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2014-12-16T06:18:01Z 10000-01-01 Mccurley, Kevin Snow 2014-12-16T06:18:01Z 1981 1981 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. Made available in DSpace on 2014-12-16T06:18:01Z (GMT). No. of bitstreams: 1 8203527.pdf: 3225365 bytes, checksum: a0759f94faae8280328006254f73a91a (MD5) Previous issue date: 1981 Embargo set by: Seth Robbins for item 71364 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 134 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981. http://hdl.handle.net/2142/71198 (UMI)AAI8203527 Mathematics Explicit Estimates for Functions of Primes in Arithmetic Progressions text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.