digits missing in their decimal form.
Proof is shown using a method called the Hardy-Littlewood circle method, which helps solve a problem involving binary numbers.
Maynard uses Harman's sieve to control and gather important information about two types of arithmetic conditions, called "Type I" and "Type II".
By separating certain mathematical conditions from digital conditions, Maynard can determine when the Fourier transform (a math concept) is large for both primes and numbers with restricted digits.
This research uses techniques like the geometry of numbers, the large sieve, and moment estimates to make educated guesses.
Overall, this study proves that there are always more prime numbers without certain digits by using advanced math methods and analyzing different types of arithmetic information.
Primes are a fundamental concept in mathematics that have fascinated mathematicians for centuries. These numbers, which can only be divided by 1 and themselves, have been studied extensively to understand their properties and patterns. In recent years, there has been a growing interest in exploring the existence of prime numbers with certain restrictions on their digits. This is where James Maynard's research paper "Primes with restricted digits" comes into play.
In this paper, Maynard delves into the question of whether there exist infinitely many prime numbers that do not contain a specific digit $a_0$ in their decimal expansion. This may seem like a simple question at first glance, but it turns out to be quite complex and requires advanced mathematical techniques to answer.
The proof presented by Maynard utilizes the Hardy-Littlewood circle method applied to a binary problem. The circle method is a powerful tool used in analytic number theory to study problems involving additive or multiplicative structures of integers. It involves representing integers as points on circles and using tools from Fourier analysis to analyze these points.
To control the minor arcs in this problem, Maynard employs Harman's sieve technique. This technique allows one to obtain suitable arithmetic information about both "Type I" (primes) and "Type II" (numbers with restricted digits) through decorrelating Diophantine conditions from digital conditions. Essentially, this means separating out the conditions that determine when the Fourier transform of primes is large from those that determine when the Fourier transform of numbers with restricted digits is large.
The estimates used in this study rely on a combination of techniques including the geometry of numbers, the large sieve, and moment estimates obtained through comparison with a Markov process. The geometry of numbers deals with questions related to lattice points (points whose coordinates are all integers) while the large sieve is an important tool for studying prime gaps - intervals between consecutive primes.
One key aspect of this research is the use of moment estimates obtained through comparison with a Markov process. A Markov process is a stochastic model that describes a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. By comparing this process to the problem at hand, Maynard is able to obtain important estimates that help prove his main result.
Overall, Maynard's research successfully demonstrates the existence of infinitely many prime numbers without a specific digit in their decimal expansion. This may seem like a small victory, but it has significant implications for our understanding of prime numbers and their distribution. It also showcases the power and versatility of advanced mathematical techniques in solving seemingly simple problems.
In conclusion, James Maynard's paper "Primes with restricted digits" provides valuable insights into an intriguing question about prime numbers. Through his use of advanced mathematical methods such as the Hardy-Littlewood circle method and Harman's sieve technique, he proves the existence of infinitely many primes without a specific digit in their decimal expansion. This research not only adds to our knowledge about prime numbers but also highlights the beauty and complexity of mathematics.
Let $a_0\in\{0,\dots,9\}$. We show there are infinitely many prime numbers which do not have the digit $a_0$ in their decimal expansion. The proof is an application of the Hardy-Littlewood circle method to a binary problem, and rests on obtaining suitable `Type I' and `Type II' arithmetic information for use in Harman's sieve to control the minor arcs. This is obtained by decorrelating Diophantine conditions which dictate when the Fourier transform of the primes is large from digital conditions which dictate when the Fourier transform of numbers with restricted digits is large. These estimates rely on a combination of the geometry of numbers, the large sieve and moment estimates obtained by comparison with a Markov process.
