Inequalities between sums over prime numbers in progressions
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Inequalities between sums over prime numbers in progressions Emre Alkan ∗ * Correspondence:
[email protected] Department of Mathematics, Koç University, Sarıyer, 34450 Istanbul, Turkey
Abstract We investigate a new type of tendency between two progressions of prime numbers which is in support of the claim that prime numbers that are congruent to 3 modulo 4 are favored over prime numbers that are congruent to 1 modulo 4. In particular, we show that the Riemann hypothesis for the corresponding L-function is equivalent to the occurrence of such a tendency. A generalization to teams of progressions of prime numbers is given, where the teams are formed by grouping according to the values of a quadratic character. In this way, it is shown that there is a tendency favoring prime numbers belonging to progressions arising from the quadratic nonresidues modulo a prime number congruent to 3 or 5 modulo 8. The scope of the tendency is extended conditionally, either by assuming the Riemann hypothesis for certain Dirichlet L-functions or by the presence of Siegel zeros. Our approach requires numerical verifications over certain ranges of the parameters, and in this respect, we freely benefit from computer software to carry out such tasks. Lastly, the divisor function is seen to be favorable over its average value along semigroups by comparing partial sums of the associated Dirichlet series. Keywords: Inequalities, Sums over progressions of prime numbers, Teams of progressions, Quadratic character, Riemann hypothesis for Dirichlet L-functions, Siegel zero Mathematics Subject Classification: 11Y35, 11M06, 11N13
1 Introduction The goal of this paper is to provide specific inequalities occurring among sums over progressions of prime numbers and powers of prime numbers. Investigation of such phenomena occupies a central place in understanding the finer distribution and irregularities of prime numbers. Historically, the origin of all such efforts rests on a curious assertion made by Chebyshev [6]. Precisely, he observed that there are more prime numbers congruent to 3 modulo 4 than to 1 modulo 4 in the sense that p−1 lim (−1) 2 e−px = −∞, (1.1) x→0+
p>2
where the summation in (1.1) is over all odd prime numbers. If true, (1.1) would clearly support the view that there is a preponderance of prime numbers that are congruent to 3 modulo 4. Interesting variants of (1.1) were first developed by Fujii [15] and further
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