In the study of the

Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...

, a Lehmer pair is a pair of zeros of the Riemann zeta function that are unusually close to each other. They are named after Derrick Henry Lehmer, who discovered the pair of zeros :

& \tfrac 1 2 + i\,7005.10056\dots \end

(the 6709th and 6710th zeros of the zeta function). More precisely, a Lehmer pair can be defined as having the property that their complex coordinates

\gamma_n

and

\gamma_

obey the inequality :

\frac \ge C\sum_
\left(\frac+\frac\right)

for a constant

C>5/4

. It is an unsolved problem whether there exist infinitely many Lehmer pairs. If so, it would imply that the

De Bruijn–Newman constant The de Bruijn–Newman constant, denoted by \Lambda and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function H(\lambda,z), where \lambda is a real parameter ...

is non-negative, a fact that has been proven unconditionally by Brad Rodgers and Terence Tao.

References

{{reflist, refs= {{citation , last1 = Csordas , first1 = George , last2 = Smith , first2 = Wayne , last3 = Varga , first3 = Richard S. , author3-link = Richard S. Varga , doi = 10.1007/BF01205170 , issue = 1 , journal = Constructive Approximation , mr = 1260363 , pages = 107–129 , title = Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann hypothesis , volume = 10 , year = 1994, s2cid = 122664556 {{citation , last = Lehmer , first = D. H. , authorlink = Derrick Henry Lehmer , doi = 10.1007/BF02401102 , journal = Acta Mathematica , mr = 0086082 , pages = 291–298 , title = On the roots of the Riemann zeta-function , volume = 95 , year = 1956, doi-access = free {{citation, last1=Rodgers , first1=Brad , last2=Tao , first2=Terence, author2-link=Terence Tao , arxiv=1801.05914 , title=The De Bruijn–Newman constant is non-negative , orig-year=2018, bibcode=2018arXiv180105914R , mr = 4089393 , doi = 10.1017/fmp.2020.6, journal = Forum Math. Pi, volume = 8, year = 2020, s2cid=119140820 {{citation, work=What's New, title=Lehmer pairs and GUE, date=January 20, 2018, first=Terence, last=Tao, authorlink=Terence Tao, url=https://terrytao.wordpress.com/2018/01/20/lehmer-pairs-and-gue/ Analytic number theory

See also

References