2017/06/18 by Yaoming Shi, Shi, Yaoming
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #General Mathematics (math.GM) #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.1706.08868
openalex publication_date 2017/06/18 · openalex created_date 2017/07/14 · openalex updated_date 2026/07/28
The Riemann Ξ(z) function (even in z) admits a Fourier transform of an even kernel Φ(t)=4e9t/2θ''(e2t)+6e5t/2θ'(e2t). Here θ(x):=θ3(0,ix) and θ3(0,z) is a Jacobi theta function, a modular form of weight (1)/(2). (A) We discover a family of functions \Φn(t)\n\geqslant 2 whose Fourier transform on compact support (-(1)/(2)log n, (1)/(2)log n), \F(n,z)\n\geqslant2, converges to Ξ(z) uniformly in the critical strip S1/2:=\|\Im(z)|< (1)/(2)\. (B) Based on this we then construct another family of functions \H(14,n,z)\n\geqslant 2 and show that it uniformly converges to Ξ(z) in the critical strip S1/2. (C) Based on this we construct another family of functions \W(n,z)\n\geqslant 8:=\H(14,n,2z/log n)\n\geqslant 8 and show that if all the zeros of \W(n,z)\n\geqslant 8 in the critical strip S1/2 are real, then all the zeros of \H(14,n,z)\n\geqslant 8 in the critical strip S1/2 are real. (D) We then show that W(n,z)=U(n,z)-V(n,z) and U(n,z1/2) and V(n,z1/2) have only real, positive and simple zeros. And there exists a positive integer N\geqslant 8 such that for all n\geqslant N, the zeros of U(n,x1/2) are strictly left-interlacing with those of V(n,x1/2). Using an entire function equivalent to Hermite-Kakeya Theorem for polynomials we show that W(n\geqslant N,z1/2) has only real, positive and simple zeros. Thus W(n\geqslant N,z) have only real and imple zeros. (E) Using a corollary of Hurwitz's theorem in complex analysis we prove that Ξ(z) has no zeros in S1/2∖ℝ, i.e., S1/2∖ ℝ is a zero-free region for Ξ(z). Since all the zeros of Ξ(z) are in S1/2, all the zeros of Ξ(z) are in ℝ, i.e., all the zeros of Ξ(z) are real.