All Zeros of the Riemann Zeta Function in the Critical Strip Are Located on the Critical Line and Are Simple ()
1. Introduction and Summary
The proof of the Riemann hypothesis is a problem that many mathematicians consider to be the most important problem of mathematics. Indeed, it is one of at most seven mathematics problems for which the Clay Institute has offered a million dollars for its solution. To this end, the pdf publication [1] of Bombieri presents an excellent summary—along with references, to papers and books, to connections with prime numbers, to Fermat’s last theorem, and to the work of authors who have shown that the first 1.5 billion zeros of the zeta function listed with increasing imaginary parts are all simple—all of which are related to the mathematics of this subject. Similarly Wikipedia of the web [2] offers an excellent summary along with references about this subject. The magazine Nature recently published a related article about a discovery by Y. Zhang, of a conjecture (see [1] ) on the spacing of prime numbers
with increasing size [3] . Physicists have also published results on the Riemann hypothesis: in [4] , Meulens compares data about the Riemann hypothesis with solutions of two dimensional Navier Stokes equations, while others [5] have compared eigenvalues of self-adjoint operators with zeros of the Riemann Zeta function. Several papers about solutions to the Riemann hypothesis have also appeared. To this end, the papers of Violi [6] , Coranson-Beaudu, [7] , Garcia-Morales, [8] , and Chen [9] are similar to ours, in that their proof of the Riemann hypothesis are for functions that are different from the zeta function, but which have the same zeros in D as the zeta function.
Castelvecchi, author of the article [3] makes the comment: “The Riemann hypothesis will probably remain at the top of mathematicians wish lists for many years to come. Despite its importance, no attempts so far have made much progress.”
We wish of course to disagree with Castelvecchi’s comment at the end of the above paragraph, since we believe that we have indeed proved the Riemann hypothesis in this self-contained paper, in which we accomplish the following:
1) In §2, defining the function G and showing in detail that G has exactly the same zeros, in the critical strip,
, including multiplicity, as the zeta function;
2) Proving the positivity of
for
and the negativity of
for
, where
(1.1)
3) Introducing the Schwarz reflection principle in §3, which the functions
and
of Equation (1.1) satisfy in D;
4) In §4, proving the Riemann hypothesis by contradiction, by use of results developed in §2, and in §3, of this paper, and by use of the trapezoidal and midordinate rules, [10] , i.e., by proving that G (and
) have no zeros in the region D\L, where L denotes the critical line,
; and
5) Proving by contradiction, via use of results developed in §2 and in §3 and by use of the trapezoidal and midordinate rules that all of the zeros of G, (i.e., all of the zeros of the zeta function) on the critical line L are simple.
Let
, and let
. In this paper we thus derive results about the function G defined by the integral,
(1.2)
which is related to the well-known integral for the Riemann zeta function, defined by
(1.3)
where
denotes the gamma function.
The operations of Schwarz reflection, the evaluation of
and
on important intervals of
, and the operations of trapezoidal and midordinate quadrature can be readily applied to the Fourier transform representation of G, which is gotten from Equation (2) defined for
, whereas an explicit Fourier transform of ζ defined by Equation (1.3) for
does not seem to be available.
2. Fourier Integral Representation of G, via κ
The function ζ has many other representations, with the best known of these given by:
(2.1)
By setting
and
in (2.1), we get the Fourier integral representation of G, namely,
(2.2)
where
,
, and where
is defined by
(2.3)
2.1. Properties of κ, ζ and G
In this section we use the definition of G given in Equation (1.2) and the identities of Equation (2.1) to derive a functional equation for G, and to derive additional properties of κ and G. We also show in detail, that ζ and G have exactly the same zeros in D, including multiplicity, that
is positive for all
and strictly decreasing as a function of σ, for
, and we determine ranges of values of
,
and their derivatives on the real line.
Let us next assign notations for the left and right half of the complex plane, the critical strip(s), and the critical line.
Definition 2.1 Let
denote the left half of the complex plane, i.e.,
, and let
denote the right half. Let the critical strip be defined by
, and let the negative and positive critical strips
be defined as follows:
, and
. The critical line is defined by
.
2.2. Relevant Gamma Function Relations
We shall require the use of the following lemma:
Lemma 2.2 (i.) Replacement of z with z/2 in the duplication formula for the Gamma function, to get:
(ii.) Both
and
are bounded by
for all
, by Equations (6.1.30) and (6.1.31) of [11] ; and
(iii.) That the function
is an entire function [11] ;
is analytic in
except for simple poles at
(
).
Proof. Item (i.) is just Equation (6.1.18) of [11] with z replaced by z/2;
Items (ii.) follow from Equations (6.1.30) and (6.1.31) of [11] ; and
Item (iii.) is just a restatement of a result found in Chapter 16. of [11] . +
2.3. Bounds on κ
The next lemma describes some asymptotic bounds on the function κ, which are obtained by inspection of Equation (2.2).
Lemma 2.3 For any
and for x real, we have
(2.4)
Hence the integral
is finite for any polynomial Q.
Proof. The bounds of κ given in (2.4) follow by inspection of the function κ as defined in (2.2). +
2.4. Analyticity Definition of Multiplicity
Definition 2.4 Let
, let m denote an integer, and let f be analytic in a neighborhood of
.
(a.) The function f is said to have multiplicity m at
if
, with finite c;
(b.) If the multiplicity of f at
is m, and if
, then we shall more specifically say that f is of exact multiplicity m at
;
(c.) If f is of exact multiplicity m at
, then
is said to be a zero (resp., a pole) of f of multiplicity m if
(resp., if
). In particular, if
, (resp., if
,) then
is said to be a simple zero (resp., a simple pole) of f.
2.5. Functional Equations for ζ and G
An important identity of the Riemann zeta function is the well known functional equations for ζ:
(2.5)
This functional equation for the Riemann zeta function has many important uses, including, e.g., the analytic continuation of the zeta function to all of
.
The function G also possesses a functional equation which is given in Lemma 2.5 below, which plays a similar role as the functional equation for ζ. the functional equation for
is gotten by substituting the right-hand-side of the third equation of (2.1) into (2.5), and by use of Lemma 2.2:
Lemma 2.5 Let
, and let G be defined as in (2.2). Then, a functional equation for the function G, valid for all
is:
(2.6)
This equation can also be written in the form:
(2.7)
where K is given by
(2.8)
and where K is non-vanishing in D.
Proof. That K is non-vanishing on D follows from Lemma 2.2. +
2.6. Zeros of G and ζ in D
We prove here the G and ζ have the same zeros with the same multiplicity in D and that these zeros are isolated.
Lemma 2.6 (i.) The functions G and ζ have exactly the same zeros in D, including multiplicity; and
(ii.) All zeros of G in
are isolated.
Proof. (i.) By inspection if the third equation of (2.1) we get, if
is a zero of G of multiplicity
, then we have the identity
, where the function
is analytic and non-vanishing in D, so that
is also a zero of ζ of multiplicity
. In addition, by taking the nth derivative of
, with non-negative integer, n, we get,
(2.9)
Hence, if
is a zero of ζ of multiplicity
, then by applying induction with respect to
to Equation (2.9), we conclude that the multiplicity of the zero
of G is also m;
(ii.) Suppose that there exists a cluster of zeros
of G in D with a sub-sequence that has a limit point z. If
, then G would have to vanish, by Vitali’s theorem. If z is on the line
, then, since
, and since G is analytic in
, it follows by use of the functional equation of G, that
where the point
is now located on the line
, i.e., we are back to the previous case of the convergence of such a sub-sequence to a point on the interior of the right half plane, where
is analytic and bounded, so that
would again have to vanish identically in
. +
2.7. Definitions of
,
and
Definition 2.7 Let G and κ be defined as in Equation (2.2), and let us define
as follows:
(2.10)
If for brevity, we write
for
, C and S for
and
, and
for
, then Equation (2) yields the following definitions for
and for
, where n denotes a non-negative integer:
(2.11)
Let
be defined for any non-negative integer m by
, so that by the Cauchy-Riemann equations,
, where these functions are
readily shown to exist, by Lemma 2.3.
In addition, by Equation (2.1), the functions
and
are related by the following identity:
(2.12)
Lemma 2.8 Let the functions
be defined as in Definition 2.7. Then, for all
, and for all
,
is analytic on the right half plane, and hence also in D. In particular given any
,
is uniformly bounded in the region
.
Proof. This result follows directly by inspection of Equation (2.2 and Lemma 2.3. We omit the straight-forward proofs. +
2.8. Restricting the Domain of
The following lemma restricts the domain of some of our inequalities:
Lemma 2.9 Let Δ be defined by
, where the functions
are defined in Definition 2.7. Then
for all
, and moreover,
is a strictly decreasing function of
for any fixed
.
Proof. We have
which shows
is a strictly decreasing function of
for all fixed
. By making the one-to-one transformation
of
in the above expressions for
, and setting
, we get
Since 1/W is positive on
, and since
is a strictly decreasing function of
for all fixed
, we need only prove that
for all
. To this end we have, by use of Taylor series expansions, that
(2.13)
By way of proceeding from the first to the second line of Equation (2.13) we used the following relations, which are valid for all
:
. The right hand side of Equation (13) then shows that
for all
, i.e., that
for all
. +
2.9. Inequalities for
The following lemma summarizes values of
that have been established.
Lemma 2.10 Let m denote a non-negative integer. Then:
(i.)
for all
;
(ii.)
for all
; and
(iii.)
for all
.
Proof. Item (i.) The proof of this Item follows by inspection of Equation (2.11);
Item (ii.) The proof of this Item follows by Lemma 2.9 and by inspection of Equation (2.11); and
Item (iii.) The proof of this item follows by inspection of Equation (2.11). +
3. Schwarz Reflection
We present the Schwarz reflection principle, which we define as follows:
Definition 3.1 Let f be analytic in D, and real on
, for some
. Then f can be continued analytically (i.e., reflected) across
from
to
by means of the formula
. (3.1)
Remark 3.2 The Schwarz reflection principle enables analytic continuation from
to all of D. For example, if n denotes a non-negative integer, so that the functions
and
are given for
, then by Lemma 2.10, and by Equation (2.8),
is a non-vanishing function of
on
, while if
, then
is a non-vanishing function of
,
changes sign as t changes sign, while
does not change sign as t changes sign.
4. Proof of the Riemann Hypothesis
Short proofs of all of all of the results which stated in the abstract of this paper are made possible by means of two well-known methods of quadrature (see e.g., [10] ), which are defined by the following lemma.
Lemma 4.1 Let f be a real-valued function that is continuous on a finite interval
of
, and twice differentiable in
, let
, and let us set
(4.1)
Then there exist points
and
in
, such that
(4.2)
The first equation of (4.2) denotes the simplest trapezoidal rule, while the second denotes the simplest midordinate rule.
Theorem 4.2 Every zero of G in the critical strip
is located on the critical line
, and it is a simple zero.
Proof. We shall now carry out the proof of Theorem 4.2 by means of the proof of the two lemmas, which follow.
Lemma 4.3 If
, with
and with
, then
.
Proof. Let
, with
and with
, and let us set
,
, and
. If for
, and 4,
vanishes at one of these points, then by the functional equation of Lemma 2.5, and by Schwarz reflection,
vanishes at all of them.
If we assume that
vanishes then
and
vanish, so that we have, by integration of
over
, that
(4.3)
Thus, in the notation of Definition 2.7 and Lemma 4.1, with
, and for some
and
, we get,
(4.4)
By taking the difference between the two equations of (4.4), forming the average of this difference and its complex conjugate, noting that
, and that
, a weighted average for some
, we get,
(4.5)
However, this is a contradiction, since the left and right hand side of Equation (4.5) are different in sign, by Lemma 2.10.
Hence we cannot allow the vanishing of
, i.e., of
, so that our above assumption of the vanishing of
is false.
This completes the proof of Lemma 4.3. +
Remark 4.4 The Riemann hypothesis is true, by Lemma 4.3.
Lemma 4.5 If
is a zero of G on the critical line, then
is a simple zero of G.
Proof. (i.) Let us assume that
vanishes with multiplicity
, where n denotes an arbitrary finite non-negative integer, and let us apply the trapezoidal rule of Lemma 4.1 to the integration of
over
. We then have
, and since by Definition 2.4, both
and
must vanish,
and
must also vanish. We thus get, for some
, that
(4.6)
By averaging of this equation and it’s complex conjugate, thus eliminating the possible imaginary part on the right hand side, we get
(4.7)
Since the left hand side of Equation (4.7) vanishes whereas, by Lemma 2.10, the right hand side does not, this equation provides a contradiction, which tells us that we cannot allow the vanishing of
, i.e., of
, with multiplicity
, where n denotes an arbitrary finite non-negative integer.
(ii.) Similarly, if
vanishes with multiplicity
, with arbitrary finite positive integer n, then both
and
must vanish, so that, by proceeding as we did to arrive at Equation (7), we get
(4.8)
The left hand side this equation vanishes whereas by Lemma 2.10, the right hand side does not, so that Equation (4.8) presents a contradiction. Together with our conclusion for Equation (4.7), this proves that we cannot allow the vanishing of
with multiplicity
, with m denoting an arbitrary positive integer.
It was shown in [12] that the Riemann Zeta function has an infinite number of zeros on the critical line L, and by Lemma 4.3 above there are no zeros of G in D\L. It thus follows, that every zero
of D must be a simple zero on L. +
Acknowledgments
The author is grateful to Drs. Paul Gauthier and Marc Stromberg for their corrections of my proofs related to Lemma 2.5.