The integral formula for secant is often introduced in a very artificial way. In this post, I look at the history of this integral and give several derivations.
One of Landau’s four problems from 1912 concerns the infinitude of primes in the values of a certain quadratic polynomial.
In this post, we show that the largest prime factors of the values of this polynomial are “relatively large” infinitely often.
Dirichlet’s Theorem on the infinitude of primes in arithmetic progressions relies on the non-vanishing of non-trivial Dirichlet characters at 1.
In this post, I’ll show how this reduction can be introduced in an intuitive way via sieve theory. If we actually sieve, we obtain estimates for the number of integers whose prime factors lie in given congruence classes.
Which integers are a multiple of the sum of their “small” divisors? In the post, we study whether this set should be finite or not.
Our partial solution relates this problem to many outstanding conjectures in number theory about the distribution of prime numbers, such as the twin prime conjecture and the infinitude of Mersenne primes.
In this note, I’ll discuss why square-root cancellation is so typical in problems in number theory and give a quick survey of important sums known or widely conjectured to satisfy bounds of this form.
This post discusses two classic problems in analytic number theory: the Gauss circle problem and the Dirichlet divisor problem.
These problems are known to be related at a deep level, a fact which is often missed at first glance because the obvious/early attacks on them look quite different.
In this post, I compare these “trivial” estimates, and show how Gauss’ estimate can be realized using a few different techniques.
For many calculus students, Riemann sums are those annoying things that show up in the derivation of the arc length formula.
In truth, these handy sums have done so much more. In this post, I’ll give some examples of Riemann sums dating from before the birth of calculus and some applications of Riemann sums that are still used today.