zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A new bound for the smallest x with π(x)>li(x). (English) Zbl 1042.11001
Summary: Let π(x) denote the number of primes x and let li(x) denote the usual integral logarithm of x. We prove that there are at least 10 153 integer values of x in the vicinity of 1·39822×10 316 with π(x)>li(x). This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of π(x)-li(x) in four different regions, including the regions discovered by Lehman, te Riele, and the authors of this paper, and a more distant region in the vicinity of 1·617×10 9608 , where π(x) appears to exceed li(x) by more than ·18x 1 2 /logx. The plots strongly suggest, although upper bounds derived to date for li(x)-π(x) are not sufficient for a proof, that π(x) exceeds li(x) for at least 10 311 integers in the vicinity of 1·398×10 316 . If it is possible to improve our bound for π(x)-li(x) by finding a sign change before 10 316 , our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of li(x)-π(x) and find that as x departs from the region in the vicinity of 1·62×10 9608 , the density is 1-2·7×10 -7 =·99999973, and that it varies from this by no more than 9×10 -8 over the next 10 30000 integers. This should be compared to Rubinstein and Sarnak.

MSC:
11-04Machine computation, programs (number theory)
11A15Power residues, reciprocity
11M26Nonreal zeros of ζ(s) and L(s,χ); Riemann and other hypotheses
11Y11Primality
11Y35Analytic computations