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A Small Taste from My New Book: Season 2 Episode 10

Posted by Nikitas Sotiropoulos on April 25, 2026

Complex Plane Applications Laplace’s method Steepest descent

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A Small Taste from My New Book: Season 2 Episode 9

Posted by Nikitas Sotiropoulos on April 17, 2026

From Finite Sums to Analytic Continuation Perron’s Formula and Vanishing Tails

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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 6

In this episode, we step from global statements about the zeros of the Riemann zeta function into their local behavior . While classical results describe how many zeros lie below a given height on average, they say very little about what happens in short vertical intervals. Yet it is precisely this local structure—clustering, spacing, and short-scale irregularity—that reveals the true analytic complexity of the zeta function. To address this, we return to one of the most versatile tools in complex analysis: Jensen’s formula . By carefully choosing the center and radius of the Jensen disk, and by exploiting regions where the zeta function (or its completed form ξ) is well-controlled, we turn analytic growth estimates into concrete bounds on the number of zeros in narrow strips. The geometry here is not incidental. The specific placement of the disk, the choice of radii, and the alignment with the interval are all tuned to capture exactly the zeros that matter—no more, n...

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

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

Mathematicians sometimes overlook the fact that their audience may not share the same familiarity with the terminology and conventions used in a lecture or written exposition. This is particularly challenging for readers encountering a topic for the first time, or for those who are not accustomed to the author’s personal notation and style. As a result, even well-presented ideas can become difficult to follow if the foundational language is not made explicit. To avoid unnecessary confusion and to make the discussion as accessible as possible, we will begin by carefully introducing the basic concepts, notation, and assumptions that will be used throughout this text. Establishing this common ground will allow us to focus on the mathematical ideas themselves, rather than on decoding terminology, and will provide a clear baseline from which more advanced material can be developed. In “ The Riemann Hypothesis Revealed ” at the section “The growth notation” I explain the idea behind Bi...

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