# A lower bound for biases amongst products of two primes

- Patrick Hough
^{1}Email author

**Received: **18 October 2016

**Accepted: **1 June 2017

**Published: **1 September 2017

## Abstract

We establish a conjectured result of Dummit, Granville and Kisilevsky for the maximum bias in P2 races, which compare the number of P\(2's\le x\) whose prime factors, evaluated at a given quadratic character, take specific values.

## Keywords

## 1 Background

The obvious generalisations of this to other moduli have been studied at great length and the most recent work on this can be found in [1]. In these works, one is concerned with the probability that, given“There is a notable difference in the splitting of the prime numbers between the two forms 4n + 3, 4n + 1: the first form contains a lot more than the second.”

*x*, there are more primes \(p\le x\) that lie in one particular residue class than lie in any other with respect to the modulus under consideration. A comprehensive exposition of the main results on these primes races can be found in [2]. Whilst these results present some of the most ‘predictable’ such prime races, it has not yet been possible to find a race in which one ‘team’ leads indefinitely for sufficiently large

*x*. It turns out that far more concrete results can be given when we consider semi-primes, integers that are a product of exactly two primes.

In 2015, Dummit et al. [3] showed that significantly more than a quarter of all odd integers of the form *pq* up to *x*, with *p* and *q* both prime, satisfy \(p\equiv q\equiv 3~\mathrm{mod}\,4\). More generally they proved the following Theorem.

### Theorem 1.1

*d*. For \(\eta =-1\) or 1 we have

We focus on just how large this bias can get if we restrict the conductor of our quadratic character by \(d\le x\). The following prediction was made in relation to this problem.

### Conjecture 1.1

It is important to note that in proving such a claim, one must prove a uniform version of Theorem 1.1 since the proof there assumes that *x* is allowed to be very large compared to *d*. Whilst this is not a straightforward task, once achieved, all that remains is to show that \(\mathcal {L}_{\chi _d}\) can be found suitably large for \(d\le x\). It turns out that this final step is easily accomplished using a slight adaptation of a result from Granville and Soundararajan’s work on extremal values of \(L(1,\chi )\) [4].

In this paper, we prove Conjecture 1.1 and indeed give a slightly stronger result in the form of the following Theorem.

### Theorem 1.2

*x*there exist at least \(D(x)^{1-\epsilon }\) integers \(d\le D(x)\) for which

In 1928, Littlewood gave [5] an upper bound on the value of \(L(1,\chi )\) as follows.

### Theorem 1.3

*q*), one has

This upper bound, when combined with a lower bound derived from [4], allows us to claim that, under GRH, we have found the maximum bias amongst these races.

### Theorem 1.4

Using a second derived result from [4], the initial legwork of proving a uniform version of Theorem 1.1 allows us to show just how *small* this bias can get.

### Theorem 1.5

*x*there exist at least \(D(x)^{1-\epsilon }\) integers \(d\le D(x)\) for which

*d*that give rise to particularly biased prime races of this kind. In order to find these large biases, is it crucial to study the quantity \(\mathcal {L}_{\chi _d}\). Indeed, if \(\chi _d(p)=-1\) for a large proportion of the small primes

*p*or alternatively \(\chi _d(p)=+1\), then we expect to have a prime race with a correspondingly large bias. Such characters are closely linked to prime generating polynomials and it was the work of Jacobson Jr. and Williams that gave us the best results in this area [6]. We are thus able to give, what we believe to be, the lowest known value of \(\mathcal {L}_{\chi _d}\) where the conductor

*d*of our character is

## 2 Proof of Theorem 1.2

*L*-function \(L(s,\chi )\) associated with the real Dirichlet character \(\chi \) of modulus

*q*. A zero of \(L(s,\chi )\) is called a Siegel zero if, for some suitable positive constant

*c*, \(L(s,\chi )\) has a real zero \(\beta \) such that

*x*, there exist many \(d\le D(x)\) corresponding to real Dirichlet characters, such that

*j*-fold iterated logarithm by \(\log _j\) where appropriate.

### Theorem 2.1

*D*be large and \(\tau =\log _2(D)-2\log _3(D)\). Then

### Proof

*D*, at least \(D^{1-\epsilon }\) fundamental discriminants \(d\le D\) for which \(L(1,\chi _d)\ge e^{\gamma }(\log _2(D)-2\log _3(D))\). This follows from the well established fact that there are \(\frac{6}{\pi ^2}x+O(x^{\frac{1}{2}+\epsilon })\) fundamental discriminants

*d*with \(|d|\le x\). Note that we may write

*D*, there are at least \(D^{1-\epsilon }\) fundamental discriminants \(d\le D\) for which \(\mathcal {L}_{\chi _d} \ge \log _3(D) +O(1)\).

In light of a result by Page that ensures there is never more than one conductor \(d\le x\) for which \(L(s,\chi _d)\) has a Siegel zero, Theorem 2.1 combines with (2.2) to give (2.1) as required.

### Corollary 2.2

*D*, there are at least \(D^{1-\epsilon }\) fundamental discriminants \(d\le D\) for which

### 2.1 Proof of uniform version of (1.1) with \(L(s,\chi )\) having no Siegel zero

*ab*and

*ba*in the sum separately, which accounts for the factor of 2 used here in contrast with the the usual formulation of this result. Let us consider the first sum here.

### Theorem 2.3

*c*is a suitable positive constant, there is at most one real primitive character \(\chi \) to a modulus \(d\le x\) for which \(L(s,\chi )\) has a real zero \(\beta \) satisfying

*L*-function has no zero lying in the region defined by (2.6) we have, after some computation,

*N*. Using (2.7), we may bound the second term of (2.4) as follows:

*a*. These techniques, by symmetry, yield the same result for the second sum of (2.3). Summarising, we have shown that

*a*and

*b*yields the same bound for the second term and the third term is dealt with simply by taking the product of the two sums after applying (2.7).

*x*and \(d\le \exp [C(\log {x})^{1/2}]\) where \(L(s,\chi _d)\) has no ‘exceptional’ zero,

*x*, \(b<1/2\) and

*c*.

### Proof

## 3 Proof of theorem 1.4

*x*, on the range \(d\le x\) if we assume the Generalised Riemann Hypothesis (GRH). The crucial advantage in making this assumption is that it allows us to write

*x*, there exists \(d\le x\) such that

## 4 Proof of Theorem 1.5

*x*and \(d\le \exp [C(\log {x})^{1/2}]\), where \(L(s,\chi _d)\) has no Siegel zero,

### Theorem 4.1

*D*, there are at least \(D^{1-\epsilon }\) fundamental discriminants \(d\le D\) for which

### Proof

*D*,

*D*, there are at least \(D^{1-\epsilon }\) fundamental discriminants \(d\le D\) such that

### Proof

## 5 Computational aspects

*x*, the value of this quadratic is divisible by a prime

*p*if and only if

*f*is related to a quantity \(C(\Delta )\). They also suggest that the larger the value of \(C(\Delta )\), the higher the asymptotic density of primes for

*any*polynomial of discriminant \(\Delta \). We thus restrict to polynomials of the form \(f(x)=x^2+x+A\). If we also denote by \(P_A(n)\), the number of primes produced by \(f_A(x)\) for \(0\le x\le n\) then Conjecture

*F*can be written as follows.

### Conjecture 5.1

*A*is given by

*d*given by \(\chi (n)=(d/n)\) where \(d=1-4A\) to yield a low value of \(\mathcal {L}_{\chi _d}\). Firstly, we note that \(d\equiv 1 ~\mathrm{mod}\,4 \) is square-free and thus (

*d*/

*n*) defines a real primitive character. A simple piece of SageMath yields

*d*is the conductor of the character \(\chi _d\), we have that, on taking \(d=1-4A\) where

*A*is given by (5.1) and \(\eta =-1\),

*d*.

*d*which were of interest when searching for extreme biases. The modulus 5 is included for comparison and was the one used to introduce the search for greater biases in [3]. The modulus \(-163\) is the discriminant of Euler’s polynomial and 5,417,453 yielded the highest bias found when performing a naive search for several hours over all integers. When searching for small values of \(L(1,\chi )\), Lehmer et al. showed in [10] that the modulus 49,107,823,133 gave rise to the (then) lowest known value of \(L(1,\chi )\approx 0.169(5)\). The final row represents the prime race corresponding to the character that we have extracted from Jacobson and Williams work.

The values of \(r_d(x)\) over small *x* for moduli of interest that arose during our investigation

Modulus, | \(r_d(10^3)\) | \(r_d(10^4)\) | \(r_d(10^5)\) | \(r_d(10^6)\) |
---|---|---|---|---|

5 | 1.860 | 1, 626 | 1.523 | 1.457 |

\({-}163\) | 2.470 | 2.020 | 1.793 | 1.687 |

5,417,453 | 3.134 | 2.383 | 2.026 | 1.860 |

49,107,823,133 | 3.528 | 2.555 | 2.105 | 1.905 |

\(1-4A\) | 3.847 | 2.974 | 2.394 | 2.067 |

## Declarations

### Open Access

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

### Acknowledgements

This paper comes as a result of my M.Sc. dissertation at University College London. I give greatest thanks to my supervisor Prof. Andrew Granville. I also thank Ronnie George and Tarquin Grossman for their general exuberance.

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## Authors’ Affiliations

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