# 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 $\pi(x)>\text{li}(x)$. (English) Zbl 1042.11001
Summary: Let $\pi(x)$ denote the number of primes $\le x$ and let $\text{li}(x)$ denote the usual integral logarithm of $x$. We prove that there are at least $10^{153}$ integer values of $x$ in the vicinity of $1.39822\times 10^{316}$ with $\pi(x)>\text{li}(x)$. This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of $\pi(x)-\text{li}(x)$ 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 $1.617\times 10^{9608}$, where $\pi(x)$ appears to exceed $\text{li}(x)$ by more than $.18x^{\frac 12}/\log x$. The plots strongly suggest, although upper bounds derived to date for $\text{li}(x)-\pi(x)$ are not sufficient for a proof, that $\pi(x)$ exceeds $\text{li}(x)$ for at least $10^{311}$ integers in the vicinity of $1.398\times 10^{316}$. If it is possible to improve our bound for $\pi(x)-\text{li}(x)$ by finding a sign change before $10^{316}$, our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of $\text{li}(x)-\pi(x)$ and find that as $x$ departs from the region in the vicinity of $1.62\times 10^{9608}$, the density is $1-2.7\times 10^{-7}=.99999973$, and that it varies from this by no more than $9\times 10^{-8}$ over the next $10^{30000}$ integers. This should be compared to Rubinstein and Sarnak.

##### MSC:
 11-04 Machine computation, programs (number theory) 11A15 Power residues, reciprocity 11M26 Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 11Y11 Primality 11Y35 Analytic computations
Full Text: