zbMATH — the first resource for mathematics

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.
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
Full Text: Link EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.