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On gigantic density of zeros of some signals defined by prime numbers. (English) Zbl 0859.11051
Denoting the sum \(\sum_{p\leq P} {1\over\sqrt p} \cos (t\ln p)\) (which occurs in the Riemann-Siegel formula) by \(G(0;t,0)\), the author considers the more general sums \[ G(x;t,\varphi) = \sum_{p\leq P} {1\over \sqrt {p+x}} \cos \{t\ln (p+x) + \varphi_p\} \] with \(x\in [0,1]\), \(t>0\), and \(p,P\) denoting primes. (For further details of definitions and related results, see the author’s paper ‘On the order of a sum of E. C. Titchmarsh in the theory of the Riemann-zeta function’ (Russian), Czech. Math. J. 41, 663–684 (1991; Zbl 0754.11023) (Russian)]. The main result in the paper is the following:
With \(x\in [0,1/9P^2]\), \(P\in [2,T^{1/9}]\), there exists a zero of \(G(x;t, \varphi)\) of odd order in the interval \((x,x+ P^6 \Delta/T \ln^4P)\), where \(\Delta\) is a sufficiently large positive number.
MSC:
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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References:
[1] Goldman, S.: Theory of Information. Constable and Company, London, 1953. · Zbl 1151.94459
[2] Moser, J.: Riemann-Ziegel formula and some analogues of the Kotelnikov-Whittaker-Neuquist theorem from the theory of information. Acta Math. Univ. Comen. 58-59 (1991), 37-74.
[3] Moser, J.: On the order of a sum of E. C. Titchmarsh in the theory of Riemann zeta function. Czechoslov. Math. J. 41(116) (1991), 663-684. · Zbl 0754.11023
[4] Titchmarsh, E.C.: The Theory of Riemann Zeta Function. IL, Moscow, 1953. · Zbl 0042.07901
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