Explicit Bounds for Some Functions of Prime Numbers

Rosser, Barkley

doi:10.2307/2371291

Cited by 147 publications

(109 citation statements)

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“…for t ≥ 2, due to R o s s e r [22]. We may take this also as an estimate for the modulus of S(t) for sufficiently large t since the difference between these two expressions is vanishing for t → ∞.…”

Section: S(t + ε) Hence S(t) Is Differentiable For T = γmentioning

confidence: 99%

The Riemann zeta-function and moment conjectures from Random Matrix Theory

Steuding

2009

Mathematica Slovaca

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ABSTRACT. On the basis of the Random Matrix Theory-model several interesting conjectures for the Riemann zeta-function were made during the recent past, in particular, asymptotic formulae for the 2kth continuous and discrete moments of the zeta-function on the critical line,by Conrey, Keating et al. and Hughes, respectively. These conjectures are known to be true only for a few values of k and, even under assumption of the Riemann hypothesis, estimates of the expected order of magnitude are only proved for a limited range of k. We put the discrete moment for k = 1, 2 in relation with the corresponding continuous moment for the derivative of Hardy's Z-function. This leads to upper bounds for the discrete moments which are off the predicted order by a factor of log T .

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Section: S(t + ε) Hence S(t) Is Differentiable For T = γmentioning

confidence: 99%

The Riemann zeta-function and moment conjectures from Random Matrix Theory

Steuding

2009

Mathematica Slovaca

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“…for n ≥ 55 [29]. We find • n log 2 (n) ≤ n ln(n)+2 since log 2 (n) ≥ ln(n) + 2 for n ≥ 92 ≥ e 2/(log 2 (e)−1) and • n ln(n)−4 ≤ 2n log 2 (n) since ln(n) − 4 ≥ 1 2 log 2 (n) for n ≥ 2 21 ≥ e 8/(2−log 2 (e)) which proves the claim.…”

Section: Preliminariesmentioning

confidence: 99%

Confined Guessing: New Signatures From Standard Assumptions

et al. 2014

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We put forward a new technique to construct very efficient and compact signature schemes. Our technique combines several instances of an only mildly secure signature scheme to obtain a fully secure scheme. Since the mild security notion we require is much easier to achieve than full security, we can combine our strategy with existing techniques to obtain a number of interesting new (stateless and fully secure) signature schemes. Concretely, we get:• A scheme based on the computational Diffie-Hellman (CDH) assumption in pairingfriendly groups. Signatures contain O(1) and verification keys O(log k) group elements, where k is the security parameter. Our scheme is the first fully secure CDH-based scheme with such compact verification keys.• A scheme based on the (non-strong) RSA assumption in which both signatures and verification keys contain O(1) group elements. Our scheme is significantly more efficient than existing RSA-based schemes.• A scheme based on the Short Integer Solutions (SIS) assumption. Signatures contain O(log(k) · m) and verification keys O(n · m) Z p -elements, where p may be polynomial in k, and n, m denote the usual SIS matrix dimensions. Compared to state-of-the-art SIS-based schemes, this gives very small verification keys, at the price of slightly larger signatures. In all cases, the involved constants are small, and the arising schemes provide significant improvements upon state-of-the-art schemes. The only price we pay is a rather large (polynomial) loss in the security reduction. However, this loss can be significantly reduced at the cost of an additive term in signature and verification key size.

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“…The method of proof is essentially due to Backlund [2], with refinements due to Rosser [9] and the author. Assuming that ±T does not coincide with the ordinate of a zero, consider the rectangle R with vertices at a, -iT, a, + iT, 1 -a, + iT, and 1 -a, -iT, where a, > 1.…”

Section: Estimates Of N(t X)mentioning

confidence: 99%

“…Henceforth we shall abbreviate N(T,x) as N(T), and furthermore we use . If x = Xo 's tne principal character, then we appeal to a result of Rosser [9], who proved that (in our notation) (2.17) Mr,xo)-Jiog¿ The left side of (2.19) is increasing in T for T s* 1467, and is positive for T = 1467.…”

Section: ¿77mentioning

confidence: 99%