Celebratio Mathematica

[1]P. T. Bateman and S. Chowla: “Averages of character sums,” Proc. Am. Math. Soc. 1 : 6 (1950), pp. 781–787. MR 0042445 Zbl 0041.01904 article

Suppose that $ \chi $ is a primitive residue character modulo $ k $, $ k > 1 $, and that for $ y $ non-negative, \[ S(y) = \sum_{0\leq l\leq y}\chi(l) .\] It is important (see, for example, [Landau 1909]) in the analytic theory of numbers to have as much information as possible about the sums $ S(y) $, in particular about their maximum order of magnitude; it is known (cf. [Pólya 1918; Schur 1918; Landau 1918]), for example, that \[ S(y) < k^{1/2}\log k ,\] but unknown whether or not \[ M(\chi) = o(k^{1/2}\log k) \] as $ k $ tends to infinity, where $ M(\chi) $ is the maximum of $ |S(1)|,\dots $, $ |S(k-1)| $. Hua [1942a; 1942b; 1942c] has shown that it is often helpful to consider the averages \[ n^{-1}\sum_{m=0}^n S(m) .\] In this paper we consider some further developments of this idea.

[2]P. T. Bateman, S. Chowla, and P. Erdős: “Remarks on the size of $L (1, χ)$ ,” Publ. Math. Debrecen 1 (1950), pp. 165–182. MR 0037322 Zbl 0036.30702 article

[3]P. T. Bateman and S. Chowla: “The equivalence of two conjectures in the theory of numbers,” J. Indian Math. Soc. (N.S.) 17 (1953), pp. 177–181. MR 0062176 Zbl 0055.27503 article

[4]P. T. Bateman: “Theorems implying the non-vanishing of $\sum χ (m) m^{- 1}$ for real residue-characters,” J. Indian Math. Soc. (N.S.) 23 (1959), pp. 101–115. MR 0124299 Zbl 0228.10021 article

[5]P. T. Bateman: “Lower bounds for $\sum h (m) / m$ for arithmetical functions $h$ similar to real residue-characters,” J. Math. Anal. Appl. 15 : 1 (July 1966), pp. 2–20. Dedicated to H. S. Vandiver on his eighty-third birthday. MR 0199164 Zbl 0144.27803 article

[6]P. T. Bateman, G. B. Purdy, and S. S. Wagstaff, Jr.: “Some numerical results on Fekete polynomials,” Math. Comput. Simul. 29 : 129 (January 1975), pp. 7–23. MR 0480293 Zbl 0301.10036 article

Derrick Henry Lehmer	Related
Samuel Standfield Wagstaff, Jr.	Related
George Purdy	Related

Paul T. Bateman

Character sums