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A Small Taste from My New Book: Season 2 Episode 8

Posted by Nikitas Sotiropoulos on April 08, 2026

The point of controlled zero distance Well-spaced ordinate estimate

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A Small Taste from My New Book: Season 2 Episode 7

Posted by Nikitas Sotiropoulos on April 03, 2026

Big‑O Techniques Critical Strip Geometry Local Zero Analysis Zeta Logarithmic Derivative Estimates

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Nikitas Sotiropoulos: Exploring math, code, and complex ideas.

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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: Season 2 Episode 5

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: Episode 2

A Small Taste from My New Book: Episode 2 Welcome to Episode 2 of " A Small Taste from My New Book." In this installment, we continue our journey through the elegant world of complex analysis, focusing on how classical contour integration techniques reveal deep connections across modern mathematics. This episode builds on the foundational ideas introduced previously, where we explored the interplay between keyhole contours and vertical line integrals. Here, we delve deeper into the art of contour deformation method that not only simplifies challenging integrals but also illuminates the underlying unity between transform calculus and analytic number theory. By examining how integrals over different contours can be related and transformed, we uncover the power of Cauchy's Theorem and the residue calculus. These tools allow us to transition seamlessly between representations, making it possible to evaluate integrals, extract coefficients, and even approach the analyti...

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