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Using the properties of theta-series and Schwarz reflection principle, a proof for Riemann hypothesis (RH) is directly presented and the first ten nontrivial zeros are easily obtained. From now on RH becomes Riemann Theorem (RT) and all its equivalent results and the consequences assuming RH are true.

The Riemann zeta function has its origin in Dirichlet series function

where n runs through all integers, and

Euler showed the production formula,

where p ranges over all primes. It converges for real s greater than 1.

Riemann extends

and Jacobi functional equation in his ground breaking paper (Riemann 1859) [

The function

The convergence of the Euler product shows that ζ(s) has no zeros in the region: Re s > 1, as none of the factors have zeros. The Riemann hypothesis discusses zeros outside the region of convergence of this series and Euler product.

The Riemann hypothesis is a deep mathematical conjecture which states that the nontrivial Riemann zeta function zeros, i.e., the values of s other than −2, −4, −6, such that

Riemann [^{st} century listed by Clay Mathematics Institute. “The attempt to solve the problem has occupied the best efforts of many of the best mathematicians of the twentieth century. It is now unquestionably the most celebrated problem in mathematics and it continues to attract the attention of the best mathematicians, not only because it has gone unsolved for more than one and half century but also because it appears tantalizingly vulnerable and because its solution would probably bring to light new techniques of far reaching importance” [

The other thing makes it is of importance is that it has deep relation with physics. The relation between the zeros of Riemann zeta function on the critical line and the properties of random matrix are described in [

Milestones and great events with the RH have been listed [

Edwards [

To prove the truth of RH, and for convenience, we list some important related results as lemmas as follows [

Lemma 1.

where

The relation of (2.1) is a surprising functional equation, and is far obvious and looks barely possible. We can find the proofs from various references. (See [

Lemma 2.

The following proof procedure is from [

We use the integral formula for the gamma-function. For Re s > 0 and n a natural number, we have

We now suppose that

Changing the order of summation and integration, we obtain

Next, for x > 0 we have

Thus, we have the equality

From (2.2), using this relation and then making the change of variables

as was to be proved.

We note that here since

The gamma-function has the first order pole at the point s = 0; we also have

Since the value of

Lemma 3. The zeros of the Riemann zeta-function (2.3) are the even negative numbers

i.e.

Proof ( [

Since

And

For its value is real on the real axis, so by Schwarz reflection principle, it follows that,

Hence, when ever

Lemma 4. Suppose that t > 0, and we have

Lemma 5. If the following integration is convergent, f is monotone

Then

Now we are ready to prove Riemann theorem (RT) and find the zeros positions.

Riemann Zeta-function

And the RH holds (RT):

Proof: Let

Then,

We consider the imaginary part of the equation (3.2) of both sides.

And

The above integral is real integral of complex parameters. Consider the exponential complex function properties, we have

The many-valued function here the logarithm of x is determined in such a way that it is real for positive value of x [

Suppose

From lemma 3, replace t by −t for t > 0, or equivalently replace

We have the following

From Lemma 1, we have

And from lemma 4, we get

And from lemma 5, we have,

So

We have finally proved RT.

Considering the real parts of (3.2), and let

and the following function

where

Its roots are the zeros on the critical line of Riemann zeta function. Function (4.2) contains the whole information of zeros of Riemann zeta function. Since

The results are well match to the known zeros of Riemann zeta function.

Simple but substantial solutions for Riemann zeta function zeros are presented by letting imaginary and real parts of both sides of the equation (1.1), or its analytic continuation (3.2) and (3.2), to equal. The key step is using the properties of theta-series and the reflection principle to replace t by ?t. The first ten nontrivial zeros positions are easily obtained. From now on RH becomes Riemann Theorem (RT) and all its equivalent results become true. And we may investigate the properties of function (4.2) to study prime distribution.

Xiang Liu,Rybachuk Ekaterina,Fasheng Liu, (2016) A Direct Proof for Riemann Hypothesis Based on Jacobi Functional Equation and Schwarz Reflection Principle. Advances in Pure Mathematics,06,193-200. doi: 10.4236/apm.2016.64016