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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 3

In this episode, we return to one of the most delicate—and revealing—interfaces between complex analysis and analytic number theory: the way zeros of the Riemann zeta function encode global information through seemingly simple identities. Rather than treating these formulas as formal manipulations, we slow down and examine why they work, which assumptions are legitimate, and where symmetry plays a decisive role. The discussion centers on the logarithmic derivative of the completed zeta function and the constants that emerge when evaluating it at special points. By carefully exploiting the functional equation and the pairing of nontrivial zeros, we see how apparently asymmetric sums reorganize into clean, meaningful expressions. What might look at first like an arbitrary rearrangement turns out to be a structurally enforced consequence of analytic continuation and absolute convergence. Along the way, classical constants such as Euler’s constant and familiar special func...

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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