On the Constant Terms of Certain Laurent Series

Preprint

Article

On the Constant Terms of Certain Laurent Series

Altmetrics

Downloads

307

Views

Comments

Barry Brent^*

This version is not peer-reviewed

Submitted:

28 June 2023

Posted:

29 June 2023

You are already at the latest version

Alerts

Abstract

We study the divisibility properties of the constant terms of certain meromorphic modular forms for Hecke groups. We relate those properties to those of some sequences that have already appeared in the literature. For possible use in later drafts, we show how to invert the map taking a Laurent series $f(x) = 1/x + \sum_{n=0}^{\infty}a_{n+1} x^n$ to the sequence of constant terms of its positive powers. At the end of the article, we construct from elementary arithmetic functions some meromorphic but not necessarily modular functions and study their constant terms.

Keywords:

Subject: Computer Science and Mathematics - Algebra and Number Theory

1. Introduction

For

g (x)

a Laurent series in x, let

C (g)

denote its constant term. Let

f (x) = 1 / x + \sum_{n = 0}^{\infty} a_{n + 1} x^{n}

(sic.) In this article, we study

C (f^{k})

, mostly in settings where (after substituting one of several exponential functions for x) f is a meromorphic modular form for some matrix group.

The constant

C (f^{k})

is a function of the coefficients

a_{1}, . . ., a_{k}

. Furthermore the numbers

C (f), C (f^{2}), . . .

determine

a_{1}, a_{2}, . . .

. To see this, let

c_{k}

be the coefficient of

x^{k}

in the polynomial

γ (x) = {(1 + \sum_{n = 1}^{k} a_{n} x^{n})}^{k}

. It is clear that

c_{k} = C (f^{k})

. We have

c_{1} = a_{1}

c_{2} = a_{1}^{2} + 2 a_{2}

c_{3} = a_{1}^{3} + 6 a_{1} a_{2} + 3 a_{3}

, etc.

The numerical coefficients on the right sides of these equations may be calculated using the multinomial theorem. An entirely straightforward application of the multinomial theorem expresses

γ (x)

in terms of powers of the monomials

a_{i} x^{i}

. To obtain the numerical coefficients of the sums for

c_{k}

, we will express

γ (x)

as a linear combination simply of the powers of x. We also show how to invert the map taking a Laurent series

f (x) = 1 / x + \sum_{n = 0}^{\infty} a_{n + 1} x^{n}

to the sequence

{c_{1}, c_{2}, . . .}

In the present draft we will not formalize the inversion procedure, but we list here the first few solutions:

a_{1} = c_{1}, a_{2} = c_{2} / 2 - c_{1}^{2} / 2, a_{3} = c_{3} / 3 - c_{1} c_{2} + 2 c_{1}^{3}

, etc. We observe that the expressions for

c_{k}

and the “reciprocal” expressions for

a_{k}

have in common that they are linear combinations of certain monomials from which it is possible to read off in a natural way certain partitions of k, and the same partitions appear in both situations.

The occasion for our interest was a problem in the theory of quadratic forms, which led us to the empirical finding that equations (1) and (2) below and corresponding equations for other meromorphic modular forms are valid for

k \leq

50 [9]. Here we test (1), (2) and several analogues for

k \leq 5000

The function

Δ (z)

occurring in Equation (2) is defined as follows. For z in the upper half plane and

q = q (z) = exp (2 π i z), Δ (z)

is the weight twelve normalized cusp form for

S L (2, Z)

with Fourier expansion

\sum_{n = 1}^{\infty} τ (n) q^{n}

, where

τ

denotes Ramanujan’s function. The reciprocal

1 / Δ (z)

appears in expressions for the dimensions of certain Lie algebras ([15], page 328; [16], page 45.) It also appears in string theory, for example, in the counting of black hole microstates ([26], Equation (14).) We will study the constant terms

C (1 / Δ^{k}) (k

a positive integer).

The Klein invariant appearing in Equation (1),

j (z) = 1 / q + \sum_{n = 0}^{\infty} c (n) q^{n}

, defined on the upper half of the complex plane with

c (0) = C (j) = 744

, is central (for example) to the classical theory of modular forms and to the moonshine phenomenon. We will also study the constant terms

C (j^{k})

Constant terms of meromorphic modular forms came into our work on quadratic forms as follows. Siegel studied the constant terms in the Fourier expansions of a particular family of meromorphic modular forms

T_{h}

for

S L (2, Z)

(“level one modular forms”) in 1969 [34,35]. Siegel demonstrated that these constant terms never vanish. He used this to establish a bound on the exponent of the first non-vanishing Fourier coefficient for a level one entire modular form f of weight h such that the constant term of f is itself non-vanishing. Theta functions fit this description, so Siegel was able to give an upper bound on the least positive integer represented by a positive-definite even unimodular quadratic form in

2 h

variables. While working on an extension of Siegel’s result on the non-vanishing of the

T_{h}

constant terms to higher-level modular forms, we came across the regularities described in equations (1) and (2). It is apparent that, if only we had proofs of these statements and their analogues, we would have known that the constant terms of

1 / Δ^{k}, j^{k}

, and their analogues were non-zero immediately.

One question we study is the nature of the special role of the primes

p = 2

and 3 in equations (1) and (2): why these primes but not others (apparently)? We searched for regularities involving other primes among the modular forms for Hecke groups (section 7.) Along the way we made observations relevant to the classical situation as well (conjecture 3.) Constant terms of meromorphic modular forms of certain kinds appear to have multiplicative structure. While seeking a level two version of Siegel’s result, the present writer found numerical evidence for divisibility properties of the constant terms for several kinds of modular form, including the

T_{h}

[9]; if these properties hold, the constant terms cannot vanish. 1 Let

d_{b} (n)

be the sum of the digits in the base b expansion of n. Then (apparently)

{o r d}_{2} (C (j^{k})) = {o r d}_{2} (C (1 / Δ^{k})) = 3 d_{2} (k)

(1)

and

{o r d}_{3} (C (j^{k})) = {o r d}_{3} (C (1 / Δ^{k})) = d_{3} (k) .

(2)

We argue (based on numerical experiments) that the

C (j^{k})

inherit the stated properties from the O.E.I.S. sequence A005148 [28], which was originally studied by Newman, Shanks and Zagier [27,44] in an article on its use in series approximations to

π

We tried to find patterns in the p-orders of constant terms of j and other modular forms for

S L (2, Z)

for p larger than three. Our search within

S L (2, Z)

seemed to fail, so we searched among the Hecke groups

G (λ_{n}), n = 3, 4, . . .

. The matrix group

S L (2, Z)

coincides with the Hecke group

G (λ_{3})

, discussed below. It is isomorphic to the product of cyclic groups

C_{2} * C_{3}

; while in general

G (λ_{m}) ≅ C_{2} * C_{m}

for

m = 3, 4, . . . .

We will state some conjectures about the constant terms, for example, of meromorphic forms for Hecke groups isomorphic to

C_{2} * C_{p^{k}}, p

prime.

Recently we found apparent regularities for

p = 5, 7, 11

in the original case of

S L (2, Z)

(conjectures 2 and 13.) They are conditions equivalent to the statement that

{o r d}_{p} (C (f))

vanishes (for

p = 5, 7, 11

when

f = j^{k}

, and for

p = 5

and 7 when

f = 1 / Δ^{k}

.) These conditions are simple restrictions on the digits in the base p expansions of k. The author’s thesis advisor2 remarked that (1) and (2) might follow from congruences of Ramanujan. We report experiments that support this suggestion in the last section.

The present article states several conjectures based on extensive computations (mainly done with SageMath). The data is available in a GitHub repository [7].

2. Constant terms

We want to write the polynomial

γ (x)

discussed in the introduction as a linear combination of powers of x. The usual multinomial theorem evaluates expressions of the form

{(y_{1} + y_{2} + . . . + y_{k})}^{k}

. The result is a linear combination over “partitions”

λ

of the products

\prod_{t} y_{t}^{λ_{t}}

(but not quite the usual partitions; see below.) To get what we want, we would need to substitute

a_{t} x^{t}

for

y_{t}

, shift the index t by one, and then sort out the powers of x. We prefer to re-derive everything from scratch. First we analyze

γ (x)

in terms of the elementary school process of multiplying polynomials (in our jargon: use of “paths”.) Then we translate this into the language of integer partitions. An integer partition

λ

is usually defined as a finite non-increasing sequence of positive integers. We loosen this a little in order to simplify our arguments.

Definition 1.

A partition $(λ_{1}, λ_{2}, . . ., λ_{N})$ is a finite non-increasing sequence of non-negative integers, which may be empty.
$D_{λ}$ is the set of distinct parts $λ_{i}$ of λ.
If λ is non-empty, $l (λ) = N$ .
$l (λ) = 0$ if λ is empty.
$| λ | = \sum_{w = 1}^{N} λ_{w}$ if λ is non-empty.
$| λ | = 0$ if λ is empty.
Setting $| λ | = n$ , we say that λ is a partition of n.
$Λ_{n}$ is the set of distinct partitions λ such that $| λ | = n$ .
$Λ [k, n]$ is the set of partitions λ such that $| λ | = n$ and $l (λ) = k$ .
(a)

$B_{λ, i}$ is the subsequence $(i, . ., i)$ of maximal length in λ.

(b)

$B_{λ}$ is the set of sequences ${B_{λ, i}}_{i \in D_{λ}}$ . The $B_{λ, i}$ are the blocks of λ and $B_{λ}$ is the block decomposition of λ.
$Ψ (λ) =$

$\frac{N!}{\prod_{w \in D_{λ}} l (B_{λ, w})!}$

Remark 1.

$B_{λ_{1}} = B_{λ_{2}}$ if and only if $λ_{1} = λ_{2}$ .
$l (λ) = \sum_{i \in D_{λ}} l (B_{λ, i})$
$| λ | = \sum_{i \in D_{λ}} | B_{λ, i} | = \sum_{i \in D_{λ}} i \cdot l (B_{λ, i})$ .
When enumerating the permutations of a partition λ, the original partition λ occurs with multiplicity $Ψ (λ)$ .

Definition 2.

Write $1 = a_{0}$ and $γ (x) = {(\sum_{n = 0}^{k} a_{n} x^{n})}^{k} = \sum_{n = 0}^{k^{2}} c_{n} x^{n}$ (say.)
Let L be the $k \times (k + 1)$ lattice of monomials

$\begin{matrix} c o l n u m 0 & c o l n u m 1 & . . . & c o l n u m k \\ r o w 1 & a_{0} x^{0} & a_{1} x^{1} & . . . & a_{k} x^{k} \\ r o w 2 & a_{0} x^{0} & a_{1} x^{1} & . . . & a_{k} x^{k} \\ . . . . . . . & . . . . . . . . & . . . & . . . . . . . \\ . . . . . . . & . . . . . . . . & . . . & . . . . . . . \\ r o w k & a_{0} x^{0} & a_{1} x^{1} & . . . & a_{k} x^{k} \end{matrix} .$
Let $P_{L}$ be the collection of all $(k + 1)$ -tuples (“paths in L”) $π = (w_{0}, w_{1}, . ., w_{k})$ such that $0 \leq w_{i} \leq k$ for each i. The path π visits each row of L once, in numerical order, and visits the columns of L in the order of, and with the multiplicity of, the terms in the sequence $(w_{0}, w_{1}, . ., w_{k})$ .
For $π = (w_{0}, w_{1}, . . ., w_{k})$ in $P_{L}$ , let

$v (π, w) = # {j s . t . w_{j} = w}$

be the number of visits π makes to column w of L.
For π in $P_{L}$ such that $π = (w_{0}, w_{1}, . . ., w_{k})$ , let

(a)

$μ (π) = \prod_{i = 0}^{k} a_{w_{i}} x^{w_{i}} .$

(b)

$α (π) = \prod_{i = 0}^{k} a_{w_{i}} .$
For π in $P_{L}$ , $π^{*}$ is the unique partition that occurs in the set of permutations of π.
$P_{n}$ is the set of paths π in $P_{L}$ such that $| π^{*} | = n$ .
$Λ_{L}$ is the set of distinct partitions in $P_{L}$ .

Remark 2.

By the ordinary distributive law, $γ (x) = \sum_{π \in P_{L}} μ (π)$ .
For π in $P_{L}$ and $0 \leq w \leq k$ , $v (π, w) = l (B_{π^{*}, w})$ .
For π in $P_{L}$ ,

(a)

$μ (π) = \prod_{w = 0}^{k} {(a_{w} x^{w})}^{v (π, w)} .$

(b)

$α (π) = \prod_{w = 0}^{k} {(a_{w})}^{v (π, w)} .$
A partition λ belongs to $Λ [k, n]$ if and only if $n = {deg}_{x} μ (λ)$ .
Therefore $μ (π) = α (π) \cdot x^{n}$ if π belongs to $P_{n}$ .
$P_{n}$ is closed under permutations.
For $π_{1}$ and $π_{2}$ both in $P_{n}$ , the following statements are equivalent:

(a)

$π_{2}$ is a permutation of $π_{1}$ .

(b)

$μ (π_{1}) = μ (π_{2})$ .

(c)

$α (π_{1}) = α (π_{2})$ .
Let $λ = π^{*}$ for any π in $P_{n}$ . The multiplicity of λ among the permutations of π is $Ψ (λ)$ . It is independent of the choice of π.
Hence

$γ (x) = \sum_{λ \in Λ_{L}} Ψ (λ) μ (λ) = \sum_{n \geq 0} (\sum_{λ \in Λ [k, n]} Ψ (λ) α (λ)) x^{n} .$

The following equation makes manifest the numerical coefficients mentioned in the introduction. Because

C (f^{k}) = γ_{k}

, we have

C (f^{k}) = \sum_{λ \in Λ [k, k]} Ψ (λ) α (λ) .

(3)

Therefore:

Theorem 1.

With f as above, the map

f (x) \mapsto {C (f), C (f^{2}), . . .}

is invertible.

Proof.

The first coefficient in the expansion of f, namely

a_{1}

, is equal to

C (f)

. Now suppose that we have in hand the coefficients

a_{n}, n < k

. By the equation above, all of the coefficients

a_{1}, . . ., a_{k}

appear in the expansion of

C (f^{k})

, and

a_{k}

appears only in the partition

λ^{*} = (k, 0, 0 . . . ., 0)

, where

l (λ^{*}) = k

and

l (B_{λ^{*}, 0}) = k - 1

. From the definitions,

α (λ^{*}) = a_{k}

and

Ψ (λ^{*}) = k

, so the only term of

C (f^{k})

including

a_{k}

k \cdot a_{k}

. Thus

C (f^{k}) = \sum_{λ \neq λ^{*}} Ψ (λ) α (λ) + k \cdot a_{k},

and we just solve this for

a_{k}

.□

3. Modular forms

3.1. Ramanujan’s congruences

Here are the congruences of Ramanujan mentioned above ([38], page 290 and elsewhere.)3 4

τ (n) \equiv σ_{11} (n) (m o d 2^{8})

(4)

for odd n.

τ (n) \equiv n σ_{1} (n) (m o d 3) .

(5)

τ (n) \equiv n^{2} σ_{1} (n) (m o d 3^{2}) .

(6)

τ (n) \equiv n^{2} σ_{7} (n) (m o d 3^{3}) .

(7)

τ (n) \equiv n σ_{1} (n) (m o d 5) .

(8)

τ (n) \equiv n σ_{9} (n) (m o d 5^{2}) .

(9)

τ (n) \equiv n σ_{3} (n) (m o d 7) .

(10)

τ (n) \equiv σ_{11} (n) (m o d 691) .

(11)

τ_{r} (n) \equiv n σ_{2 r - 1} (n) (m o d 11)

(12)

for

r = 2, 3

and 4.5

Remark 3.

Equation (3) extends to all of the positive integers as follows: let

o = {o r d}_{2} (n)

and

g (n) = 8^{o} \cdot σ_{11} (n / 2^{o}) .

Then

τ (n) \equiv g (n) (m o d 2^{8}) .

To see this, recall Ramanujan’s conjecture (proved by Mordell [25]) that, for

n \geq 1

and p prime:

τ (p) τ (p^{n}) = τ (p^{n + 1}) + p^{11} τ (p^{n - 1}) .

6 Setting

p = 2

, an easy induction argument shows that

{o r d}_{2} (τ (2^{o})) = 3 o

, and the claim follows from the multiplicativity of

τ (n)

3.2. Modular forms for Hecke groups

For

m = 3, 4, . . .

, let

λ_{m} = 2 cos π / m

and let

J_{m}

be a certain meromorphic modular form for the Hecke group

G (λ_{m})

, built from triangle functions, with Fourier expansion

J_{m} (z) = \sum_{n = - 1}^{\infty} a_{n} (m) q_{m}^{n},

where

q_{m} (z) = exp 2 π i z / λ_{m}

. (For further details, the reader is referred to the books by Carathéodory [13,14] and by Berndt and Knopp [3], the articles of Lehner and Raleigh [23,29], to the dissertation of Leo [24], and to a summary, including pertinent references to that material, in the 2021 article [8].)

Raleigh gave polynomials

P_{n} (x)

such that

a_{- 1} {(m)}^{n} q_{m}^{2 n + 2} a_{n} (m) = P_{n} (m)

for

n = - 1, 0, 1, 2

and 3. He conjectured that similar relations hold for all positive integers n [29]. 7 Akiyama proved Raleigh’s conjectures in 1992 [1].

Using the weight-raising properties of differentiation and the

J_{m}

, Hecke constructed families

H

comprising modular forms of positive weight for each

G (λ_{m})

sharing certain properties [3,18]. It seems apparent that Akiyama’s result can be extended: there should exist polynomials

Q_{H, n} (x)

interpolating the coefficient of

X_{m}^{n}

in the Fourier expansions of the members of Hecke families

H

. 8

In section 4 of the 2021 article, we made use of a certain uniformizing variable

X_{m} (z)

for z in the upper half plane [8]. By Akiyama’s theorem, we have a series of the form

J_{m} (x) : = \sum_{n = - 1}^{\infty} {\tilde{P}}_{n} (x) X_{m}^{n}

for polynomials

{\tilde{P}}_{n} (x)

Q [x]

with the property that

J_{m} = J_{m} (m)

. We will make use of the change of variables

X_{m} \mapsto 2^{6} m^{3} X_{m}

for a

G (λ_{m})

-modular form (originally employed, apparently, by Leo ([24], page 31). It has the effect when

m = 3

of recovering the Fourier series of a variety of standard modular forms. This is set up as a

Definition 3.

For z in the upper half plane and

k_{a} \neq 0

, let

f (z) : = \sum_{n = a}^{\infty} k_{n} X_{m} {(z)}^{n}

and

g (z) = \sum_{n = a}^{\infty} k_{n} 2^{6 n} m^{3 n} X_{m} {(z)}^{n} .

If the last expansion is written as

g (z) = \sum_{n = a}^{\infty} {\tilde{k}}_{n} X_{m} {(z)}^{n}

, then let

\bar{f} (z) : = g (z) / {\tilde{k}}_{a} .

Also, for

m = 3, 4, . . .

, let

j_{m} (z) : = \bar{J_{m}} (z)

The Fourier expansion of

j_{3}

is 9

j_{3} (z) = 1 / X_{3} (z) + 744 + 196884 X_{3} (z) + 21493760 X_{3} {(z)}^{2} + . . .,

which matches the standard expansion

j (z) =

1 / exp (2 π i z) + 744 + 196884 exp (2 π i \cdot z) + 21493760 exp (2 π i \cdot 2 \cdot z) + . . . .

Definition 4.

Let

F = {f_{3}, . . ., f_{m}, . . .}

where

f_{m}

is modular for

G (λ_{m})

. Then let the Fourier expansion of

f_{m}^{k}

in powers of

X_{m}

be written

f_{m} {(z)}^{k} = \sum_{n} a (f_{m}^{k}, n) X_{m}^{n} .

(Thus

C (f_{m}^{k}) = a (f_{m}^{k}, 0)

Proposition 1.

Let

K = {J_{3},, J_{4}, . . .}

and

\bar{K} = {j_{3}, j_{4}, . . . .}

Then there exist polynomials

Q_{K, k, n} (x)

and

Q_{\bar{K}, k, n} (x)

Q [x]

such that

a (J_{m}^{k}, n) = Q_{K, k, n} (m)

and

a (j_{m}^{k}, n) = Q_{\bar{K}, k, n} (m)

for

k = 1, 2, . . ., m = 3, 4, . . .

, and

n = - k, 1 - k, . . . .

For k equal to one, the first claim is just Akiyama’s theorem and the claim for k not equal to one is then obvious. The second statement follows immediately.

3.3. Polynomial interpolation of Fourier coefficients

When, given a sequence of functions

f_{m}

modular for

G (λ_{m})

in a family

F

, we looked for polynomials

Q_{F, n} (x)

such that each

f_{m}

with Fourier expansion

f_{m} (τ) = \sum_{n} a_{m, n} X_{m}^{n} (τ),

satisfied

Q_{F, n} (m) = a_{m, n}

, We evaluated finite sequences

{a_{m, n}}_{m = 1, 2, 3, 4, . . ., M_{n}}

(with n held constant) and generated candidates

g_{n} (x)

for

Q_{F, n} (x)

by Lagrange interpolation. The bounds

M_{n}

were linear in n and chosen large enough that the degrees of the

g_{n} (x)

produced in this way also appeared to be linear in n. Over the course of experiments described in the article [8], this linearity seemed to be associated with systematic behavior. For example, if a polynomial

g_{n} (x)

was factored as

g_{n} (x) = r_{n} \cdot p_{1} (x) \cdot p_{2} (x) . . . \cdot p_{a} (x)

where each of the

p_{i}

was monic,

r_{n}

was rational, and the degree of

g_{n} (x)

was linear in n, then often the sequence

{r_{3}, r_{4}, . . .}

was readily identifiable (sometimes after resorting to Sloane’s encyclopedia.) We take such regularities as evidence that

g_{n} (m) = a_{m, n}

for all m. Thus, when formulating conjectures about the

C (J_{m}^{k})

and

C (j_{m}^{k})

10, we did not always use tables of the

C (J_{m}^{k})

and

C (j_{m}^{k})

directly. Instead (for example), we used Lagrange interpolation to identify polynomials

h_{k} (x)

and

{\bar{h}}_{k} (x)

such that

C (J_{m}^{k}) = h_{k} (m)

and

C (j_{m}^{k}) = {\bar{h}}_{k} (m)

by letting m run through a small set of values sufficient to produce the linearity behavior mentioned above; so we assumed (in this example) that

h_{k} (x) \equiv Q_{K, k, 0} (x)

and

{\bar{h}}_{k} (x) \equiv Q_{\bar{K}, k, 0} (x)

identically. We made tables of p orders of the

h_{k} (m)

and the

{\bar{h}}_{k} (m)

. In this way we checked larger sets of m values than would have been practicable if we had checked the constant terms themselves.

Unlike the later conjectures, conjecture 1 is not a way of summarizing patterns in experimental data. Rather it codifies our assumption that the linearity behavior is a reliable signal.

Conjecture 1.

$h_{k} (x) \equiv Q_{K, k, 0} (x)$ identically; consequently, $h_{k} (m) = C (J_{m}^{k})$ identically.
${\bar{h}}_{k} (x) \equiv Q_{\bar{K}, k, 0} (x)$ identically; consequently, ${\bar{h}}_{k} (m) = C (j_{m}^{k})$ identically.

4. The reciprocals of cusp forms for $S L (2, Z)$

Let

E_{2 r}

denote the weight

2 r

Eisenstein series with q-series

1 + γ_{r} \sum_{r = 1}^{\infty} σ_{r - 1} (n) q^{n}

for certain rational numbers

γ_{r}

; this is Rankin’s notation. In our experiments, including the case

r = 1

, which is not in Rankin’s list, we rely on SageMathto pick out the unique normalized cusp form of weight

12 + 2 r

, so there is no need to specify

γ_{r}

by hand. Recall several facts:11 Setting

E_{0} (z) = 1, τ_{0} (n) = τ (n)

, and

r = 0, 2, 3, 4, 5

or 7:

$Δ (z) E_{2 r} (z)$ generates the space of weight $12 + 2 r$ cusp forms for $S L (2, Z)$ .
Writing $Δ_{r} = Δ (z) E_{2 r} (z)$ and $Δ_{r} = \sum_{n = 1}^{\infty} τ_{r} (n) q^{n}$ : the functions $n \mapsto τ_{r} (n)$ are multiplicative.

Conjecture 2.

Suppressing the dependence upon k and r, let

d_{p} = d_{p} (k), C = C (1 / Δ_{r}^{k})

and

o_{p} = {o r d}_{p} (C)

1.

Let

r = 0

(a)

o_{2} = 3 d_{2}

and

o_{3} = d_{3}

(b)

C / 3^{o_{3}} \equiv 1 (m o d 3)

if and only if k is even.

(c)

C / 3^{o_{3}} \equiv 2 (m o d 3)

if and only if k is odd.

(d)

i.: $C \equiv 0, 1, or 4 (m o d 5)$ .
ii.: $o_{5} = 0$ if and only if the set of digits in the base 5 expansion of k is a subset of ${0, 1, 2}$ .12

(e)

o_{7} = 0

if and only if the set of digits in the base 7 expansion of k is a subset of

{0, 1, 2, 3, 4}

2.

Let

r = 2

.13

(a)

o_{2} = 3 d_{2}

(b)

i.

o_{3} = d_{3}

if and only if

k \equiv 0 (m o d 3)

ii.

If D is a positive integer such that

D \equiv 2 (m o d 3)

k > D

, and, for some positive

n, k = D + 3^{n}

, then

o_{3}

is constant for large n. Let

L_{_{D}} = {lim}_{n \to \infty} o_{3}

and

N_{_{D}}

be the smallest value of n such that

n \geq N_{_{D}} \Rightarrow o_{3} = L_{_{D}}

. Below is a table for small D. More extensive tables are posted on GitHub [7].

D	2	5	8	11	14	17	20	23
$L_{_{D}}$	4	5	8	5	6	8	7	8
$N_{_{D}}$	1	2	3	3	3	3	4	3

(c)

If k is even and

k \equiv 0 (m o d 3)

, then

C / 3^{o_{3}} \equiv 1 (m o d 3)

(d)

If k is odd and

k \equiv 0 (m o d 3)

, then

C / 3^{o_{3}} \equiv 2 (m o d 3)

(e)

o_{5} = 0

if and only if the set of digits in the base 5 expansion of k is a subset of

{0, 1, 2}

3.

Let

r = 3

.14

(a)

o_{2} = 3 d_{2}

if and only if k is even.

(b)

i.: $o_{3} = d_{3}$ .
ii.: $C / 3^{o_{3}} \equiv 1 (m o d 3)$ if and only if k is even.
iii.: $C / 3^{o_{3}} \equiv 2 (m o d 3)$ if and only if k is odd.

(c)

o_{5} = 0

, then the set of digits in the base 5 expansion of k is a subset of

{0, 1, 2, 3}

(d)

o_{7} = 0

, then the set of digits in the base 7 expansion of k is a subset of

{0, 1, 2, 3, 4}

4.

Let

r = 4

(a)

For all positive

k, o_{2} = 3 d_{2}

(b)

i.: For all positive k, $C \equiv 0 (m o d 3)$ .
ii.: If $k \equiv 0 (m o d 3)$ , then $o_{3} = d_{3}$ .
iii.: If $k \equiv 1 (m o d 3)$ , then $o_{3} = d_{3}$ if and only if k belongs to O.E.I.S sequence A191107 [20] 15 ${1, 4, 10, . . .}$ ,
iv.: If $k \equiv 2 (m o d 3)$ and $d_{3}$ divides $o_{3}$ , then $o_{3} / d_{3} = 2$ .

(c)

i.: $c \equiv 0, 1$ , or $4 (m o d 5)$ .
ii.: $o_{5} = 0$ if and only if the digits in the base 5 expansion of k is a subset of ${0, 1, 2}$ .
iii.: If $o_{5} = 0$ , then $C / 5^{o_{5}} \equiv 1$ or $4 (m o d 5)$ . 16

(5)

Let

r = 5

(a)

If k is even, then

o_{2} = 3 d_{2} .

(b)

If D is a positive odd integer,

k > D

and

k = D + 2^{n}

, then

o_{2}

is constant for large n. Let

L_{_{D}} = {lim}_{n \to \infty} o_{2}

and

N_{_{D}}

be the smallest value of n such that

n \geq N_{_{D}} \Rightarrow o_{2} = L

. Below is a table for small D. More extensive tables are posted on GitHub [7].

D	1	3	5	7	9	11	13	15	17	19
$L_{_{D}}$	10	13	17	19	15	17	23	27	17	17
$N_{_{D}}$	3	3	6	5	6	5	8	9	6	6

Remark 4.

1.: A p-adic geometric view of conjectures 2.2.b (ii) and 2.5.b is that the function $k \mapsto C (1 / Δ^{k})$ takes certain units k in sufficiently small disks around certain other units to circles around zero.
2.: Conjectures 2.2.b (ii) and 2.5.b have only limited empirical support because the mentioned p-adic units k grow exponentially with n and, on account of drastic slowdowns for large k, our experiments tested only $k \leq 5000$ . Thus for $p = 2$ and 3, we could only check $n \leq 12$ and 7, respectively. We will include tables of what empirical data we do have in the appendix.

5. Constant terms for $j^{k}, k = 1, 2, . . .$

Recall that

h_{k} (x)

and

{\bar{h}}_{k} (x)

are polynomials identified from numerical data by Lagrange interpolation conjectured to satisfy

C (J_{m}^{k}) = h_{k} (m)

and

C (j_{m}^{k}) = {\bar{h}}_{k} (m)

. In this section, we illustrate connections between the divisibility patterns (described in the introduction) for the constant terms of the

j {(τ)}^{k} = j_{3} {(τ)}^{k}

Fourier expansions on one side, and the

h_{k} (x)

on the other. Let

{\bar{h}}_{k} (x)

factor as

{\bar{h}}_{k} (x) = ν_{k} \cdot p_{k, 1} (x) \times p_{k, 2} (x) \times . . . \times p_{k, α} (x) =

(say)

ν_{k} \cdot {\tilde{p}}_{k} (x)

where each of the

p_{k, n} (n = 1, 2, . . ., α

) is monic and

ν_{k}

is rational. We represent O.E.I.S. sequence A005148 [28]

{0, 1, 47, 2488, 138799, . . .}

{a_{0}, a_{1}, . . .}

Conjecture 3.

1.

ν_{k} = 24 a_{k}

2.

{\tilde{p}}_{k} (3)

is always odd.

3.

{o r d}_{2} (a_{k}) = 3 d_{2} (k) - 3

4.

{o r d}_{3} ({\tilde{p}}_{k} (3)) = d_{3} (k) - 1

5.

From the introduction:

{o r d}_{2} (C (j_{3}^{k})) = 3 d_{2} (k)

and

{o r d}_{3} (C (j_{3}^{k})) = d_{3} (k)

6.

We restate another observation from the article [9]. Let

o_{k} = {o r d}_{3} (C (j_{3}^{k})), κ = C (j_{3}^{k}) / 3^{o_{k}},

and

ρ_{k} = m o d (κ, 3)

. Then

ρ_{k} = 1

or 2, according as k is even or odd, respectively.

7.

(a): Let $p = 5$ or 7 and let $o = {o r d}_{p} (C (j_{3}^{k}))$ . Then $o = 0$ if and only if the set of digits in the base p expansion of k is a subset of ${0, 1, 2}$ .
(b): Let $p = 11$ . With notation as above, $o = 0$ if and only if the set of digits in the base p expansion of k is a subset of ${0, 1, 2, 3, 4}$ .

Remark 5.

Clause 5 of the conjecture follows from the earlier clauses. First claim:

{o r d}_{2} (C (j_{3}^{k})) = {o r d}_{2} ({\bar{h}}_{k} (3)) = {o r d}_{2} (ν_{k} \cdot {\tilde{p}}_{k} (3)) = {o r d}_{2} (24 a_{k}) \cdot {\tilde{p}}_{k} (3)) = {o r d}_{2} (24) + {o r d}_{2} (a_{k}) + {o r d}_{2} ({\tilde{p}}_{k} (3)) = 3 + 3 d_{2} (k) - 3 + 0 = 3 d_{2} (k) .

Second claim: In their 1984 article [27], Newman, Shanks and Zagier demonstrated that

{o r d}_{3} (a_{k}) = 0

for all k. Therefore (under the previous clauses)

{o r d}_{3} (C (j_{3}^{k})) = {o r d}_{3} ({\bar{h}}_{k} (3)) = {o r d}_{3} (ν_{k}) + {o r d}_{3} ({\tilde{p}}_{k} (3)) = 1 + {o r d}_{3} (a_{k}) + d_{3} (k) - 1 = d_{3} (k)

6. Sufficient conditions for equations (1), (2)

We construct some Laurent series (not necessarily modular, even after an appropriate substitution) such that their constant terms satisfy analogues of Equation (1) or Equation (2). Some conjectures in this section were tested with Monte Carlo methods.

Conjecture 4.

1.: Let $A_{n} = l c m ({2 \cdot 8^{d_{2} (k)}}_{k = 1, . . ., n + 1})$ . If

$f (x) = \sum_{k = 1}^{\infty} a_{k} x^{k}$

is in $Z [[x]], a_{k} \equiv τ (k) (m o d A_{n})$ for $k = 1, 2, . . ., n + 1$ , then

${o r d}_{2} (C (1 / f {(x)}^{n})) = 3 d_{2} (n) .$
2.: Let $B_{n} = l c m ({3 \cdot 3^{d_{3} (k)}}_{k = 1, . . ., n + 1})$ . If

$f (x) = \sum_{k = 1}^{\infty} a_{k} x^{k}$

is in $Z [[x]]$ , $a_{k} \equiv τ (k) (m o d B_{n})$ for $k = 1, 2, . . ., n + 1$ , then

${o r d}_{3} (C (1 / f {(x)}^{n})) = d_{3} (n) .$
3.: Let $C_{n} = l c m ({6 \cdot 8^{d_{2} (k)} \cdot 3^{d_{3} (k)}}_{k = 1, . ., n + 1})$ . If

$f (x) = \sum_{k = 1}^{\infty} a_{k} x^{k}$

is in $Z [[x]]$ , $a_{k} \equiv τ (k) (m o d C_{n})$ for $k = 1, 2, . . ., n + 1$ , then

${o r d}_{2} (C (1 / f {(x)}^{n})) = 3 d_{2} (n)$

and

${o r d}_{3} (C (1 / f {(x)}^{n})) = d_{3} (n) .$

In the following conjectures, analogues to the series expansion of

Δ (z)

from the right sides of Ramanujan’s congruences (3) – (11) are constructed. Graphical tests indicate that they are not modular forms, but they each appear to have some of the behaviors conjectured for

Δ

Conjecture 5.

1.

Let

o_{k} = {o r d}_{2} (k), g_{k} = 8^{o_{k}} \cdot σ_{11} (k / 2^{o_{k}}),

and

f (x) = \sum_{k = 1}^{\infty} g_{k} x^{k} .

Then

(a): ${o r d}_{2} C (1 / f {(x)}^{n})) = 3 d_{2} (n) .$
(b): $C (1 / f {(x)}^{n}) \equiv 1 (m o d 3)$ .

2.

Let

A_{n}

be as in the previous conjecture,

g_{k}

be as above, and let

f (x) = \sum_{k = 1}^{\infty} a_{k} x^{k},

where

a_{k} \equiv g_{k} (m o d A_{n})

. Then

{o r d}_{2} (C (1 / f {(x)}^{n})) = 3 d_{2} (n) .

Conjecture 6.

Let

o_{2} = {o r d}_{2} (k), o_{3} = {o r d}_{3} (k), g_{k} = k \cdot σ_{1} (k),

and

f (x) = \sum_{k = 1}^{\infty} g_{k} x^{k} .

1.: If n is divisible by 4, then ${o r d}_{2} (C (1 / f {(x)}^{n})) = 3 d_{2} (n) .$
2.: If n is divisible by 3, then ${o r d}_{3} (C (1 / f {(x)}^{n})) = d_{3} (n) .$
3.: If $n - 1$ is divisible by 3 and $n - 2$ is a power of 3 or twice a power of 3, then once again ${o r d}_{3} (C (1 / f {(x)}^{n})) = d_{3} (n) .$ 18

Conjecture 7.

Let

g_{k} = k^{2} \cdot σ_{1} (k)

and

f (x) = \sum_{k = 1}^{\infty} g_{k} x^{k} .

1.: If n is even, then ${o r d}_{2} (C (1 / f {(x)}^{n})) = 3 d_{2} (n) .$
2.: For $n = 1, 2, . . .$ , ${o r d}_{3} (C (1 / f {(x)}^{n})) = d_{3} (n) .$

7. Powers of reciprocals of generating functions of certain other arithmetic functions

The functions studied in this section are constructed from certain multiplicative or additive (in the sense that

f (a b) = f (a) + f (b)

when

gcd (a, b) = 1)

arithmetic functions. They are not necessarily modular or consistent with analogues of equations (1) and (2).

Conjecture 8.

Let

f_{r} (x) = \sum_{k = 1}^{\infty} σ_{r} (k) x^{k} .

1.: $C (1 / f_{0} {(x)}^{n}))$ is odd if and only if n is divisible by three.
2.: For all positive integers n, $C (1 / f_{1} {(x)}^{n}))$ is odd.

In the following conjecture we study divisor sums with multiplicity.

Definition 5.

1.: For $n = \prod_{i} p_{i}^{n_{i}}$ , $d = \prod_{i} p_{i}^{d_{i}}$ with $0 \leq d_{i} \leq n_{i}$ , and $(\binom{a}{b})$ the usual binomial coefficient, the multiplicity of d in n is

$μ (d, n) : = \prod_{d | n} (\binom{n_{i}}{d_{i}}) .$
2.: $σ_{r}^{μ} (n) : = \sum_{d | n} μ (d, n) d^{r} .$

Conjecture 9.

19 Let

f_{r}^{μ} (x) = \sum_{k = 1}^{\infty} σ_{r}^{μ} (k) x^{k}

and

C_{r, n} = C (1 / f_{r}^{μ} {(x)}^{n})

1.

(a): $C_{r, n}$ is odd for all positive integers r and n.
(b): If r is odd and n is even, then $C_{r, n} \equiv 1 (m o d 3)$ .

2.

C_{0, n}

is even for all positive integers n.

3.

(a): $C_{1, n}$ is odd for all positive integers n.
(b): $C_{1, n} \equiv 0 (m o d 3)$ if and only if n is even.

8. Constant terms for $j_{m}^{k}, k = 1, 2, . . .$

By imposing restrictions on k and m, we found several narrow conjectures about constant term p orders for various primes p.

8.1. m a prime power

Conjecture 10.

If p is prime and a is an integer that is larger than 2, then

{o r d}_{p} (C (j_{p^{a}}^{k})) = (a - 3) k + {o r d}_{p} (C (j_{p^{3}}^{k})) .

Conjecture 11.

Let

a \geq 2

. Then

{o r d}_{2} (C (j_{2^{a}}^{2})) = 2 a + 7

Conjecture 12.

Let p be a prime number larger than 2 and let a be a positive integer. Then

{o r d}_{p} (C (j_{p^{a}}^{p})) = a p - 2

8.2. Other m

Conjecture 13.

d_{2} (k) = 1

a = {o r d}_{2} (m), a \geq 2

, and

o = {o r d}_{2} (C (j_{m}^{k}))

, then

o = k (a + 2) + 3

Conjecture 14.

Let

d_{2} (k) = 1

m \equiv 2 (m o d 4)

, and

a = {o r d}_{2} (m) (= 1

, of course.) Then

{o r d}_{2} (C (j_{m}^{k})) = k (a + 6) + 1 = 7 k + 1

Now let

K_{n}, n = 0, 1, 2, . . .

be the

n^{t h}

Catalan number. (We depart from the standard notation because we have been using the letter “c” in so many other contexts.) One of several explicit formulas for

K_{n}

K_{n} = \frac{(2 n)!}{(n + 1)! n!} .

For n positive let

K_{1, n}

denote the

n^{t h}

Catalan number K such that

K \neq K_{0}

and

{o r d}_{2} (K) = 1

.21

Conjecture 15.

Let k be the

n^{t h}

positive integer such that

d_{2} (k) = 2

; also,

m = 4 j, (j = 1, 2, . . .)

, and

a = {o r d}_{2} (m)

. Furthermore, let

o = {o r d}_{2} (C (j_{m}^{k}))

and

t = ((a + 6) k + 2 - o) / 4

. Then

t = K_{1, n}

Conjecture 16.

Let

d_{2} (k) = 2

m = 4 j + 2, j = 1, 2, . . .

, and

a = {o r d}_{2} (m)

(again,

a = 1

.) Then

{o r d}_{2} (C (j_{m}^{k})) = (a + 6) k + 2 = 7 k + 2

Conjecture 17.

m \equiv 0 (m o d 3)

, then

{o r d}_{3} (C (j_{m}^{k})) = k \cdot {o r d}_{3} (m) + d_{3} (k) - k .

8.3. The constant terms $c (J_{m}^{k})$

The Fourier coefficients of the

J_{m}

are rational numbers, but typically they are not integers.

Conjecture 18.

22 Let p be a prime number greater than two and let

c (J_{p}^{p}) = a / b

(

a, b

relatively prime integers, b positive.) Then

b = 2^{6 p - 3 d_{2} (p)} p^{2 p + 2}

References

Akiyama, S. A note on Hecke’s absolute invariants. J. Ramanujan Math. Soc 1992, 7, 65–81. [Google Scholar]
M. H. Ashworth. “Congruence and identical properties of modular forms”. Ph.D. thesis supervised by A. O. L. Atkin, cited in [39]. University of Oxford, 1968.
Berndt, B.C.; Knopp, M.I. Hecke’s theory of modular forms and Dirichlet series; Vol. 5, World Scientific, 2008.
Berndt, B.C.; Ono, K. Ramanujan’s unpublished manuscript on the partition and tau functions with proofs and commentary. Sém. Lotharingien de Combinatoire 1999, 42, 63. [Google Scholar] [CrossRef]
Boas, R.P.; Buck, R.C. Polynomial expansions of analytic functions; Vol. 19, Springer Science & Business Media, 2013.
Bottomley, H. The On-Line Encyclopedia of Integer Sequences, A099628. http://oeis.org/A099628.
Brent, B. GitHub files for this article. https://github.com/barry314159a/NewmanShanks, 2023.
Brent, B. Polynomial interpolation of modular forms for Hecke groups. http://math.colgate.edu/~integers/v118/v118.pdf. [CrossRef]
Brent, B. Quadratic minima and modular forms. Experimental Mathematics 1998, 7, 257–274. [Google Scholar] [CrossRef]
Brockhaus, K. The On-Line Encyclopedia of Integer Sequences, A164123. http://oeis.org/A164123.
Buckholtz, J.D. Series expansions of analytic functions. Journal of Mathematical Analysis and Applications 1973, 41, 673–684. [Google Scholar] [CrossRef]
Byrd, P.F. Expansion of analytic functions in polynomials associated with Fibonacci numbers. Fibonacci Q. 1963, 1, 16. [Google Scholar]
Carathéodory, C. Theory of functions of a complex variable, Second English Edition; Vol. 1, Chelsea Publishing Company, 1958. Translated by F. Steinhardt.
Carathéodory, C. Theory of functions of a complex variable, Second English Edition; Vol. 2, Chelsea Publishing Company, 1981. Translated by F. Steinhardt.
Frenkel, I.B. Representations of Kac-Moody algebras and dual resonance models. Applications of group theory in physics and mathematical physics 1985, 21, 325–354. [Google Scholar]
Frenkel, I.B.; Lepowsky, J.; Meurman, A. Vertex operator algebras and the Monster; Academic press, 1989.
Gerasimov, J.S. The On-Line Encyclopedia of Integer Sequences, A176003. http://oeis.org/A176003.
Hecke, E. Über die bestimmung dirichletscher reihen durch ihre funktionalgleichung. Mathematische Annalen 1936, 112, 664–699. [Google Scholar] [CrossRef]
Kimberling, C. The On-Line Encyclopedia of Integer Sequences, A037453. http://oeis.org/A037453.
Kimberling, C. The On-Line Encyclopedia of Integer Sequences, A191107. http://oeis.org/A191107.
Kolberg, O. Congruences for Ramanujan’s Function τ(n); Norwegian Universities Press, 1962.
Lehmer, D.H. Note on some arithmetical properties of elliptic modular functions. Duplicated notes, University of California at Berkeley, cited in [39].
Lehner, J. Note on the Schwarz triangle functions. Pacific Journal of Mathematics 1954, 4, 243–249. [Google Scholar] [CrossRef]
Leo, J.G. Fourier coefficients of triangle functions, Ph.D. thesis. http://halfaya.org/ucla/research/thesis.pdf, 2008.
Mordell, L.J. Note on certain modular relations considered by Messrs. Ramanujan, Darling, and Rogers. Proceedings of the London Mathematical Society 1922, 2, 408–416. [Google Scholar] [CrossRef]
Murthy, S. Black holes and modular forms in string theory. https://arxiv.org/pdf/2305.11732.pdf, 2023. [CrossRef]
Newman, M.; Shanks, D. On a Sequence Arising in Series for π. In Pi: A Source Book; Springer, 2004; pp. 462–480.
Plouffe, S.; Sloane, N.J.A. The On-Line Encyclopedia of Integer Sequences, A005148. http://oeis.org/A005148.
Raleigh, J. On the Fourier coefficients of triangle functions. Acta Arithmetica 1962, 8, 107–111. [Google Scholar] [CrossRef]
Ramanujan, S. On certain arithmetical functions. Trans. Cambridge Philos. Soc 1916, 22, 159–184. [Google Scholar]
Ramanujan, S. On certain arithmetical functions. In Collected papers of Srinivasa Ramanujan; Cambridge University Press, 2015; pp. 136–162.
Rankin, R.A. Ramanujan’s unpublished work on congruences. Modular Functions of one Variable V: Proceedings International Conference, University of Bonn, Sonderforschungsbereich Theoretische Mathematik July 2–14, 1976. Springer, 2006, pp. 3–15. [CrossRef]
Serre, J.P. A course in arithmetic; Springer-Verlag, 1970.
Siegel, C.L. Berechnung von Zetafunktionen an ganzzahligen Stellen. Akad. Wiss. 1969, 10, 87–102. [Google Scholar]
Siegel, C.L. Evaluation of zeta functions for integral values of arguments. Advanced Analytic Number Theory, Tata Institute of Fundamental Research, Bombay 1980, 9, 249–268. [Google Scholar]
Sloane, N.J.A. The On-Line Encyclopedia of Integer Sequences, A049001. http://oeis.org/A049001.
Sloane, N.J.A.; Wilks, A. The On-Line Encyclopedia of Integer Sequences, A005187. http://oeis.org/A0005187.
Swinnerton-Dyer, H.P.F. Congruence properties of τ(n). Ramanujan revisited: proceedings of the [Ramanujan] Centenary Conference, University of Illinois at Urbana-Champaign, June 1–5, 1987. Harcourt Brace Jovanovich, 1988, pp. 289–311.
Swinnerton-Dyer, H.P.F. On l-adic representations and congruences for coefficients of modular forms. Modular Functions of One Variable III: Proceedings International Summer School University of Antwerp, RUCA July 17–August 3, 1972. Springer, 1973, pp. 1–55. [CrossRef]
Swinnerton-Dyer, H.P.F. On l-adic representations and congruences for coefficients of modular forms (II). Modular Functions of one Variable V: Proceedings International Conference, University of Bonn, Sonderforschungsbereich Theoretische Mathematik July 2–14, 1976. Springer, 2006, pp. 63–90. [CrossRef]
Turpel, A. The On-Line Encyclopedia of Integer Sequences, A037168. http://oeis.org/A037168.
Wikipedia. j-invariant. https://en.wikipedia.org/wiki/J-invariant, 2022.
Wilton, J.R. Congruence properties of Ramanujan’s function τ(n). Proceedings of the London Mathematical Society 1930, 2, 1–10. [Google Scholar] [CrossRef]
Zagier, D. Appendix to `On a Sequence Arising in Series for π’ by Newman and Shanks. In Pi: A Source Book; Springer, 2004; pp. 462–480.
Zumkeller, R. The On-Line Encyclopedia of Integer Sequences, A084920. http://oeis.org/A084920, 2013.

1	For example, see Serre [33], section 3.3, Equation (22), or the Wikipedia page [42].
2	Glenn Stevens
3	The following congruences appear in Ramanujan’s unpublished (before Berndt and Ono did publish it) manuscript [4]: (4) is Ramanujan’s (11.8), (5) is Ramanujan’s (12.3), (6) is also Ramanujan’s (12.3), (7) is Ramanujan’s (2.1), (8) is Ramanujan’s (4.2), and (10) is Ramanujan’s (12.7).
4	It is well known that they have been strengthened; see the articles [2,4,21,22,30,31,38,39,40,43].
5	The congruences in (11) are displayed in the table at the top of page SwD-32 (page 32 of the proceedings [39]), and, in the form shown here, as Equation (13) on page Ran-6 (page 8 of the proceedings [32].)
6	See Equation (53) of proposition 14 in section 5.5 of Serre’s book [33].
7	For more on expansions over polynomial fields, see, for example, the book of Boas and Buck [5] and the articles by Buckholtz and Byrd ([11,12].)
8	See the paper [8].
9	See Equation (23) of Serre’s book [33], section 3, and the SageMath notebook “jpower constant term NewmanShanks 26oct22.ipynb” in [7].
10	See the SageMath notebooks in the repository [7], in the folder “conjectures”.
11	See page ran-4 (page six in the proceedings volume) of Rankin’s article [32].
12	See O.E.I.S. page [19].
13	The converses of clauses (c) and (d) are false.
14	Again, the converses of clauses (c) and (d) are false.
15	Description: “Increasing sequence generated by these rules: $a (1) = 1$ , and if x is in a then $3 x - 2$ and $3 x + 1$ are in a.” Mathematicacode: `h = 3; i = -2; j = 3; k = 1; f = 1; g = 7; a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]`.
16	The converse is false.
17	See the folder “conjectures” in the repository [7].
18	For this sequence, see the O.E.I.S. page [10] of K. Brockhaus.
19	Clause 1 is based on substantially less data than the clauses that specify particular values of r.
20	Again, see the SageMathnotebooks in the folder “conjectures” in the repository [7]. Also see O.E.I.S. pages [17,36,41,45].
21	See Bottomley’s O.E.I.S. page [6].
22	See [37] and other O.E.I.S. pages cited within it.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

MDPI Initiatives

Important Links

Choose an area of interest and we will send you notifications of new preprints at your preferred frequency.

Disclaimer

On the Constant Terms of Certain Laurent Series

Abstract

1. Introduction

2. Constant terms

3. Modular forms

3.1. Ramanujan’s congruences

3.2. Modular forms for Hecke groups

3.3. Polynomial interpolation of Fourier coefficients

4. The reciprocals of cusp forms for S L ( 2 , Z )

5. Constant terms for j k , k = 1 , 2 , . . .

6. Sufficient conditions for equations (1), (2)

7. Powers of reciprocals of generating functions of certain other arithmetic functions

8. Constant terms for j m k , k = 1 , 2 , . . .

8.1. m a prime power

8.2. Other m

8.3. The constant terms c ( J m k )

References

MDPI Initiatives

Important Links

Subscribe

4. The reciprocals of cusp forms for $S L (2, Z)$

5. Constant terms for $j^{k}, k = 1, 2, . . .$

8. Constant terms for $j_{m}^{k}, k = 1, 2, . . .$

8.3. The constant terms $c (J_{m}^{k})$