Nt.number theory – Strong uniqueness of Euler global function

Let $ f: mathbb N to mathbb C $ to be an arithmetic function. To define $ varphi_f (n) $ by the following formula:

$$
varphi_f (n) = sum _ { substack {k leq n \ (k, n) = 1}} f (k).
$$

In other words, $ varphi_f (n) $ is the sum of $ f $ on the totals of $ n $. For example, if $ f = delta_1 (n) $ then $ varphi_f (n) = $ 1, if $ f = 1 $ then $ varphi_f (n) = varphi (n) – $ The total function of Euler. Suppose that $ f $ is completely multiplicative. Review of the first values ​​of $ f $ (Up & # 39; to $ approx $ 40) shows that if $ varphi_f $ is multiplicative only if $ f = 1 $ or $ f = delta_1 $.

Obviously, the direct analysis of some forty cases is not the most enlightening way of proving this type of proposition. This brings us to a more general question. Let's call an arithmetic function $ g $ possibly multiplicative if there is a multiplicative function $ G $ such as $ g (n) = G (n) $ for $ n $ wide enough. Is it true that if $ f $ is completely multiplicative and $ varphi_f $ is ultimately multiplicative then either $ f = delta_1 $ or $ f = 1 $?

nt.number theory – Which peer networks have a theta series with this property?

This is a slight generalization of a question I asked in Math StackExchange and which remains unanswered after a month. So I decided to post it here. I am sorry in advance if this is inappropriate for this site.

assume $ Lambda $ is an even trellis. Consider his theta series

$$ theta _ { Lambda} (q) = sum_ {a in Lambda} q ^ {(a, a) / 2}, $$

or $ ( cdot, cdot) $ means the Euclidean domestic product.

My question is:

For who $ Lambda $ have we

$$ theta _ { Lambda} (q) = 1 + m sum_ {n> 0} frac {f (n) : q ^ n} {1-q ^ n} $$

or $ m $ is not zero and $ f $ is a fully multiplicative arithmetic function?


Examples

I only know two types of networks with this property:

  1. Maximum orders in rational division algebras with class number 1, scaled by $ sqrt {2} $:

    • Dimension 1: The integers, with $ m = $ 2 and $ f (n) = lambda (n) $ is the function of Liouville.

    • Dimension 2: The rings of imaginary quadratic fields of discriminants $ D = -3, -4, -7, -8, -11, -19, -43, -67, -163 $. Right here $ m = frac {2} {L (0, f)} $ and $ f (n) = left ( frac {D} {n} right) $ is a Kronecker symbol, and $ L (0, f) $ is given by $ sum_ {n = 0} ^ {| D |} frac {n} {D} left ( frac {D} {n} right) $.

    • Dimension 4: The maximal orders of fully defined quaternion discriminant algebras $ D = 4, 9, 25, 49, 169 $. Right here $ m = frac {24} { sqrt {D} -1} $ and $ f (n) = n left ( frac {D} {n} right) $.

    • Dimension 8: The order of Coxeter in rational octonions, with $ m = $ 240 and $ f (n) = n ^ 3 $.

  2. The two 16-dimensional networks of Heterotic String Theory, $ E_8 times E_8 $ and $ D_ {16} ^ + $. Both networks have the same theta series, with $ m = $ 480 and $ f (n) = n ^ 7 $.

These include in particular all the root networks I've mentioned in the original Math.SE article.


Attempt

(Do not hesitate to skip this part)

I do not know much about modular forms, so it can contain errors. Theorem 4 in these notes implies that in an even dimension there is a level $ N $ and a character $ chi $ take values ​​in $ {- 1,0,1 } $ For who $ theta _ { Lambda} $ is a modular form of weight $ k = ( mathrm {dim} : Lambda) / $ 2. The requested property in turn implies that the Epstein zeta function of the network has an Euler product.

$$ zeta _ { Lambda} (s) propo prod_p frac {1} {1- (1 + f (p)) p ^ {- s} + f (p) p ^ {- 2s}} = zeta (s) prod_p frac {1} {1-f (p) p ^ {- s}}, $$

which in the very dimension means that $ theta _ { Lambda} $ is a proper form of Hecke (non cumulate, given the main coefficient 1); so we see that it has to act from a series of weights from Eisenstein $ klevel $ N $ and the character $ chi $, by the decomposition of the space of modular forms into subspaces Eisenstein + cuspidal.

This series Eisenstein has Fourier expansion $ E_ {k, chi} (q) = 1- (2k / B_ {k, chi}) sum ( cdots) $ or $ B_ {k, chi} $ is a generalized number of Bernoulli and the $ ( cdots) $ the part has integral coefficients. So, one possible solution would be to find those generalized Bernoulli numbers for which $ 2k / B_ {k, chi} = -m $ is a negative integer (since in $ Lambda $ there must be an even number of vectors of the standard 2) and check on a case-by-case basis if the associated Eisenstein series is the theta series of a network.

If this approach is correct, we can then use Tables 1 to 3 of this article, which show that the only cases of this type with $ mathrm {dim} : Lambda ge $ 4 are those given in the Examples section, as well as a certain series of Eisenstein weights 2 and 42, which do not appear to correspond to a network.

Moreover, I do not understand what happens in the case of odd dimensions, where the modular forms involved have a semi-integral weight. It seems that the concept of Hecke eigenform is defined a little differently, so the above approach may not work here. I found this answer which says that zeta functions associated with modular forms of an entire half-weight are usually devoid of Euler products. Here are also some potentially relevant questions (1, 2) about particular cases.

nt.number theory – Number of ringtones with standard items $ -1 $

Is there a nice characterization of numbered ringtones? $ mathcal {O} $ such as there is an element $ x in mathcal {O} $ whose standard is -1 $?

An obvious necessary condition is that $ mathcal {O} $ must have a real integration. For real quadratic number ringtones, standard -1 $ elements sometimes exist and sometimes not. For example:

Example: In $ mathbb {Z}[sqrt{2}]$, the standard of $ 1 + sqrt {2} $ is -1 $.

Example: In $ mathbb {Z}[sqrt{3}]$there is no element of standard -1 $. Indeed, the standard of $ a + b sqrt {3} $ is $ a ^ 2-3b ^ 2 $. If it was -1 $, then we would have that $ a ^ 2 equiv -1 $ mod $ 3 $, which is impossible.

nt.number theory – integers with a Hamiltonian square path

Let $ n> $ 1 to be an integer and a set $[n]= {1, ldots, n } $. We say that $ n $ has a "Hamiltonian Square Path" where there is a bijection $ varphi:[n]at[n]$ as for all $ k in [n-1]$ we have $ varphi (k) + varphi (k + 1) $ is a square number.

for example $ 15 and $ 16 have this property.

Question. Is there an integer $ N> $ 1 such as each integer $ n geq N $ has a Hamiltonian Square Road?


Note. This problem can be formulated in the language of the theory of graphs and Hamiltonian paths. We say that $ a neq b in [n]$ form an edge if their sum is square, and the above question concerns integers such that the resulting graph has a Hamiltonian path.

nt.number theory – Articles containing the avoidance arguments of Ihara

I'm trying to understand some of the recent research in number theory. There is apparently a certain lemma, called lharama lemma, that can be established in some contexts and is unknown in others. Sometimes one can still prove its consequences unconditionally. This acrobatics is called Ihara avoidance. What are the important documents containing arguments like this? Also, what are the documents containing Ihara avoidance type arguments that are technically easy to understand?

I feel that I will never be able to get into this sea of ​​clues, so if there are any
Easy paper based on this idea, I could try to understand it well and other papers would then become less scary.

nt.number theory – Convergence speed of the first zeta function P (2)

For an application to the statistical theory of groups, we need explicit upper and lower limits that an expert in number theory (I am not one) knows how to prove.

Question 1: What are the "good" limits $ f_1 (x)x} frac {1} {p ^ 2} <f_2 (x) $ or $ p> x $ is prime?

For our application, limits sharper than the following $ f_1 (x), f_2 (x) $ are desirable:
$$ frac {1} {12 left ( frac {x} { log (x) -4} +1 right) ^ 4}x} frac {1} {p ^ 2} < frac {1} {x-1}. $$
The lower limit can hold for $ x> x_1 $ and the upper limit for $ x> x_2 $ provided $ x_1, x_2 $ are small ".

Question 2: Is there a function $ f (x) $ and explicit positive constants $ c_1, c_2 $ such as
$$ c_1f (x)x} frac {1} {p ^ 2} <c_2f (x)? $$

nt.number Theory – "Mathematics Made Difficult": Show that $ 17 times 17 = $ 289. Generalize this result

The second exercise of the book Mathematics made difficult by Carl E. Linderholm is as follows:

CA watch $ 17 times 17 = $ 289. Generalize this result.

What is the meaning of this statement? Is there really a more general principle hidden in elementary calculus?


I can only guess the following pattern: $ 17 is first and $ 2 = 1 + 1 $, $ 8 = $ 7 + 1, $ 9 = 7 + 1 + 1 $.

nt.number theory – Looking for proof that $ pi $ is irrational using a serial representation

This was requested on MSE but no answers.

I'm looking for proof that $ pi $ is irrational using a representation of the series for $ pi $but do not find it.

However, on this page of Wikipedia, we see that Apery's evidence on the irrationality of $ zeta (3) $ can be simplified to apply on $ zeta (2) $ which is better than what I'm looking for because it shows that $ pi ^ 2 $ is irrational. But I do not find that proof either. If this has been done, maybe nobody has ever published it.

nt.number theory – The greatest common divisor of two specified number sequences (search for equality)

I have a sequence of primitive numbers $ A = {a_1, …, a_n } $, that is to say: $ gcd (a_1, …, a_n) = $ 1, or $ a_1 the a_2 the … the a_n $.

Plus, I have the second sequence of numbers: $ B = {k-a_1, …, k-a_n } $, or $ k ge a_i, i = 1, …, n $.

I'm looking for such conditions in which: $ gcd (a_1, …, a_n) equiv gcd (k-a_1, …, k-a_n) = $ 1.

In more general form: $ gcd (a_1, …, a_n) equiv gcd (k-a_1, …, k-a_n) $ 1 $.


It can be seen that the problem is solved only in certain forms.

I found only four particular solutions.

  1. If there is such a number $ exists a_s in A: k-a_t = a_s $, or $ a_t in A $ then $ gcd (a_1, …, a_n) equiv gcd (k-a_1, …, k-a_n) $.

  2. Let $ gcd (a_1, …, a_n) = e $ and $ gcd (a_n-a_1, …, a_2-a_1) = E $. Yes $ e = E $ and $ e | k $then $ gcd (a_1, …, a_n) equiv gcd (k-a_1, …, k-a_n) $.

  3. Let $ P = p_1 cdot … cdot p_n $ denotes the primorial equaling the product of the first $ n $ prime numbers and $ p_i $ is the $ i ^ {th} $ Prime number. Let $ a_i = frac {P} {p_i} $ and $ k = P $then $ gcd (a_1, …, a_n) equiv gcd (k-a_1, …, k-a_n) = $ 1.

  4. Let $ gcd (k-a_1, …, k-a_n) = $ 1 and $ a_i | k, forall a_i in A $then $ gcd (a_1, …, a_n) = $ 1.

I am convinced that there are other solutions, but I do not find them yet.
I will be grateful for any help.

Theory nt.number – The correct determining exponent of the weight $ k $ – operator for the definition of operators Hecke / adélisantes modular forms

For $ g in operatorname {SL} _2 ( mathbb R) $, and $ mathbb H $ the top half of the plane, and $ k geq 1 $ an integer, the weight $ k-operator on the functions $ f: mathbb H rightarrow mathbb C $ is defined by

$$ f[g](z) = f (g.z) j (g, z) ^ {- k} $$

or $ j (g, z) = (cz + d) ^ {- 1} $, $ g = begin {pmatrix} a & b \ c & d end {pmatrix} $.

In order to define the Hecke operators, to adapt the modular forms or to identify the modular forms as functions on $ operatorname {GL} _2 ( mathbb R) ^ + $, it is necessary to extend this definition to $ g in operatorname {GL} _2 ^ + ( mathbb R) $. In A first course in modular form, Chapter 5.1 Set of Diamonds and Shurman

$$ f[g](z) = f (g.z) j (g, z) ^ {- k} det (g) ^ {k-1} $$

In Automorphic forms and representations, chapter 1.4 Bump sets

$$ f[g](z) = f (g.z) j (g, z) ^ {- k} det (g) ^ {k / 2} $$

Which exponent of the determinant is best to use, and why? If we adele an eigenform of Hecke for $ operatorname {SL} _2 ( mathbb Z) $ and look at the corresponding automorphic representation $ pi = otimes_p pi_p $, which normalization is better to define Hecke operators with, if we want the standard operator Hecke $ T_p $ naturally coincide with an action of spherical Hecke algebra $ mathscr H ( operatorname {GL} _2 ( mathbb Q), operatorname {GL} _2 ( mathbb Z_p)) $ on the local component $ pi_p $?

Recall that to adelise a modular form $ f $ of $ operatorname {SL} _2 ( mathbb Z) $ of a given weight, we would identify first of all $ f $ with a function $ phi $ sure $ operatorname {GL} _2 ^ + ( mathbb R) $ putting

$$ phi (g) = f[g](i) $$

and then we would define an automorphic form $ varphi $ sure $ operatorname {GL} _2 ( mathbb Q) backslash operatorname {GL} _2 ( mathbb A) $ using decomposition $ operatorname {GL} _2 ( mathbb A) = operatorname {GL} _2 ( mathbb Q) operatorname {GL} _2 ^ + ( mathbb R) K $ for K $ a suitable compact subgroup, writing $ g = alpha g _ { infty} k $and setting $ varphi (g) = phi (g _ { infty}) $.