mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Fekete polynomial is a

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

f_p(t):=\sum_^ \left (\frac\right )t^a\,

where

\left(\frac\right)\,

is the

Legendre symbol In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residu ...

modulo some integer ''p'' > 1. These polynomials were known in nineteenth-century studies of

Dirichlet L-function In mathematics, a Dirichlet ''L''-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By a ...

s, and indeed to

Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...

himself. They have acquired the name of

Michael Fekete Michael (Mihály) Fekete ( he, מיכאל פקטה; 19 July 1886 – 13 May 1957) was a Hungarian-Israeli mathematician. Biography Fekete was born in 1886 in Zenta, Austria-Hungary (today Senta, Serbia). He received his PhD in 1909 from ...

, who observed that the absence of real zeroes ''t'' of the Fekete polynomial with 0 < ''t'' < 1 implies an absence of the same kind for the

L-function In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ris ...

L\left(s,\dfrac\right).\,

This is of considerable potential interest in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, in connection with the hypothetical

Siegel zero Siegel (also Segal or Segel), is a German and Ashkenazi Jewish surname. it can be traced to 11th century Bavaria and was used by people who made wax seals for or sealed official documents (each such male being described as a ''Siegelbeamter''). A ...

near ''s'' = 1. While numerical results for small cases had indicated that there were few such real zeroes, further analysis reveals that this may indeed be a 'small number' effect.

References

Peter Borwein Peter Benjamin Borwein (born St. Andrews, Scotland, May 10, 1953 – 23 August 2020) was a Canadian mathematician and a professor at Simon Fraser University. He is known as a co-author of the paper which presented the Bailey–Borwein–Plo ...

: ''Computational excursions in analysis and number theory.'' Springer, 2002, , Chap.5.

External links

* Brian Conrey, Andrew Granville,

Bjorn Poonen Bjorn Mikhail Poonen (born 27 July 1968 in Boston, Massachusetts) is a mathematician, four-time Putnam Competition winner, and a Distinguished Professor in Science in the Department of Mathematics at the Massachusetts Institute of Technology. Hi ...

and

Kannan Soundararajan Kannan Soundararajan (born December 27, 1973) is an India-born American mathematician and a professor of mathematics at Stanford University. Before moving to Stanford in 2006, he was a faculty member at University of Michigan where he pursued hi ...

,
Zeros of Fekete polynomials
',

arXiv arXiv (pronounced "archive"—the X represents the Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not peer review. It consists of ...

e-print math.NT/9906214, June 16, 1999. Polynomials Zeta and L-functions {{polynomial-stub