1. The Injectivity Criteria Theorem

As we mentioned in the previous page, the problem of solving RH has been reduced to the proof of the injectivity of the integral operator defined in (2). To solve this problem, first we introduced our Injectivity Criteria Theorem on separable Hilbert spaces, then we applied the Criteria to the integral operator used in the RH equivalent formulations choosing the most appropriate family of subspaces from the point of view of the computation efforts. Here is a description of the steps made in obtaining the proof of the Alcantara-Bode hypothesis.

The theory

Our Injectivity Criteria Theorem states: a linear bounded operator on a separable Hilbert space, strict positive definite on a dense family of including finite dimension subspaces and having the injectivity parameters bounded by a strict positive constant, is injective (Injectivity Criteria Theorem [3], [4]). The injectivity parameters associated with the operator restrictions on each subspace are defined as the ratio between the min and max eigenvalues when the operator is Hermitian. We provide a formula for non Hermitian operators using the positivity parameters and the optimal bounds of the adjoint of the operator restrictions. An outline of the proof of our Theorem for non Hermitian operators follows.

Let $T$ be a linear bounded operator on a separable Hilbert space H strict positive definite on a dense family of finite dimension including subspaces, name it $F$ in H, and consider $L_F$ the class of such operators: if $T \in L_F$ then there exist strict positive constants $\alpha_n$ such that $\langle Tv,v\rangle \geq \alpha_n\|v\|^2$ for every $v \in S_n$ , n $\geq 1$ . Denoting $\omega_n$ as the closest approximation of the norm of the restriction of the adjoint of T on $S_n$ ( $\|T^*v\| \leq \omega_n\|v\|$ for every $v \in S_n)$ , we then define the injectivity parameters $\mu_n = \alpha_n/w_n \quad n \geq 1,$ associated with the linear operator $T$ , relative to the dense family $F$ . For $T = T^*$ on every finite dimension subspace $S_n$ is easy to obtain $\mu_n = \lambda_{min}(T_{|S_n})/\lambda_{max}(T_{|S_n})$ . In applied mathematics, the parameter $\mu_n^{-1}$ is named sometimes the condition number of the restriction on the subspace $S_n$ of the Hermitian operator $T$ , terminology which we used often in our paper.

Observation: if $T \in L_F$ then $T$ has no zeros in any subspace of $F$ – otherwise, if there exists an $u \in S_n$ for some n such that $Tu = 0$ , the strict positivity is violated: we could not have $0 = \langle Tu,u\rangle \geq \alpha_n\|u\|^2$ for u not null. It follows that, if $T$ has a zero, it could be only outside the set $\cup_1^\infty S_n$ . The elements of norm equal to one from H that are outside of the infinite union, are eligible elements (of norm 1) to be zeros of a linear operator strict positive definite on the family $F$ . Lets observe that $u$ and $u/\|u\|$ are either both in the null space of $T$ or both outside of it. Thus, working with elements of norm 1 from outside of our dense family, (i.e. elegible elements), does not restrict generality.

For every u eligible, let $u_n = (P_{\mid_{S_n}} u)$ be the orthogonal projection of u on $S_n$ and define $\beta_n(u) := \|u - u_n\|$ , its normed residuum. If u is a zero of $T, (Tu = 0, u \in H \setminus \cup_{i=1}^\infty S_i, \|u\| = 1)$ , then (Lemma 1, [3], [4]):

$(3) \qquad \qquad \qquad \mu_n \|u_n\| \leq \beta_n(u), \quad n \geq n_0 := n_0(u)$ ,

$n_0(u)$ being the subspace index starting from which the projections $u_n$ are not null – in fact, the inequality is true also for every subspace $S_n$ for which $\quad n < n_0$ . For every u eligible, the existence of the index $n_0(u)$ is a result of the density of the family of including subspaces. $%The inequality (3) is obtained by combining the positivity property of the linear operator on latex S_n with the identity latex \langle Tv, v \rangle = \langle T(v-u), v\rangle, valid for latex u a zero of latex T and for every latex v \in H$

Rewriting the previous inequality, an eligible zero of $T$ , name it $u$ , verifies:

$(4) \qquad \qquad \qquad \mu_n \leq \beta_n(u)/\sqrt{1-\beta_n^2(u)}$ ,

where the left side of the inequality is the operator injectivity parameter defined on $S_n$ that is independent from the set of eligible elements and the right side is an expression, independent from any linear operator, of the residuum of $u$ converging to 0 for n -> $\infty$ .

Obviously, an eligible u verifying

$(5) \qquad \qquad \qquad \mu_n > \beta_n(u)/\sqrt{1-\beta_n^2(u)}$ for $n \geq n_1(u), n_1 \geq n_0$ ,

is not a zero of $T (u \notin N_T)$ .

Remark: if the inequality (5) is valid for every eligible element, then T is injective.

Follows: a sufficient condition for T to be injective is given by the Injectivity Criteria Theorem ([3], [4]):

if the sequence $\mu_n, n \geq 1$ is inferior bounded by a strict positive constant, then $N_T = \lbrace 0\rbrace$ , equivalently T is injective.

Another elementary property of linear operators on Hilbert spaces allows us to simplify our framework: $T$ and $T^*T$ have the same null spaces. Since $T^*T$ is positive definite on the whole space ( $\langle T^*Tu,u\rangle \geq 0, \forall u \in$ H), it follows that when no information on the positivity of $T$ is known, we could switch $T$ with its associated Hermitian $T^*T$ for continuing the investigation with the new operator.

The application of the Criteria to Alcantara-Bode’s equivalent formulation of RH.

In order to apply the Injectivity Criteria theorem to a linear operator on a separable Hilbert space, we have to find the most appropriate dense family of finite dimension including subspaces. For the space of functions $L^2(0,1)$ we considered the family of subspaces spanned by the indicator interval functions, (or characteristic interval functions), with disjoint support covering the domain (0,1). To ease the computation effort $%and to meet the method's requirements,$ we split the domain in equal subintervals and use their indicator functions $\chi_{h,k}(x) = 1$ when $x\in ((k-1)h, kh]$ and 0 in the rest of the domain, k=1,n, nh=1, for building the corresponding subspace $S_h = span \{\chi_{h,k}, k=1,n, nh=1\}$ . The including property of the subspaces is obtained by halving the subintervals, for obtaining the new subspace including the previous one. The family of subspaces obtained by this process, denoted by $F_\chi$ , is dense in $L^2(0,1)$ , a result well known in literature and, with the orthogonal projections built using the finite rank (integral) operators technique based on indicator interval functions we meet the requests in order to apply the Injectivity Criteria.

From the point of the view of the theoretical approach of discretizations on separable Hilbert space H, let $S_h$ be the subspace spanned by the family of functions ${\chi_{h,k}, k = 1,n}$ , $nh =1$ . The family of functions defines a finite rank integral operator with the kernel function:

$(6) \qquad \qquad \qquad r_h(y,x) = h^{-1}\sum_{k=1}^n\chi_{h,k}(y)\chi_{h,k}(x)$

which is an orthogonal projection in $L^2(0,1)$ on $S_h$ $% having the orthogonal eigenfunctions latex {\chi_{h,k}, k = 1,n}$ (see [5]). In fact, each $S_h \in F_\chi$ is generated by the families of the orthogonal eigenfunctions of the projection operator $P_h (:= P_{|S_h})$ . This approximation method based on orthogonal projections defined by finite rank operators has been used (in [5],[6]) in the approximation of positive integral operators having Mercer type kernel functions.

Now, taking a look at the kernel function $r_h$ , (6), we could observe that it is a sum of n functions (nh=1) with disjoint compact support. Consequently, the corresponding projection of a kernel function will maintain the structure of a sum of kernel pieces with disjoint compact support. By applying the projection operator having the kernel function $r_h$ to the kernel function $\rho$ , it follows that the discretization form on $S_h$ of $\rho$ is given by the formula:

$(7) \qquad \qquad \qquad \rho_h(y,x) = h^{-1}\sum_{k=1,n}\chi_{h,k}(y) \rho(y,x) \chi_{h,k}(x)$ , nh=1, $n \geq 2$ .

Thus the matrix representation of the corresponding integral operator restriction is a 1-diagonal sparse matrix.

In other words: the orthogonal projections in $L^2(0,1)$ on $S_n, nh = 1, n \geq 2$ having the orthogonal eigenfunctions ${\chi_{h,k}, k = 1,n}$ produce 1-diagonal sparse matrix representations for the integral operator retrictions on these subspaces. Thus, the family of subspaces $S_h, nh = 1, n \geq 2$ obtained by halving the sub-intervals in (0,1) on each level of the multi-level domain discretization is dense in $L^2(0,1)$ , and is built on including finite dimension subspaces.

In consequence, the matrix entries of the integral operator restriction to a subspace $S_h$ , nh=1, have the form:

$(8) \qquad \qquad \qquad d_{kk}^h = \int_{(k-1)h}^{kh} \int_{(k-1)h}^{kh} \rho(y,x)dxdy$ , k=1,n

and zero outside the diagonal. The entries are strict positive valued, inferior bounded by $d_{11}^h$ and superior by $d_{22}^h$ , showing the strict positivity of the integral operator on the family of subspaces ${S_h, nh=1, n \geq 2}$ . Moreover, for n -> $\infty$ , the sequence of entries in (8) modulo $h^2$ , is monotonically convergent to 1/2. Thus, it has been easy to determine the operator injectivity parameters sequence:

$(9) \qquad \qquad \qquad \mu_h (T_\rho) \geq$ ( 3 − 2 $\gamma$ )/( 2(3ln2 – 1) ) $> 0$ , nh=1, $n \geq 2$

where $\gamma$ is the Euler-Mascheroni constant. Because the kernel function of the adjoint operator is given by $\rho(x,y)$ , the adjoint operator restrictions on family subspaces are bounded and, their matrix representations coincide with those of our operator justifying the formula (9). Consequently, it follows from the Injectivity Criteria (Theorem 1, [3], [4]) that the Hilbert-Schmidt integral operator $T_\rho$ on $L^2(0,1)$ having the kernel function $\rho(y,x) = \lbrace y/x\rbrace$ has the null space containing only the element 0 – equivalently, the integral operator is injective on H. Based on this last result, according to Alcantara-Bode’s equivalent formulation of the Riemann Hypothesis [2[, the Riemann Hypothesis holds.

Observations:

1) We applied our method to the integral operator $T_\rho$ which is neither Hermitian, nor symmetric, nor self-adjoint, because these properties are not mandatory requirements in our theory – how they are instead in the studies related to the Mercer like kernels. On the other hand, we observed that the positivity parameters of the associated Hermitian operator could be computed from the positivity parameters of our operator. In addition, because both our operators are bounded, we obtained an estimate for the injectivity parameters of the associated Hermitian, showing the obviously, expected result: the Hermitian operator $T_\rho^* T_\rho$ is injective like $T_\rho$ .

2) The use of orthogonal projections, (6), in the case of the integral operator in (2) from RH equivalences, is justified by observing that its kernel, the fractional part function, is continuous almost everywhere excepting a set of measure Lebesgue zero. Based on these observations, we notice that we have all the ingredients needed to apply the injectivity criteria (Theorem 1, [3], [4]).

[1] Beurling, A.(1955) “A closure problem related to the Riemann zeta function”, $\emph{Proc. Nat. Acad. Sci. 41}$ pg. 312-314, 1955.

[2] Alcantara-Bode, J. (1993) “An Integral Equation Formulation of the Riemann Hypothesis”, $\emph{Integr Equat Oper Th, Vol. 17}$ , 1993.

[3] Adam D., “On the Injectivity of an Integral Operator Connected to Riemann Hypothesis”, J. Pure Appl Math. 2022; 6(4):19-23 , DOI: 10.37532/2752-8081.22.6(4).19-23

[4] Adam D., “On the Injectivity of an Integral Operator Connected to Riemann Hypothesis”, Research Square https://doi.org/10.21203/rs.3.rs-1159792/v9 (crossref) – v1 dated Dec 2021

[5] Buescu, J., Paixao A. C., (2007) ”Eigenvalue distribution of Mercer-like kernels”, Math.Nachr.280,No.9–10, pg. 984 – 995, 2007.

[6] Chang, C.H., Ha, C.W. (1999) ”On eigenvalues of differentiable positive definite kernels”, Integr. Equ. Oper. Theory 33 pg. 1-7, 1999

[7] Adam, D. (1994) “Mesh Independence of Galerkin Approach by Preconditioning”, Preconditioned Iterative Methods – Johns Hopkins Libraries, Lausanne, Switzerland; [Langhome, Pa.] : Gordon and Breach, ©1994

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