A Small Taste from My New Book: Episode 12

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From Numbers to Asymptotic Summation

How did mathematicians bridge the gap between simple sums and the powerful tools of modern analysis?

In the early days of calculus, mathematicians like Pierre de Fermat and Johann Faulhaber tackled the challenge of summing powers of integers—a problem that would eventually lead to the development of integral calculus. Their work laid the groundwork for what we now call Riemann sums, the method of approximating areas under curves by summing up rectangles. This approach not only helped define the integral but also connected discrete mathematics with continuous analysis.

But the story doesn’t end there. Jacob Bernoulli discovered that these sums could be expressed using special numbers—now known as Bernoulli numbers—and polynomials that bear his name. These Bernoulli numbers and polynomials appear everywhere in number theory and combinatorics, and they even have their own beautiful symmetry properties.

Euler-Maclaurin formula is a bridge between sums and integrals that introduces correction terms involving Bernoulli numbers. This formula is a cornerstone of asymptotic analysis, allowing mathematicians to approximate sums with remarkable precision and to understand the behavior of functions as their arguments grow large.

Mathematicians often downplay the importance of the Euler-Maclaurin formula in textbooks—mentioning, for example, the trapezoidal rule without explaining its deeper origins. Authors typically introduce these formulas as practical tools for approximating integrals, but rarely emphasize that such rules are direct consequences of the Euler-Maclaurin formula—a powerful bridge between discrete sums and continuous integrals. This tendency to compartmentalize techniques can obscure the unity and elegance of mathematical analysis, leaving readers unaware of the profound connections that underpin even the simplest computational recipes.

You’ll find a detailed and insightful exploration of Bernoulli numbers, summation formulas and techniques in my book, “The Riemann Hypothesis Revealed,” specifically in the section titled: “A Path from Numbers to Asymptotic Summation”.

 Why does this matter?

Because these techniques—born from the need to sum simple powers—are now essential tools in analytic number theory, the study of the Riemann zeta function, and the quest to understand the distribution of prime numbers.

If you’ve ever wondered how mathematicians turn the infinite into the manageable, or how the humble sum connects to the deepest mysteries of mathematics, this is the path that leads there.

Let the exploration begin.

Stirling’s formula from Euler–Maclaurin (the canonical asymptotic bridge)

We start from the most innocent discrete object:

Apply Euler–Maclaurin to

You get

Since

this becomes

Then regrouping the constant terms gives

Set

In the limit

So

As established in my book “The Riemann Hypothesis Revealed” 

Hence

Therefore,

or

This is not just “Stirling’s formula” it is a full asymptotic expansion whose coefficients are precisely Bernoulli numbers.

Why is this interesting for our theme? Because the discrete object  is turned into

  • a smooth integral,
    • plus a purely arithmetic correction series.

    This is probably the cleanest historical example of: Euler–Maclaurin as a machine for asymptotic summation.

    The harmonic numbers and the appearance of

    Take

    Apply Euler–Maclaurin to .

    You obtain

    Again set

    It is well known that

    Subtract  from

    Take the limit  and immediately the result is  .

    Thus,

    Using the identity (as detailed in “The Riemann Hypothesis Revealed”)

    the expansion becomes

    So:

    • Euler–Maclaurin produces Bernoulli numbers,
    • analytic number theory rewrites them as special values of the zeta function.

    This is exactly where asymptotic summation meets the analytic continuation of ζ(s).

    It is one of the first places where the zeta function is forced into an apparently elementary sum.

    Epilogue to Episode 12

    Episode 12 explores the development from basic summation formulas to the Euler–Maclaurin formula, as presented in the section “A Path from Numbers to Asymptotic Summation” of the book: ”The Riemann Hypothesis Revealed.” It demonstrates how analytic techniques connect discrete sums with continuous integrals, providing a unified perspective on classical and modern mathematical analysis.

     By examining the role of Bernoulli numbers and the emergence of asymptotic expansions, the episode clarified how classical tools like Euler–Maclaurin underpin results such as Stirling’s formula and the asymptotics of harmonic numbers. These methods are not just theoretical curiosities—they remain central in analytic number theory and the study of special functions, including the Riemann zeta function.

    The main takeaway is that the bridge between sums and integrals, built by mathematicians over centuries, is both practical and conceptually unifying. The techniques discussed provide a reliable framework for approximating sums, understanding growth rates, and connecting discrete and continuous mathematics. For readers interested in deeper details or further examples, the referenced book offers a more comprehensive treatment of these ideas.

     

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