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Title:Explicit Estimates for Functions of Primes in Arithmetic Progressions
Author(s):Mccurley, Kevin Snow
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
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
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.
Issue Date:1981
Description:134 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981.
Other Identifier(s):(UMI)AAI8203527
Date Available in IDEALS:2014-12-16
Date Deposited:1981

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