University of Illinois Urbana-Champaign

Explicit Estimates for Functions of Primes in Arithmetic Progressions

Mccurley, Kevin Snow

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Description

Title
Explicit Estimates for Functions of Primes in Arithmetic Progressions
Author(s)
Mccurley, Kevin Snow
Issue Date
1981
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
  • This thesis is concerned with essentially three topics: Explicit zero-free regions for Dirichlet L-functions, numerical estimates for the error term in the prime number theorem for arithmetic progressions, and Waring's problem for cubes.
  • In chapter 1 the following result is proved: Among the (phi)(k) characters (chi) modulo k there is at most one character for which the Dirichlet L-function L(s,(chi)) has a zero (rho) = (beta) + i(gamma) with (beta) > 1 - 1/(Rlogq), where R = 9.645908801, and q = max{k, k(VBAR)(gamma)(VBAR), 30}. If such a zero exists it is a real zero of an L-function formed with a real non-principal character. Several methods of proof are discussed for showing that a given modulus k does not admit an exceptional zero.
  • In chapter 2 explicit numerical values are given for constants C(,1) and C(,2) with the property that
  • (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)
  • where (chi) is a primitive character modulo k, and N(T,(chi)) counts the number of zeros of L(s,(chi)) with 0 < (beta) < 1 and (VBAR)(gamma)(VBAR) (LESSTHEQ) T.
  • The object of chapter 3 is the estimation of the Chebyshev functions (theta)(x;k,l) and (psi)(x;k,l). For various values of (epsilon), tables and c and b are given for which it can be asserted that
  • provided that (k,l) = 1, x (GREATERTHEQ) exp(clog('2)k), k (GREATERTHEQ) 10('b), and the modulus k does not admit an exceptional zero. The method used in the proof is similar to that used by Rosser and Schoenfeld in the case k = 1, where an integral average of the function (psi)(x;k,l) is expressed in an explicit formula involving the zeros of Dirichlet L-functions. The explicit formula can then be estimated directly with the use of results from chapters 1 and 2.
  • Chapter 4 considers the case k = 3 in more detail. In this case the results of chapter 3 can be sharpened by making use of extensive numerical information concerning the zeros of the two Dirichlet L-functions modulo 3.
  • Waring's problem for cubes is the topic of chapter 5. It is proved that every integer exceeding exp(1.1 x 10('6)) is a sum of seven non-negative integral cubes. Previous proofs of the seven cube theorem were ineffective due to the use of the Siegel-Walfisz theorem. Numerical evidence is presented for the conjecture that every integer exceeding 1290740 is a sum of five non-negative integral cubes.
Graduation Semester
1981
Type of Resource
text
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http://hdl.handle.net/2142/71198

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