Category Archives: Math.NT

Notes from a talk at Dartmouth on the Fibonacci zeta function

I recently gave a talk “at Dartmouth”1. The focus of the talk was the (odd-indexed) Fibonacci zeta function:
$$ \sum_{n \geq 1} \frac{1}{F(2n-1)^s},$$
where $F(n)$ is the nth Fibonacci number. The theme is that the Fibonacci zeta function can be recognized as coming from an inner product of automorphic forms, and the continuation of the zeta function can be understood in terms of the spectral expansion of the associated automorphic forms.

This is a talk from ongoing research. I do not yet understand “what’s really going on”. But within the talk I describe a few different generalizations; firstly, there is a generalization to other zeta functions that can be viewed as traces of units on quadratic number fields, and secondly there is a generalization to quadratic forms recognizing solutions to Pell’s equation.

I intend to describe additional ideas from this talk in the coming months, as I figure out how pieces fit together. But for now, here are the slides.

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Pictures of equidistribution – the line

In my previous note, we considered equidistribution of rational points on the circle $X^2 + Y^2 = 2$. This is but one of a large family of equidistribution results that I’m not particularly familiar with.

This note is the first in a series of notes dedicated to exploring this type of equidistribution visually. In this note, we will investigate a simpler case — rational points on the line.

(more…)

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Points on X^2 + Y^2 = 2 equidistribute with respect to height

When you order rational points on the circle $X^2 + Y^2 = 2$ by height, these points equidistribute.

Stated differently, suppose that $I$ is an arc on the circle $X^2 + Y^2 = 2$. Then asymptotically, the number of rational points on the arc $I$ with height bounded by a number $H$ is equal to what you would expect if $\lvert I\rvert /2\sqrt{2}\pi$ of all points with height up to $H$ were on this arc. Here, $\lvert I\rvert /2\sqrt{2}\pi$ the ratio of the arclength of the arc $I$ with the total circumference of the circle.

This only makes sense if we define the height of a rational point on the circle. Given a point $(a/c, b/c)$ (written in least terms) on the circle, we define the height of this point to be $c$.

In forthcoming work with my frequent collaborators Chan Ieong Kuan, Thomas Hulse, and Alexander Walker, we count three term arithmetic progressions of squares. If $C^2 – B^2 = B^2 – A^2$, then clearly $A^2 + C^2 = 2B^2$, and thus a 3AP of squares corresponds to a rational point on the circle $X^2 + Y^2 = 2$. We compare one of our results to what you would expect from equidistribution. From general principles, we expected such equidistribution to be true. But I wasn’t sure how to prove it.

With helpful assistance from Noam Elkies, Emmanuel Peyre, and John Voight (who each immediately knew how to prove this), I learned how to prove this fact.

The rest of this note contains this proof.

(more…)

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Notes behind a talk: visualizing modular forms

Today, I’ll be at Bowdoin College giving a talk on visualizing modular forms. This is a talk about the actual process and choices involved in illustrating a modular form; it’s not about what little lies one might hold in their head in order to form some mental image of a modular form.1

This is a talk heavily inspired by the ICERM semester program on Illustrating Mathematics (currently wrapping up). In particular, I draw on2 conversations with Frank Farris (about using color to highlight desired features), Elias Wegert (about using logarithmically scaling contours), Ed Harriss (about the choice of colorscheme), and Brendan Hassett (about overall design choices).

There are very many pictures in the talk!

Here are the slides for the talk.

I wrote a few different complex-plotting routines for this project. At their core, they are based on sage’s complex_plot. There are two major variants that I use.

The first (currently called “ccomplex_plot”. Not a good name) overwrites how sage handles lightness in complex_plot in order to produce “contours” at spots where the magnitude is a two-power. These contours are actually a sudden jump in brightness.

The second (currently called “raw_complex_plot”, also not a good name) is even less formal. It vectorizes the computation and produces an object containing the magnitude and argument information for each pixel to be drawn. It then uses numpy and matplotlib to convert these magnitudes and phases into RGB colors according to a matplotlib-compatible colormap.

I am happy to send either of these pieces of code to anyone who wants to see them, but they are very much written for my own use at the moment. I intend to improve them for general use later, after I’ve experimented further.

In addition, I generated all the images for this talk in a single sagemath jupyter notebook (with the two .spyx cython dependencies I allude to above). This is also available here. (Note that using a service like nbviewer or nbconvert to view or convert it to html might be a reasonable idea).

As a final note, I’ll add that I mistyped several times in the preparation of the images for this talk. Included below are a few of the interesting-looking mistakes. The first two resulted from incorrectly applied conformal mappings, while the third came from incorrectly applied color correction.

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Non-real poles and irregularity of distribution I

$\DeclareMathOperator{\SL}{SL}$ $\DeclareMathOperator{\MT}{MT}$After the positive feedback from the Maine-Quebec Number Theory conference, I have taken some time to write (and slightly strengthen) these results.

We study the general theory of Dirichlet series $D(s) = \sum_{n \geq 1} a(n) n^{-s}$ and the associated summatory function of the coefficients, $A(x) = \sum_{n \leq x}’ a(n)$ (where the prime over the summation means the last term is to be multiplied by $1/2$ if $x$ is an integer). For convenience, we will suppose that the coefficients $a(n)$ are real, that not all $a(n)$ are zero, that each Dirichlet series converges in some half-plane, and that each Dirichlet series has meromorphic continuation to $\mathbb{C}$. Perron’s formula (or more generally, the forward and inverse Mellin transforms) show that $D(s)$ and $A(x)$ are duals and satisfy \begin{equation}\label{eq:basic_duality} \frac{D(s)}{s} = \int_1^\infty \frac{A(x)}{x^{s+1}} dx, \quad A(x) = \frac{1}{2 \pi i} \int_{\sigma – i \infty}^{\sigma + i \infty} \frac{D(s)}{s} x^s ds \end{equation} for an appropriate choice of $\sigma$.

Many results in analytic number theory take the form of showing that $A(x) = \MT(x) + E(x)$ for a “Main Term” $\MT(x)$ and an “Error Term” $E(x)$. Roughly speaking, the terms in the main term $\MT(x)$ correspond to poles from $D(s)$, while $E(x)$ is hard to understand. Upper bounds for the error term give bounds for how much $A(x)$ can deviate from the expected size, and thus describe the regularity in the distribution of the coefficients ${a(n)}$. In this article, we investigate lower bounds for the error term, corresponding to irregularity in the distribution of the coefficients.

To get the best understanding of the error terms, it is often necessary to work with smoothed sums $A_v(x) = \sum_{n \geq 1} a(n) v(n/x)$ for a weight function $v(\cdot)$. In this article, we consider nice weight functions, i.e.\ weight functions with good behavior and whose Mellin transforms have good behavior. For almost all applications, it suffices to consider weight function $v(x)$ that are piecewise smooth on the positive real numbers, and which take values halfway between jump discontinuities.

For a weight function $v(\cdot)$, denote its Mellin transform by \begin{equation} V(s) = \int_0^\infty v(x)x^{s} \frac{dx}{x}. \end{equation} Then we can study the more general dual family \begin{equation}\label{eq:general_duality} D(s) V(s) = \int_1^\infty \frac{A_v(x)}{x^{s+1}} dx, \quad A_v(x) = \frac{1}{2 \pi i} \int_{\sigma – i \infty}^{\sigma + i \infty} D(s) V(s) x^s ds. \end{equation}

We prove two results governing the irregularity of distribution of weighted sums. Firstly, we prove that a non-real pole of $D(s)V(s)$ guarantees an oscillatory error term for $A_v(x)$.

Theorem 1

Suppose $D(s)V(s)$ has a pole at $s = \sigma_0 + it_0$ with $t_0 \neq 0$ of order $r$. Let $\MT(x)$ be the sum of the residues of $D(s)V(s)X^s$ at all real poles $s = \sigma$ with $\sigma \geq \sigma_0$.Then \begin{equation} \sum_{n \geq 1} a(n) v(\tfrac{n}{x}) – \MT(x) = \Omega_\pm\big( x^{\sigma_0} \log^{r-1} x\big). \end{equation}


Here and below, we use the notation $f(x) = \Omega_+ g(x)$ to mean that there is a constant $k > 0$ such that $\limsup f(x)/\lvert g(x) \rvert > k$ and $f(x) = \Omega_- g(x)$ to mean that $\liminf f(x)/\lvert g(x) \rvert < -k$. When both are true, we write $f(x) = \Omega_\pm g(x)$. This means that $f(x)$ is at least as positive as $\lvert g(x) \rvert$ and at least as negative as $-\lvert g(x) \rvert$ infinitely often.

Theorem 2

Suppose $D(s)V(s)$ has at least one non-real pole, and that the supremum of the real parts of the non-real poles of $D(s)V(s)$ is $\sigma_0$. Let $\MT(x)$ be the sum of the residues of $D(s)V(s)X^s$ at all real poles $s = \sigma$ with $\sigma \geq \sigma_0$.Then for any $\epsilon > 0$, \begin{equation} \sum_{n \geq 1} a(n) v(\tfrac{n}{x}) – \MT(x) = \Omega_\pm( x^{\sigma_0 – \epsilon} ). \end{equation}


The idea at the core of these theorems is old, and was first noticed during the investigation of the error term in the prime number theorem. To prove them, we generalize proofs given in Chapter 5 of Ingham’s Distribution of Prime Numbers (originally published in 1932, but recently republished). There, Ingham proves that $\psi(x) – x = \Omega_\pm(x^{\Theta – \epsilon})$ and $\psi(x) – x = \Omega_\pm(x^{1/2})$, where $\psi(x) = \sum_{p^n \leq x} \log p$ is Chebyshev’s second function and $\Theta \geq \frac{1}{2}$ is the supremum of the real parts of the non-trivial zeros of $\zeta(s)$. (Peter Humphries let me know that chapter 15 of Montgomery and Vaughan’s text also has these. This text might be more readily available and perhaps in more modern notation. In fact, I have a copy — but I suppose I either never got to chapter 15 or didn’t have it nicely digested when I needed it).

Motivation and Application

Infinite lines of poorly understood poles appear regularly while studying shifted convolution series of the shape \begin{equation} D(s) = \sum_{n \geq 1} \frac{a(n) a(n \pm h)}{n^s} \end{equation} for a fixed $h$. When $a(n)$ denotes the (non-normalized) coefficients of a weight $k$ cuspidal Hecke eigenform on a congruence subgroup of $\SL(2, \mathbb{Z})$, for instance, meromorphic continuation can be gotten for the shifted convolution series $D(s)$ through spectral expansion in terms of Maass forms and Eisenstein series, and the Maass forms contribute infinite lines of poles.

Explicit asymptotics take the form \begin{equation} \sum_{n \geq 1} a(n)a(n-h) e^{-n/X} = \sum_j C_j X^{\frac{1}{2} + \sigma_j + it_j} \log^m X \end{equation} where neither the residues nor the imaginary parts $it_j$ are well-understood. Might it be possible for these infinitely many rapidly oscillating terms to experience massive cancellation for all $X$? The theorems above prove that this is not possible.

In this case, applying Theorem 2 with the Perron-weight \begin{equation} v(x) = \begin{cases} 1 & x < 1 \\ \frac{1}{2} & x = 1 \\ 0 & x > 1 \end{cases} \end{equation} shows that \begin{equation} \sideset{}{‘}\sum_{n \leq X} \frac{a(n)a(n-h)}{n^{k-1}} = \Omega_\pm(\sqrt X). \end{equation} Similarly, Theorem 1 shows that \begin{equation} \sideset{}{‘}\sum_{n \leq X} \frac{a(n)a(n-h)}{n^{k-1}} = \Omega_\pm(X^{\frac{1}{2} + \Theta – \epsilon}), \end{equation} where $\Theta < 7/64$ is the supremum of the deviations to Selberg’s Eigenvalue Conjecture (sometimes called the the non-arithmetic Ramanujan Conjecture).

More generally, these shifted convolution series appear when studying the sizes of sums of coefficients of modular forms. A few years ago, Hulse, Kuan, Walker, and I began an investigation of the Dirichlet series whose coefficients were themselves $\lvert A(n) \rvert^2$ (where $A(n)$ is the sum of the first $n$ coefficients of a modular form) was shown to have meromorphic continuation to $\mathbb{C}$. The behavior of the infinite lines of poles in the discrete spectrum played an important role in the analysis, but we did not yet understand how they affected the resulting asymptotics. I plan on revisiting these results, and others, with these results in mind.

Proofs

The proofs of these results will soon appear on the arXiv.

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Notes from a talk at the Maine-Quebec Number Theory Conference

Today I will be giving a talk at the Maine-Quebec Number Theory conference. Each year that I attend this conference, I marvel at how friendly and inviting an environment it is — I highly recommend checking the conference out (and perhaps modelling other conferences after it).

The theme of my talk is about spectral poles and their contribution towards asymptotics (especially of error terms). I describe a few problems in which spectral poles appear in asymptotics. Unlike the nice simple cases where a single pole (or possibly a few poles) appear, in these cases infinite lines of poles appear.

For a bit over a year, I have encountered these and not known what to make of them. Could you have the pathological case that residues of these poles generically cancel? Could they combine to be larger than expected? How do we make sense of them?

The resolution came only very recently.1

I will later write a dedicated note to this new idea (involving Dirichlet integrals and Landau’s theorem in this context), but for now — here are the slides for my talk.

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Talk: Finding Congruent Numbers, Arithmetic Progressions of Squares, and Triangles

Here are some notes for my talk Finding Congruent Numbers, Arithmetic Progressions of Squares, and Triangles (an invitation to analytic number theory), which I’m giving on Tuesday 26 February at Macalester College.

The slides for my talk are available here.

The overarching idea of the talk is to explore the deep relationship between

  1. right triangles with rational side lengths and area $n$,
  2. three-term arithmetic progressions of squares with common difference $n$, and
  3. rational points on the elliptic curve $Y^2 = X^3 – n^2 X$.

If one of these exist, then all three exist, and in fact there are one-to-one correspondences between each of them. Such an $n$ is called a congruent number.

By understanding this relationship, we also describe the ideas and results in the paper A Shifted Sum for the Congruent Number Problem, which I wrote jointly with Tom Hulse, Chan Ieong Kuan, and Alex Walker.

Towards the end of the talk, I say that in practice, the best way to decide if a (reasonably sized) number is congruent is through elliptic curves. Given a computer, we can investigate whether the number $n$ is congruent through a computer algebra system like sage.1

For the rest of this note, I’ll describe how one can use sage to determine whether a number is congruent, and how to use sage to add points on elliptic curves to generate more triangles corresponding to a particular congruent number.

Firstly, one needs access to sage. It’s free to install, but it’s quite large. The easiest way to begin using sage immediately is to use cocalc.com,  a free interface to sage (and other tools) that was created by William Stein, who also created sage.

In a sage session, we can create an elliptic curve through


> E6 = EllipticCurve([-36, 0])
> E6
Elliptic Curve defined by y^2 = x^3 - 36*x over Rational Field

More generally, to create the curve corresponding to whether or not $n$ is congruent, you can use


> n = 6   # (or anything you want)
> E = EllipticCurve([-n**2, 0])

We can ask sage whether our curve has many rational points by asking it to (try to) compute the rank.


> E6.rank()
1

If the rank is at least $1$, then there are infinitely many rational points on the curve and $n$ is a congruent number. If the rank is $0$, then $n$ is not congruent.2

For the curve $Y^2 = X^3 – 36 X$ corresponding to whether $6$ is congruent, sage returns that the rank is $1$. We can ask sage to try to find a rational point on the elliptic curve through


> E6.point_search(10)
[(-3 : 9 : 1)]

The 10 in this code is a limit on the complexity of the point. The precise definition isn’t important — using $10$ is a reasonable limit for us.

We see that this output something. When sage examines the elliptic curve, it uses the equation $Y^2 Z = X^3 – 36 X Z^2$ — it turns out that in many cases, it’s easier to perform computations when every term is a polynomial of the same degree. The coordinates it’s giving us are of the form $(X : Y : Z)$, which looks a bit odd. We can ask sage to return just the XY coordinates as well.


> Pt = E6.point_search(10)[0]  # The [0] means to return the first element of the list
> Pt.xy()
(-3, 9)

In my talk, I describe a correspondence between points on elliptic curves and rational right triangles. In the talk, it arises as the choice of coordinates. But what matters for us right now is that the correspondence taking a point $(x, y)$ on an elliptic curve to a triangle $(a, b, c)$ is given by
$$(x, y) \mapsto \Big( \frac{n^2-x^2}{y}, \frac{-2 \cdot x \cdot y}{y}, \frac{n^2 + x^2}{y} \Big).$$

We can write a sage function to perform this map for us, through


> def pt_to_triangle(P):
    x, y = P.xy()
    return (36 - x**2)/y, (-2*x*6/y), (36+x**2)/y

> pt_to_triangle(Pt)
(3, 4, 5)

This returns the $(3, 4, 5)$ triangle!

Of course, we knew this triangle the whole time. But we can use sage to get more points. A very cool fact is that rational points on elliptic curves form a group under a sort of addition — we can add points on elliptic curves together and get more rational points. Sage is very happy to perform this addition for us, and then to see what triangle results.


> Pt2 = Pt + Pt
> Pt2.xy()
(25/4, -35/8)
> pt_to_triangle(Pt2)
(7/10, 120/7, -1201/70)

Another rational triangle with area $6$ is the $(7/10, 120/7, 1201/70)$ triangle. (You might notice that sage returned a negative hypotenuse, but it’s the absolute values that matter for the area). After scaling this to an integer triangle, we get the integer right triangle $(49, 1200, 1201)$ (and we can check that the squarefree part of the area is $6$).

Let’s do one more.


> Pt3 = Pt + Pt + Pt
> Pt3.xy()
(-1587/1369, -321057/50653)
> pt_to_triangle(Pt3)
(-4653/851, -3404/1551, -7776485/1319901)

That’s a complicated triangle! It may be fun to experiment some more — the triangles rapidly become very, very complicated. In fact, it was very important to the main result of our paper that these triangles become so complicated so quickly!

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Slides for a talk at JMM 2019

Today, I’m giving a talk on Zeroes of L-functions associated to half-integral weight modular forms, which includes some joint work with Li-Mei Lim and Tom Hulse, and which alludes to other joint work touched on previously with Jeff Hoffstein and Min Lee (and which perhaps should have been finished a few years ago).

Here are the slides for my talk.

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Update to Second Moments in the Generalized Gauss Circle Problem

Last year, my coauthors Tom Hulse, Chan Ieong Kuan, and Alex Walker posted a paper to the arXiv called “Second Moments in the Generalized Gauss Circle Problem”. I’ve briefly described its contents before.

This paper has been accepted and will appear in Forum of Mathematics: Sigma.More randomized squares art

This is the first time I’ve submitted to the Forum of Mathematics, and I must say that this has been a very good journal experience. One interesting aspect about FoM: Sigma is that they are immediate (gold) open access, and they don’t release in issues. Instead, articles become available (for free) from them once the submission process is done. I was reviewing a publication-proof of the paper yesterday, and they appear to be very quick with regards to editing. Perhaps the paper will appear before the end of the year.

An updated version (the version from before the handling of proofs at the journal, so there will be a number of mostly aesthetic differences with the published version) of the paper will appear on the arXiv on Monday 10 December.1

A new appendix has appeared

There is one major addition to the paper that didn’t appear in the original preprint. At one of the referee’s suggestions, Chan and I wrote an appendix. The major content of this appendix concerns a technical detail about Rankin-Selberg convolutions.

If $f$ and $g$ are weight $k$ cusp forms on $\mathrm{SL}(2, \mathbb{Z})$ with expansions $$ f(z) = \sum_ {n \geq 1} a(n) e(nz), \quad g(z) = \sum_ {n \geq 1} b(n) e(nz), $$ then one can use a (real analytic) Eisenstein series $$ E(s, z) = \sum_ {\gamma \in \mathrm{SL}(2, \mathbb{Z})_ \infty \backslash \mathrm{SL}(2, \mathbb{Q})} \mathrm{Im}(\gamma z)^s $$ to recognize the Rankin-Selberg $L$-function \begin{equation}\label{RS} L(s, f \otimes g) := \zeta(s) \sum_ {n \geq 1} \frac{a(n)b(n)}{n^{s + k – 1}} = h(s) \langle f g y^k, E(s, z) \rangle, \end{equation} where $h(s)$ is an easily-understandable function of $s$ and where $\langle \cdot, \cdot \rangle$ denotes the Petersson inner product.

When $f$ and $g$ are not cusp forms, or when $f$ and $g$ are modular with respect to a congruence subgroup of $\mathrm{SL}(2, \mathbb{Z})$, then there are adjustments that must be made to the typical construction of $L(s, f \otimes g)$.

When $f$ and $g$ are not cusp forms, then Zagier2 provided a way to recognize $L(s, f \otimes g)$ when $f$ and $g$ are modular on the full modular group $\mathrm{SL}(2, \mathbb{Z})$. And under certain conditions that he describes, he shows that one can still recognize $L(s, f \otimes g)$ as an inner product with an Eisenstein series as in \eqref{RS}.

In principle, his method of proof would apply for non-cuspidal forms defined on congruence subgroups, but in practice this becomes too annoying and bogged down with details to work with. Fortunately, in 2000, Gupta3 gave a different construction of $L(s, f \otimes g)$ that generalizes more readily to non-cuspidal forms on congruence subgroups. His construction is very convenient, and it shows that $L(s, f \otimes g)$ has all of the properties expected of it.

However Gupta does not show that there are certain conditions under which one can recognize $L(s, f \otimes g)$ as an inner product against an Eisenstein series.4 For this paper, we need to deal very explicitly and concretely with $L(s, \theta^2 \otimes \overline{\theta^2})$, which is formed from the modular form $\theta^2$, non-cuspidal on a congruence subgroup.

The Appendix to the paper can be thought of as an extension of Gupta’s paper: it uses Gupta’s ideas and techniques to prove a result analogous to \eqref{RS}. We then use this to get the explicit understanding necessary to tackle the Gauss Sphere problem.

There is more to this story. I’ll return to it in a later note.

Other submission details for FoM: Sigma

I should say that there are many other revisions between the original preprint and the final one. These are mainly due to the extraordinary efforts of two Referees. One Referee was kind enough to give us approximately 10 pages of itemized suggestions and comments.

When I first opened these comments, I was a bit afraid. Having so many comments was daunting. But this Referee really took his or her time to point us in the right direction, and the resulting paper is vastly improved (and in many cases shortened, although the appendix has hidden the simplified arguments cut in length).

More broadly, the Referee acted as a sort of mentor with respect to my technical writing. I have a lot of opinions on technical writing,5 but this process changed and helped sharpen my ideas concerning good technical math writing.

I sometimes hear lots of negative aspects about peer review, but this particular pair of Referees turned the publication process into an opportunity to learn about good mathematical exposition — I didn’t expect this.

I was also surprised by the infrastructure that existed at the University of Warwick for handling a gold open access submission. As part of their open access funding, Forum of Math: Sigma has an author-pays model. Or rather, the author’s institution pays. It took essentially no time at all for Warwick to arrange the payment (about 500 pounds).

This is a not-inconsequential amount of money, but it is much less than the 1500 dollars that PLoS One uses. The comparison with PLoS One is perhaps apt. PLoS is older, and perhaps paved the way for modern gold open access journals like FoM. PLoS was started by group of established biologists and chemists, including a Nobel prize winner; FoM was started by a group of established mathematicians, including multiple Fields medalists.6

I will certainly consider Forum of Mathematics in the future.

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The wrong way to compute a sum: addendum

Cellular Automata from Rule 106 (random initial configuration)In my previous note, I looked at an amusing but inefficient way to compute the sum $$ \sum_{n \geq 1} \frac{\varphi(n)}{2^n – 1}$$ using Mellin and inverse Mellin transforms. This was great fun, but the amount of work required was more intense than the more straightforward approach offered immediately by using Lambert series.

However, Adam Harper suggested that there is a nice shortcut that we can use (although coming up with this shortcut requires either a lot of familiarity with Mellin transforms or knowledge of the answer).

In the Lambert series approach, one shows quickly that $$ \sum_{n \geq 1} \frac{\varphi(n)}{2^n – 1} = \sum_{n \geq 1} \frac{n}{2^n},$$ and then evaluates this last sum directly. For the Mellin transform approach, we might ask: do the two functions $$ f(x) = \sum_{n \geq 1} \frac{\varphi(n)}{2^{nx} – 1}$$ and $$ g(x) = \sum_{n \geq 1} \frac{n}{2^{nx}}$$ have the same Mellin transforms? From the previous note, we know that they have the same values at $1$.

We also showed very quickly that $$ \mathcal{M} [f] = \frac{1}{(\log 2)^2} \Gamma(s) \zeta(s-1). $$ The more difficult parts from the previous note arose in the evaluation of the inverse Mellin transform at $x=1$.

Let us compute the Mellin transform of $g$. We find that $$ \begin{align}
\mathcal{M}[g] &= \sum_{n \geq 1} n \int_0^\infty \frac{1}{2^{nx}} x^s \frac{dx}{x} \notag \\
&= \sum_{n \geq 1} n \int_0^\infty \frac{1}{e^{nx \log 2}} x^s \frac{dx}{x} \notag \\
&= \sum_{n \geq 1} \frac{n}{(n \log 2)^s} \int_0^\infty x^s e^{-x} \frac{dx}{x} \notag \\
&= \frac{1}{(\log 2)^2} \zeta(s-1)\Gamma(s). \notag
\end{align}$$ To go from the second line to the third line, we did the change of variables $x \mapsto x/(n \log 2)$, yielding an integral which is precisely the definition of the Gamma function.

Thus we see that $$ \mathcal{M}[g] = \frac{1}{(\log 2)^s} \Gamma(s) \zeta(s-1) = \mathcal{M}[f],$$ and thus $f(x) = g(x)$. (“Nice” functions with the same “nice” Mellin transforms are also the same, exactly as with Fourier transforms).

This shows that not only is $$ \sum_{n \geq 1} \frac{\varphi(n)}{2^n – 1} = \sum_{n \geq 1} \frac{n}{2^n},$$ but in fact $$ \sum_{n \geq 1} \frac{\varphi(n)}{2^{nx} – 1} = \sum_{n \geq 1} \frac{n}{2^{nx}}$$ for all $x > 1$.

I think that’s sort of slick.

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