# Jacob’s ladders and some new consequences from A. Selberg’s formula

###### Abstract.

It is proved in this paper that the Jacob’s ladders together with the A. Selberg’s classical formula (1942) lead to a new kind of formulae for some short trigonometric sums. These formulae cannot be obtained in the classical theory of A. Selberg, and all the less, in the theories of Balasubramanian, Heath-Brown and Ivic.

###### Key words and phrases:

Riemann zeta-function## 1. The A. Selberg’s formula

A. Selberg has proved in 1942 the following formula

(1.1) |

(see [19], p. 55), where

(1.2) |

(comp. [19], pp. 10, 18, ) and is the Euler’s constant. Since (see [19], p. 10, [20], p. 79)

i.e.

(1.3) |

where

(1.4) |

is the signal defined by the Riemann zeta-function . Following eqs. (1.1) and (1.3) we obtain

(1.5) |

###### Remark 1.

###### Remark 2.

Let us remind that the A. Selberg’s formula (1.5) played the main role in proving the fundamental Selberg’s result

where stands for the number of zeroes of the function .

In this paper it is proved that the Jacob’s ladders together with the A. Selberg’s classical formula lead to a new kind of results for some short trigonometric sums.

## 2. The result

### 2.1.

Let us remind some notions. First of all

(2.1) |

where

(2.2) |

(see [3], (3.9); [5], (1.3); [9], (1.1), (3.1), (3.2)) and is the Jacob’s ladder, i.e. the solution of the following nonlinear integral equation

that was introduced in our paper [3]. Next, we have (see [1], comp. [18])

(2.3) |

and the collection of sequences is defined by the equation (see [1], [18], (6))

where (comp. (1.4))

### 2.2.

In this paper we obtain some new integrals containing the following short trigonometric sums

where is the prime, and is the number of divisors of . In this direction, the following theorem holds true.

###### Theorem.

Let

(2.4) |

Then we have

(2.5) |

(2.6) |

(2.7) |

where

(2.8) |

and is the prime-counting function.

###### Remark 3.

## 3. New asymptotic formulae for the short trigonometric sums: their dependence on

We obtain, putting in (2.5)

Using successively the mean-value theorem (since is a segment), we have

(3.1) |

Since

(3.2) |

(comp. Remark 8), and

(3.3) |

(see [2], (13), stands for the measure) then we obtain from (2.5) (see (3.1) - (3.3)) the following

###### Corollary 1.

For every there are the values such that

(3.4) |

where .

Similarly, we obtain from (2.6)

###### Corollary 2.

For every there are the values such that

(3.5) |

where .

## 4. New asymptotic formulae on two collections of disconnected sets

From (2.7), similarly to p. 3, we obtain

###### Corollary 3.

(4.1) |

where denote the mean-value of on .

###### Remark 6.

It follows from (4.1) that the short trigonometric sum

has an infinitely many zeroes of the odd order.

## 5. Law of the asymptotic equality of areas

###### Corollary 4.

(5.1) |

###### Remark 7.

The formula (5.1) represents the law of the asymptotic equality of the areas (measures) of complicated figures corresponding to the positive part and the negative part, respectively, of the graph of the function

(5.2) |

where . This is one of the laws governing the *chaotic* behaviour of the positive and negative values of the signal (5.2).
This signal is created by the complicated modulation of the fundamental signal , (comp. (1.4), (2.2)).

## 6. Proof of the Theorem

### 6.1.

### 6.2.

### 6.3.

Similarly, from the formulae

and

(see [2], (13) and Corollaries 8 and 9) we obtain (2.6) and (2.7), respectively.

I would like to thank Michal Demetrian for helping me with the electronic version of this work.

## References

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