# A variant of the Bombieri-Vinogradov theorem with explicit constants

###### Abstract

In this paper we improve the result of [1] with getting instead of . In particular we obtain a better version of Vaughan’s inequality by applying the explicit variant of an inequality connected to the Möbius function from [2].

## 1 Introduction

For integer number and , let

where is the von Mangoldt function. The Bombieri-Vinogradov theorem is an estimate for the error terms in the prime number theorem for arithmetic progressions averaged over all .

###### Theorem.

(Bombieri-Vinogradov) Let be a given positive number and where , then

The implied constant in this theorem is not effective, since we have to take care of characters, associated with those that have small prime factors. The main result of this paper is

###### Theorem 1.

(Bombieri-Vinogradov theorem with explicit constants) Let , . Let also denote the least prime divisor of . Define by

Then

where

Previously the best result obtained by these methods in the literature is due to Akbary, Hambrook (see [1, Theorem 1.3]), where they proved that under assumptions of Theorem 1 we have.

where is defined by

Here we reduce this power to by applying an explicit version for an upper bound for

where is Mobius function, is a given number. This version can be found in [2], namely we have

###### Lemma 1.

(Helfgott, [2, (6.9), (6.10)]) For large enough we have

and means that it is less in absolute value than .

This Lemma is a variant of the sum considered in [3], where it is shown that

tends to a positive constant as . It is also suggested without proving that can be about .

Notice, that by sharpening the inequality in Lemma 1 we will not be able to reduce the power of , since the upper bound is optimal there, so by these methods the power is the best possible. Going further seems to be a hard problem which involves among simpler things a very careful analysis of the logarithmic mean of Möbius function twisted by a Dirichlet character.

###### Remark.

Let

###### Remark.

It would be very good for applications to get in Theorem 1, however it seems impossible to get by present methods.

###### Remark 1.

Proof of the remark is exactly the same as in [1], we just have to change the power of .

The key tool for the proof of Theorem 1 is Vaughan’s identity, which we have to get in an explicit version for our goal. Define

the twisted summatory function for the von Mangoldt function and a Dirichlet character modulo . One of two main results of this paper is

###### Proposition 1.

(Vaughan’s inequality in an explicit form) For

where is any positive real number and means a sum over all primitive characters .

## 2 Proof of Proposition 1

Fix arbitrary real numbers and . In this section, we shall establish Proposition 1, which is the main ingredient in the proof of Theorem 1. Here we follow the ideas of [1] and applying the results from [2]. The main tool in the proof is the large sieve inequality (see, for example [5, p.561])

(1) |

from which it follows (see [1, Lemma 6.1]) that

(2) |

where , and , are the number of terms in the sums over and respectively. Here the , are arbitrary complex numbers.

### 2.1 Sieving and Vaughan’s identity

We reduce to the case . If , then the sum on the left-hand side of (1) is empty and we are done. Next, then only the term exists and we have

(3) |

which is better than the theorem. Finally, if , Theorem 1 follows from (2) with , , , by the estimate

From now on we assume . Notice that the fact that we can restrict ourselves to the range allows us to apply Lemma 1(otherwise it would make less sense, since the main term in Lemma 1 would be smaller than -term). As in [1] we will use Vaughan’s identity (see also [6])

where

Assume , , and is a character mod . We use the above decomposition to write

where

Let , be non-negative functions of and to be set later and denote the contributions to our main sum by

Easily we obtain

The heart of the proof of Theorem 1.3 in [1] are the following estimates:

###### Lemma.

We estimate contribution with the use of Lemma 1. Writing as a dyadic sum we have

Using the triangle inequality

where , and, as it was defined in the introduction . Now apply the large sieve inequality (2) to get

where

where and denote the number of terms in the sums over and , respectively. From the definition of and we conclude

By Chebyshev estimate we have an upper bound

Thus by Cauchy inequality

(4) |

Further

and

Using Lemma 1 we get

that implies

Since

then

Now let’s specify and . If . Then putting that into previous expression we get for the factor , then

where we used the fact that and keeping in mind the condition we find that . Working in the same manner with and

If , we let and get

where we used ,

Similarly we get for

## 3 Proof of Theorem 1

Let ,. By orthogonality of characters modulo , we have

Define if and otherwise, is the principal character mod . Then

For a character (mod ), we let be the primitive character modulo inducing . Follow the way of [1] we obtain

If then , and hence . If then . Therefore

Denote the quantity we want to estimate as

Since

then

We have to take care just of the second term in the inequality above, since the first one is smaller than the desired bound. It remains to prove

where is the function from Theorem 1. A primitive character induces characters of moduli and for principal, we observe

As it was noted in [1] for

and as , and , we have

For and primitive character (), we know that is non-principal and . Since we assumed then we can can replace by inside the internal sum for . Combining it with an expression for we get

Thus it remains to show that

Let

Partial summation gives us

Now we apply Theorem 1

where

where

Calculating the integrals gives us

Finally

### 3.1 Proof of Remark 1

Define two functions

Since

where we used the fact that for (see for example [1, Lemma 3.1])

Similarly, Thus by partial summation we obtain the bound