One Residue at a Time

In the previous episodes, we developed the analytic machinery needed to connect prime numbers with the complex zeros of the Riemann zeta function. In this episode, we bring those ideas to life. Our focus is the explicit formula for the Chebyshev function ψ(t), a central object in analytic number theory that encodes the distribution of prime powers. This episode marks a shift from purely theoretical derivations to concrete computation and visualization . We evaluate ψ(t) directly from its arithmetic definition and then reconstruct it using the explicit formula, where primes emerge as a superposition of oscillatory contributions arising from the nontrivial zeros of ζ(s). Under the assumption of the Riemann Hypothesis, these oscillations take a particularly transparent and wave-like form. By comparing the exact Chebyshev function with truncated versions of the explicit formula, we gain intuition for how much information is carried by finitely many zeta zeros and how their colle...

A Small Taste from My New Book: Season 2 Episode 4

In this episode, we continue our exploration of arithmetic functions from an analytic perspective. Building on ideas introduced earlier in the season, we focus on techniques that reveal how analytic tools can encode deep number‑theoretic information. These methods will serve as a bridge between theory and computation, preparing us for a more algorithmic approach in the episodes to come. In “ The Riemann Hypothesis Revealed ”, the reader encounters a number of exercises/tasks that can be worked out without difficulty. Beyond completeness, the results obtained below play a central role in the narrative that follows. In particular, they prepare the ground for the classical explicit formula: toward which we have been gradually building since the previous episode. This “crude” form is the Explicit Formula for the Chebyshev function , which is the fundamental bridge between the prime numbers and the zeros of the Riemann zeta function . It relates ...

A Small Taste from My New Book: Episode 8

Welcome to Episode 8 of A Small Taste from My New Book . In this episode, we venture into the heart of number theory by exploring Euler’s totient function and its deep relationship with the roots of unity. The totient function, which counts the positive integers up to that are coprime to , is not only a fundamental object in arithmetic but also reveals surprising connections to the geometry of the complex plane through primitive roots of unity. But the story doesn’t end there. The totient function is intimately linked to one of the most celebrated objects in mathematics: the Riemann zeta function. Through elegant identities and Dirichlet series, we’ll see how the sum of totients over all positive integers can be expressed in terms of products involving the zeta function, bridging combinatorics, analysis, and the deep structure of the integers. This connection underpins powerful results in analytic number theory and highlights the unity between seemingly disparate areas of ...

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