Some results on Theory of Infinite Series and Divisor Sums

where x > 0 and µ is the Moebius-µ arithmetic function (see [1]). The function µ(n) take the values (-1) r when n is square free and product of r primes, else is 0. Also µ(1) = 1.

Because f (0) = 0 and f analytic in (-1, 1), the integral

x ∞ f (e -t )dt exists for every x > 0. We assume that exists arithmetic function X(n) such that:

we will determine this function X.

Taking logarithms in both sides of (2) we have But from analytic property of f in (-1, 1) we have

and consequently

Therefore from (B) and the above relation it must be

By applying the Moebius inversion theorem (see [1]) we get

This completes the proof. Note also that holds the following similar expression

Examples on Proposition 1. 1) If f (x) = x then f (n) (0) n! = δ n , n = 1, 2, 3, . . . i.e δ 1 = 1, 0 else. Hence

)

= n, n = 1, 2, 3, . . ., then f (x) = x (x-1) 2 and X(n

-ν (n) n

, i.e σ ν (n) = d|n d ν is the sum of the ν-th power of divisors of n and σ (-1) ν is its arithmetic inverse. This means d|n σ ν (d)σ

where σ

Proof.

Easy consequence of Proposition 1.

If a is positive real number then

Proof.

Set x = a > 0 in (1) and take the logarithmic derivative in both sides with respect to a.

Proof.

Set x = a and x = 2a in (1) to take two relations, divide them. Take the logarithms and derivate. After a few simplifications we get (9).

If A(n) is arbitrary arithmetic function, then for x > 0 we have

Proof.

Using again Proposition 1 we get the result.

Lemma 1.

Proof.

, then from Moebius inversion theorem we have

Using Proposition 2 we get the result.

and |q| < 1, then for every f we have

Summing with respect to m we have

and the result follows.

Let d|n X(d) = g (n) (0) n! , then for every f and |q| < 1 we have

where

2. Also setting

or the equivalent

where Li ν (x) = ∞ n=1

x n n ν .

3. With h n = δ n in Theorem 2 and f (n) → a(n), we get

Differentiating with respect to q and setting q = e 2πiz , Im(z) > 0, we get

The case

lead us to some kind of “Eisenstein series” (we have set

where

and

Then from Hecke theorem for modular forms we have

If M k+1 denotes the k + 1-th space of modular forms (that is of weight k + 1) and

then we can say f (z) ∈ M iff f (z) can be written as a sum of p different weight modular forms.

The same thing happens and with f in which

Then f is a sum of modular forms of different weights.

Hence we can state that if a function is a “mixed” modular form

and

then

where

and

Moreover holds the next Hecke-type theorem for derivatives of modular forms:

then we have

where

and Λ * (s) satisfies the functional equation Λ * (s)

Remarks. i) Here the weight is k. ii) The proof of the theorem is based on Proposition 11 and Hecke’s theorem.

Assume that

Set also

and hence

From the above we get the following

2 Results in the theory of theta functions Set (n, m) = gcd(n, m) to be the greatest common divisor of n, m. Then one can easily see, using arguments of [1] chapter 2, that

From the relation d|n φ(d) = n and relation (11) we get

From Proposition 5 we have for general arithmetic function F :

integrating the above relation we get:

), then we have the next version of Jacobi triple product identity (see [4]):

Setting e -aπ = q, a > 0 and F (x) = cos(2tx) in (50) and using (51) we get

where

is Jacobi’s 4th theta function (see [2]).

Setting t = π 2 in (52) we get

Hence we can state the next

Proof.

We use (see [6] pg.488):

and (see [2] pg.107 and related theory):

and relation (53). The functions k = k r , K = K(k) are the elliptic singular modulus and elliptic integral of the first kind at singular values respectively. The function

Then using the summable version of Jacobi triple product identity (relation (51)), we get after differentiating with respect to t:

Hence from (50) we have

is a modular form of weight 2ν, where

are the known Eisenstein series (see [18]).

Theorem 6.

For ν = 1, 2, . . ., we have

Proof.

Easy. From (60) we immediately get the result.

Let F (n) = a 2 n 2 + a 4 n 4 + . . . + a 2k n 2k + . . . be even function with

Another interesting result which follows from relation (59

. This can be done by writing

Expanding the logarithm on the left into power series we get

From this we arrive to (66) and we have the next

with

Relation (67) can also be found if we use directly the Jacobi’s formula

This can be done integrating (68) from z = 0 to z = t. Then we arive to (67) expanding q n 1-q 2n = x 1-x 2 into Taylor series of x = q n and making the double infinite series into single using divisor sumation.

The above formula (67) for t = π 2 gives Proposition 9.

where

Theorem 9.

The above transformation T a(n) = b n is given from relation

where b 0 = 1. Hence given the general form of a n , we can construct all T a(n).

Notes. In general hold the following relations

This happens because exp -

and

with lim N →∞ (T

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