A Small Taste from My New Book: Episode 10

 

Welcome to Episode 10 of A Small Taste from My New Book. In this episode, we journey into the elegant world of asymptotic formulas, focusing on Euler’s constant and the harmonic series. Drawing from the detailed expositions in my book, The Riemann Hypothesis Revealed, we’ll explore how integral representations and analytic techniques illuminate the subtle behavior of these classical mathematical objects. Whether you’re seeking rigorous proofs or insightful connections, this episode offers a clear and accessible pathway to understanding the deep structure behind Euler’s constant and related series.

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As detailed in my book, The Riemann Hypothesis Revealed, within the section on asymptotic formulas for Euler’s constant and the harmonic series, I present the following results as exercises.

1. Show that

and deduce that

2. Next, consider

and prove that .

3. Use the above to prove

4. Show that Euler’s constant can also be represented as

Exercises 1–3 are straightforward to address (in fact, for exercise 3, simply differentiating with respect to n yields the result). Therefore, let’s turn our attention to exercise 4 and provide a detailed proof of the requested representation:

Define

We want to show that

Rewrite  in a more useful form.

Use

so

Group the integrals over :

Take the limit as .

The integrand

is continuous on  and has a finite limit as :

So

is integrable on  and we can pass to the limit:

Therefore

where  is the indicator of .

Thus,

Identify this limit with Euler’s constant.

From exercises (1-3), you should have already shown that

If you split this integral at

Since we have

it follows

Now add and subtract  inside the integrand

We claim that

In fact, since

if ,

Incidentally, this expansion reveals the presence of Bernoulli numbers—a fascinating connection that emerges naturally in the analysis. Have you noticed their appearance in this context as well?

So, at this point we have established

Hence

and

Rewrite  to pair with the first part of .

For ,

and

Since for

and

we have

where

But   and in fact there is a very clean way to see it if you interpret the difference in the only meaningful way, namely as a joint limit at the singularity.

Define

I’ll show you that

The key point is that the first integral can be computed exactly.

Make the change of variables

Then

and therefore

Hence

On the other hand,

So the difference is

Now take the limit.

Since

we obtain

and therefore

So, in the correct regularized sense,

In particular, the formal expression

is equal to  when both divergent integrals are coupled at the common singularity .

Thus, we have proved the desired representation

 

Epilogue

The exploration in Episode 10 highlights the remarkable interplay between asymptotic analysis, special functions, and classical series. By delving into integral representations for Euler’s constant and the harmonic series, we have seen how analytic techniques can reveal the subtle structure underlying familiar mathematical constants. These results not only provide rigorous proofs and elegant formulas but also demonstrate the enduring power of complex analysis to unify diverse areas of mathematics.

As we reflect on the journey through asymptotic formulas, it becomes clear that the beauty of mathematics often lies in the connections between seemingly disparate ideas. From the behavior of integrals at infinity to the precise evaluation of limits, each step deepens our understanding and appreciation for the subject. If you found this episode insightful, I encourage you to explore the broader discussions in my book, The Riemann Hypothesis Revealed, where these themes are developed with full transparency and step-by-step clarity.

Let this episode serve as a reminder that the pursuit of mathematical knowledge is not just about solving problems, but about uncovering the elegant structures that bind the mathematical universe together.