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Title:  Explicit Estimates for Functions of Primes in Arithmetic Progressions 
Author(s):  Mccurley, Kevin Snow 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  This thesis is concerned with essentially three topics: Explicit zerofree regions for Dirichlet Lfunctions, 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 Lfunction 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 Lfunction formed with a real nonprincipal 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 (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) 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 Lfunctions. 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 Lfunctions 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 nonnegative integral cubes. Previous proofs of the seven cube theorem were ineffective due to the use of the SiegelWalfisz theorem. Numerical evidence is presented for the conjecture that every integer exceeding 1290740 is a sum of five nonnegative integral cubes. 
Issue Date:  1981 
Type:  Text 
Description:  134 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1981. 
URI:  http://hdl.handle.net/2142/71198 
Other Identifier(s):  (UMI)AAI8203527 
Date Available in IDEALS:  20141216 
Date Deposited:  1981 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois