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Integral equalities involving integrals of the logarithm of the Riemann

In recent papers [

In this paper we establish new integral equalities equivalent to RH. We use exponential weight functions, and, in our opinion, the resulting equations are especially interesting. In particular, we were able to rigorously estimate the possible contribution of the Riemann function zeroes nonlying on the critical line which were shown to be extremely small, for example, smalle8r than nine milliards of decimals for the “maximal possible”; in a sense (see below), weight function

The main tool for our work here is the following generalized Littlewood theorem about contour integrals involving logarithm of an analytical function.

Let

Then

Our proof closely follows the well-known proof of the Littlewood theorem (or lemma) given, for example, in [

Illustrating the proof of the generalized Littlewood theorem.

From the proof of Theorem

Now let us consider a rectangular contour

If

Now we should remind the reader that the function

Using (

Equations (

Equalities

For all possible Riemann

In (

Equalities

Next, two similar theorems can be proved when we select left contour border line

Theorem 2a.

In conditions of the theorem, all contributions of the Riemann function zeroes lying on the critical line,

Quite similarly, we have Theorem 2b whose proof one-to-one follows that of Theorem 2a.

Theorem 2b.

Unfortunately, similar theorems cannot be formulated for

Exponential weight function appearing in the integrals considered in the previous section makes the problem of estimation of the maximal possible contribution of remaining Riemann function zeroes nonlying on the critical line a rather simple one. For example, for the real part, we know ((

To summarize, we may say that, combining a rigorous analytical treatment with the known numerical results on the Riemann function zeros on the critical line, we have found an equation of the form

There is a more recent calculation of Gourdon where it is reported that the first

It is also worthwhile to note that if we put the question what is the attainable precision of equalities pertinent to check up whether there are no Riemann function zeroes with

We now discuss what is the situation with

Unfortunately, we do not see how the constant

The other way around, we can select

For any real positive

If we move the left border of the contour further to the left, then, contrary to the case

One of such unconditionally true equalities, namely, that obtained from (

Interesting new possibilities appear if we take certain value of

Let us now analyze both integrals of (

Integral involving an argument of Riemann

Now we need to analyze second integral in (

First,

To finish the consideration of the contribution of the logarithm of gamma-function we need to add the contribution of

Finally we need to analyze the contribution of the sine factor in (

Collecting everything together we see the cancelling of

Repeating what has been said above about the “maximality” of the sum

In this paper we have established a number of new criteria involving the integrals of the logarithm of the Riemann

We also show how certain Fourier transforms of the logarithm of the Riemann zeta-function taken along the real (demi)axis are expressible via elementary functions plus logarithm of the gamma-function and definite integrals thereof, as well as certain sums over trivial and nontrivial Riemann function zeroes. In our opinion, further study of similar Fourier transforms is interesting and might be useful in the Riemann researches.

The authors declare that there is no conflict of interests regarding the publication of this paper.

^{13}first zeroes of the Riemann Zeta Function, and zeroes computation at very large height