MB
M.A.M. Bus
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Real Zeros in Finite Models of the Completed Riemann Zeta Function
A Conditional de Branges Approach
The Riemann Hypothesis can be reformulated as a question about the real zeros of a recentered version of the completed Riemann zeta function. In this thesis, this function is approximated by truncated theta-kernel models obtained by cutting off both the theta kernel and the integral in its cosine-transform representation. These finite models converge locally uniformly to the recentered completed zeta function.
The main goal of the thesis is to study a conditional de Branges mechanism for these finite models. Starting from a truncated model F_{a,N}, we form E_{a,N,τ} = F_{a,N} + iτF'*{a,N}. Under the assumption that E*{a,N,τ} satisfies the Hermite–Biehler condition, one obtains a de Branges space H(E_{a,N,τ}). In this space, multiplication by the variable z is studied as a symmetric operator, and its self-adjoint extensions provide a spectral interpretation of the zeros of F_{a,N}.
The Hermite–Biehler assumption is very strong: it already forces the zeros of F_{a,N} to be real and simple. Therefore, the operator-theoretic construction should not be seen as an independent proof of the real-zero property. Its role is to place the zeros in a spectral framework. If the Hermite–Biehler condition could be proved for a cofinal sequence of truncated theta-kernel models, then local uniform convergence and Rouché’s theorem would transfer the real-zero property to the limiting completed zeta function, implying the Riemann Hypothesis.
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The main goal of the thesis is to study a conditional de Branges mechanism for these finite models. Starting from a truncated model F_{a,N}, we form E_{a,N,τ} = F_{a,N} + iτF'*{a,N}. Under the assumption that E*{a,N,τ} satisfies the Hermite–Biehler condition, one obtains a de Branges space H(E_{a,N,τ}). In this space, multiplication by the variable z is studied as a symmetric operator, and its self-adjoint extensions provide a spectral interpretation of the zeros of F_{a,N}.
The Hermite–Biehler assumption is very strong: it already forces the zeros of F_{a,N} to be real and simple. Therefore, the operator-theoretic construction should not be seen as an independent proof of the real-zero property. Its role is to place the zeros in a spectral framework. If the Hermite–Biehler condition could be proved for a cofinal sequence of truncated theta-kernel models, then local uniform convergence and Rouché’s theorem would transfer the real-zero property to the limiting completed zeta function, implying the Riemann Hypothesis.
...
The Riemann Hypothesis can be reformulated as a question about the real zeros of a recentered version of the completed Riemann zeta function. In this thesis, this function is approximated by truncated theta-kernel models obtained by cutting off both the theta kernel and the integral in its cosine-transform representation. These finite models converge locally uniformly to the recentered completed zeta function.
The main goal of the thesis is to study a conditional de Branges mechanism for these finite models. Starting from a truncated model F_{a,N}, we form E_{a,N,τ} = F_{a,N} + iτF'*{a,N}. Under the assumption that E*{a,N,τ} satisfies the Hermite–Biehler condition, one obtains a de Branges space H(E_{a,N,τ}). In this space, multiplication by the variable z is studied as a symmetric operator, and its self-adjoint extensions provide a spectral interpretation of the zeros of F_{a,N}.
The Hermite–Biehler assumption is very strong: it already forces the zeros of F_{a,N} to be real and simple. Therefore, the operator-theoretic construction should not be seen as an independent proof of the real-zero property. Its role is to place the zeros in a spectral framework. If the Hermite–Biehler condition could be proved for a cofinal sequence of truncated theta-kernel models, then local uniform convergence and Rouché’s theorem would transfer the real-zero property to the limiting completed zeta function, implying the Riemann Hypothesis.
The main goal of the thesis is to study a conditional de Branges mechanism for these finite models. Starting from a truncated model F_{a,N}, we form E_{a,N,τ} = F_{a,N} + iτF'*{a,N}. Under the assumption that E*{a,N,τ} satisfies the Hermite–Biehler condition, one obtains a de Branges space H(E_{a,N,τ}). In this space, multiplication by the variable z is studied as a symmetric operator, and its self-adjoint extensions provide a spectral interpretation of the zeros of F_{a,N}.
The Hermite–Biehler assumption is very strong: it already forces the zeros of F_{a,N} to be real and simple. Therefore, the operator-theoretic construction should not be seen as an independent proof of the real-zero property. Its role is to place the zeros in a spectral framework. If the Hermite–Biehler condition could be proved for a cofinal sequence of truncated theta-kernel models, then local uniform convergence and Rouché’s theorem would transfer the real-zero property to the limiting completed zeta function, implying the Riemann Hypothesis.