﻿ number of fractions (1/4)
```/*CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C                                                                             C
C  COMPUTE DIFFERENCES IN NUMBER OF FRACTIONS (0 to 1/4, 1/4 to 1/2)	      C
C  03/22/15 (DKC)							      C
C                                                                             C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC*/
#include <stdio.h>
#include <math.h>
#include <stdlib.h>
int liouvile(unsigned int a);
double numdiv(unsigned int a, unsigned int flag);
double dirchar(unsigned int a, unsigned int p);
mertens9(unsigned int, unsigned int *count, unsigned int *S, unsigned int flag);
//
unsigned int prime=97;	// for Legendre symbol
unsigned int limit=0;	// partial sum flag, normally set to 0
unsigned int norm=0;	// normalize, normally set to 0
unsigned int hflag=0;	// histogram flag
unsigned int delta=1;	// increment by 1/6 if flag=0 or 1/4 if flag=1
unsigned int flag=0;   // x=3 flag
unsigned int MAXN=1000; // up to 50000
unsigned int BEGINN=2;
unsigned int out=37;
// 10, sgn(n(x)-m(x))+1.781072418*sum(sgn(n(x/i)-m(x/i))*i*log(log(i)))
// 11, sum of m(x/i)
// 12, sum of n(x/i)
// 13, sum of n(x/i)-m(x/i)
// 14, sum of (n(x/i)-m(x/i))*i
// 15, |n(x)-m(x)|+1.781072418*sum(|n(x/i)-m(x/i)|*i*log(log(i)))
// 16, sum of (n(x/i)-m(x/i))*(h+exp(h)*log(h))
// 17, differences in number of fractions before, after 1/4
// 18, number of fractions before 1/4
// 19, number of fractions after 1/4
// 20, n(x)-m(x)+1.781072418*sum((n(x/i)-m(x/i))*i*log(log(i)))
// 21, sum of (n(x/i)-m(x/i))*2*d(i) (see #24)
// 22, sum of (n(x/i)-m(x/i))*sigma(i)
// 23, sum of (n(x/i)-m(x/i)^2*L(i)
// 24, sum of (n(x/i)-m(x/i))*d(i) (see #21)
// 25, sum of n(x/i)-m(x/i) when i is a perfect square
// 26, sum of (n(x/i)-m(x/i))^2
// 27, sum of (n(x/i)-m(x/i))*log(i)
// 28, sum of sgn(n(x/i)-m(x/i))
// 29, sum of (n(x/i)-m(x/i)*log(i)*d(i)
// 30, sum of (n(x/i)-m(x/i)^2*L(i)^2
// 31, sum of sgn(n(x/i)-m(x/i))*i
// 32, sum of |sgn(n(x/i)-m(x/i))|
// 33, sum of (n(x/i)-m(x/i))*L(i)
// 34, sum of (n(x/i)-m(x/i)*M(i) (or n(x/i)M(i), m(x/i)M(i))
// 35, sum of (n(x/i)-m(x/i))*L(i)*lambda(i)
// 36, sum of (n(x/i)-m(x/i))*sigma2(i)
// 37, sum of (n(x/i)-m(x/i))*char(i)
//
void main() {
int t,tsum,sum;
unsigned int N,i,temp,count[2],iters;
double sum6,sum7,sum8,sum9,sum10,sum11,sum12,sum13,sum14,sum15,sum16,sum17,sum18;
double sum1,sum2,sum3,sum4,sum5,h,temp1;
unsigned int *S;
short M[60001];
unsigned int histo[201];
double mean,std,msum,msum2,dsum,d2sum;
FILE *Outfp;
Outfp = fopen("out1bf.dat","w");
S=(unsigned int*) malloc(40000000);
if (S==NULL)
return;
if ((out==17)&&(hflag!=0)&&(BEGINN!=2)) {
printf("parameter error \n");
goto zskip;
}
//
// compute Mertens function
//
if (out==34) {
M[0]=1;
for (N=2; N<=(MAXN+1); N++) {
sum=0;
for (i=2; i<=(N/3); i++)
sum=sum+M[N/i-1];
sum=sum+(N+1)/2;
t=1-sum;
M[N-1]=(short)t;
}
}
for (i=0; i<201; i++)
histo[i]=0;
iters=0;
dsum=0.0;
d2sum=0.0;
for (N=BEGINN; N<=MAXN; N++) {
printf(" %d \n",N);
if ((out==17)||(out==18)||(out==19))
mertens9(N, count, S, flag);
if (out==17) {
t=(int)count[0]-(int)count[1];
if (hflag==2) {
dsum=dsum+(double)t;
d2sum=d2sum+(double)t*(double)t;
iters=iters+1;
std=d2sum-dsum*dsum/(double)iters;
if (iters!=1)
std=sqrt(std/(double)(iters-1));
else
std=0.0;
fprintf(Outfp," %e \n",std);
}
if (hflag==0)
fprintf(Outfp," %d\n",t);
if (hflag==1) {
if ((t<-100)||(t>100)) {
printf("difference out of range \n");
goto zskip;
}
histo[t+100]=histo[t+100]+1;
}
continue;
}
if (out==18) {
fprintf(Outfp," %d\n",count[0]);
continue;
}
if (out==19) {
fprintf(Outfp," %d\n",count[1]);
continue;
}
if ((out==11)||(out==12))  {
sum6=0.0;
sum7=0.0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
sum6=sum6+(double)count[0];
sum7=sum7+(double)count[1];
}
if (out==11)
fprintf(Outfp," %e\n",sum6);
if (out==12)
fprintf(Outfp," %e\n",sum7);
continue;
}
if ((out==13)||(out==14)) {
sum8=0.0;
sum9=0.0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
//	 printf(" %d %d \n",count[0],count[1]);
if (delta==0)
sum8=sum8+(double)count[1]-(double)count[0];	   // h(x)
else {
if (flag==0)
sum8=sum8+(double)count[0]-(double)count[1]+1.0/6.0;   // h(x)
else
sum8=sum8+(double)count[0]-(double)count[1]+1.0/4.0;   // h(x)
}
sum9=sum9+(double)i*((double)count[1]-(double)count[0]);
}
if (out==13) {
if (norm==0)
fprintf(Outfp," %e\n",sum8);
else
fprintf(Outfp," %e\n",sum8/(double)N);
}
if (out==14)
fprintf(Outfp," %e\n",sum9);
continue;
}
if (out==10) {
sum2=0.0;
for (i=2; i<=N; i++) {
mertens9(N/i, count, S, flag);
if (delta==0)
temp1=(double)count[1]-(double)count[0];
else
temp1=(double)count[0]-(double)count[1]+1.0/6.0;
if (temp1<0.0)
sum2=sum2-(double)i*log(log((double)i));
if (temp1>0.0)
sum2=sum2+(double)i*log(log((double)i));
}
sum2=1.781072418*sum2;
mertens9(N, count, S, flag);
if (delta==0)
temp1=(double)count[1]-(double)count[0];
else
temp1=(double)count[0]-(double)count[1]+1.0/6.0;
if (temp1<0.0)
sum2=sum2-1;
if (temp1>0.0)
sum2=sum2+1;
fprintf(Outfp," %e\n",sum2);
continue;
}
if (out==15) {
sum2=0.0;
for (i=2; i<=N; i++) {
mertens9(N/i, count, S, flag);
if (delta==0)
temp1=(double)count[1]-(double)count[0];
else
temp1=(double)count[0]-(double)count[1]+1.0/6.0;
if (temp1<0.0)
temp1=-temp1;
//	 printf(" %e %e \n",temp1,(double)i*log(log((double)i)));
sum2=sum2+temp1*(double)i*log(log((double)i));
}
mertens9(N, count, S, flag);
if (delta==0)
temp1=(double)count[1]-(double)count[0];
else
temp1=(double)count[0]-(double)count[1]+1.0/6.0;
if (temp1<0.0)
temp1=-temp1;
sum2=1.781072418*sum2+temp1;
fprintf(Outfp," %e\n",sum2);
continue;
}
if (out==16) {
sum1=0.0;
h=0.0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
h=h+1.0/(double)i;
if (delta==0)
sum1=sum1+((double)count[1]-(double)count[0])*(h+exp(h)*log(h));
else {
if (flag==0)
sum1=sum1+((double)count[0]-(double)count[1]+1.0/6.0)*(h+exp(h)*log(h));
else
sum1=sum1+((double)count[0]-(double)count[1]+1.0/4.0)*(h+exp(h)*log(h));
}
}
fprintf(Outfp," %e\n",sum1);
continue;
}
if (out==20) {
sum2=0.0;
for (i=2; i<=N; i++) {
mertens9(N/i, count, S, flag);
//	 printf(" %e %e \n",(double)count[1]-(double)count[0],(double)i*log(log((double)i)));
if (delta==0)
sum2=sum2+((double)count[1]-(double)count[0])*(double)i*log(log((double)i));
else
sum2=sum2+((double)count[0]-(double)count[1]+1.0/6.0)*(double)i*log(log((double)i));
}
mertens9(N, count, S, flag);
if (delta==0)
sum2=1.781072418*sum2+(double)count[1]-(double)count[0];
else
sum2=1.781072418*sum2+(double)count[0]-(double)count[1]+1.0/6.0;
fprintf(Outfp," %e\n",sum2);
continue;
}
if (out==21) {
sum3=0.0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
if (delta==0)
sum3=sum3+((double)count[1]-(double)count[0])*2.0*numdiv(i,1);
else {
if (flag==0)
sum3=sum3+((double)count[0]-(double)count[1]+1.0/6.0)*2.0*numdiv(i,1);
else
sum3=sum3+((double)count[0]-(double)count[1]+1.0/4.0)*2.0*numdiv(i,1);
}
}
fprintf(Outfp," %e\n",sum3);
continue;
}
if (out==22) {
sum4=0.0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
if (delta==0)
sum4=sum4+((double)count[1]-(double)count[0])*numdiv(i,4);
else
sum4=sum4+((double)count[0]-(double)count[1]+1.0/6.0)*numdiv(i,4);
}
fprintf(Outfp," %e\n",sum4);
continue;
}
if (out==23) {
sum5=0.0;
tsum=0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
t=liouvile(i);
tsum=tsum+t;
sum5=sum5+((double)count[1]-(double)count[0])*((double)count[1]-(double)count[0])*(double)tsum;
}
if (norm!=0)
sum5=sum5/(double)N;
fprintf(Outfp," %e\n",sum5);
continue;
}
if (out==24) {
sum14=0.0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
if (delta==0)
sum14=sum14+((double)count[1]-(double)count[0])*numdiv(i,1);
else {
if (flag==0)
sum14=sum14+((double)count[0]-(double)count[1]+1.0/6.0)*numdiv(i,1);
else
sum14=sum14+((double)count[0]-(double)count[1]+1.0/4.0)*numdiv(i,1);
}
}
fprintf(Outfp," %e\n",sum14);
continue;
}
if (out==25) {
sum13=0.0;
for (i=1; i<=N; i++) {
temp=(unsigned int)(sqrt((double)i)+0.001);
if ((temp*temp)==i) {
mertens9(N/i, count, S, flag);
if (delta==0)
sum13=sum13+(double)count[1]-(double)count[0];
else {
if (flag==0)
sum13=sum13+(double)count[0]-(double)count[1]+1.0/6.0;
else
sum13=sum13+(double)count[0]-(double)count[1]+1.0/4.0;
}
}
}
fprintf(Outfp," %e\n",sum13);
continue;
}
if (out==26) {
sum10=0.0;
for (i=1; i<=N; i++) {
if ((N/i)<=limit)
break;
mertens9(N/i, count, S, flag);
if (delta==0)
sum10=sum10+((double)count[1]-(double)count[0])*((double)count[1]-(double)count[0]);
else
sum10=sum10+((double)count[0]-(double)count[1]+1.0/6.0)*((double)count[0]-(double)count[1]+1.0/6.0);
}
fprintf(Outfp," %e\n",sum10);
continue;
}
if (out==27) {
sum11=0.0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
if (delta==0)
sum11=sum11+((double)count[1]-(double)count[0])*log((double)i);
else {
if (flag==0)
sum11=sum11+((double)count[0]-(double)count[1]+1.0/6.0)*log((double)i);
else
sum11=sum11+((double)count[0]-(double)count[1]+1.0/4.0)*log((double)i);
}
}
fprintf(Outfp," %e\n",sum11);
continue;
}
if (out==28) {
sum12=0.0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
if (count[0]>count[1])       // sum of sgn(n(x/i)-m(x/i))
sum12=sum12-1.0;
if (count[0]<count[1])
sum12=sum12+1.0;
}
if (norm==0)
fprintf(Outfp," %e\n",sum12);
else
fprintf(Outfp," %e\n",sum12/(double)N);
continue;
}
if (out==29) {
sum18=0.0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
if (delta==0)
sum18=sum18+((double)count[1]-(double)count[0])*log((double)i)*numdiv(i,1);
else {
if (flag==0)
sum18=sum18+((double)count[0]-(double)count[1]+1.0/6.0)*log((double)i)*numdiv(i,1);
else
sum18=sum18+((double)count[0]-(double)count[1]+1.0/4.0)*log((double)i)*numdiv(i,1);
}
}
fprintf(Outfp," %e\n",sum18);
continue;
}
if (out==30) {
sum17=0.0;
tsum=0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
t=liouvile(i);
tsum=tsum+t;
if (delta==0)
sum17=sum17+((double)count[1]-(double)count[0])*((double)count[1]-(double)count[0])*(double)tsum*(double)tsum;
else {
if (flag==0)
sum17=sum17+((double)count[0]-(double)count[1]+1.0/6.0)*((double)count[1]-(double)count[0])*(double)tsum*(double)tsum;
else
sum17=sum17+((double)count[0]-(double)count[1]+1.0/4.0)*((double)count[1]-(double)count[0])*(double)tsum*(double)tsum;
}
}
if (norm!=0)
sum17=sum17/(double)N;
fprintf(Outfp," %e\n",sum17);
continue;
}
if (out==31) {
sum15=0.0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
if (count[0]>count[1])        // sum of sgn(n(x/i)-m(x/i))*i
sum15=sum15-(double)i;
if (count[0]<count[1])
sum15=sum15+(double)i;
}
fprintf(Outfp," %e\n",sum15);
continue;
}
if (out==32) {
sum16=0.0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
if (count[0]!=count[1])      // sum of |sgn(n(x/i)-m(x/i))|
sum16=sum16+1.0;
}
if (norm==0)
fprintf(Outfp," %e\n",sum16);
else
fprintf(Outfp," %e\n",sum16/(double)N);
}
if (out==33) {
sum17=0.0;
tsum=0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
t=liouvile(i);
tsum=tsum+t;
if (delta==0)
sum17=sum17+((double)count[1]-(double)count[0])*(double)tsum;
else {
if (flag==0)
sum17=sum17+((double)count[0]-(double)count[1]+1.0/6.0)*(double)tsum;
else
sum17=sum17+((double)count[0]-(double)count[1]+1.0/4.0)*(double)tsum;
}
}
if (norm!=0)
sum17=sum17/(double)N;
fprintf(Outfp," %e\n",sum17);
continue;
}
if (out==34) {
sum18=0.0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
if (delta==0)
sum18=sum18+((double)count[1]-(double)count[0])*(double)M[i-1];
else {
if (delta==1)
sum18=sum18+((double)count[0]-(double)count[1]+1.0/6.0)*(double)M[i-1];
else {
if (delta==2)
sum18=sum18+(double)count[0]*(double)M[i-1];
else {
if (delta==3)
sum18=sum18+(double)count[1]*(double)M[i-1];
else {
if (delta==4) {
sum18=sum18+((double)count[1]-(double)count[0])*(double)M[N/i-1];
//			printf(" %d %d \n",(int)count[1]-(int)count[0],M[N/i-1]);
}
else
sum18=sum18+((double)count[0]-(double)count[1]+1.0/6.0)*(double)M[N/i-1];
}
}
}
}
}
fprintf(Outfp," %e\n",sum18);
continue;
}
if (out==35) {
sum17=0.0;
tsum=0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
t=liouvile(i);
tsum=tsum+t;
if (delta==0)
sum17=sum17+((double)count[1]-(double)count[0])*((double)count[1]-(double)count[0])*(double)tsum*(double)t;
else {
if (flag==0)
sum17=sum17+((double)count[0]-(double)count[1]+1.0/6.0)*((double)count[1]-(double)count[0])*(double)tsum*(double)t;
else
sum17=sum17+((double)count[0]-(double)count[1]+1.0/4.0)*((double)count[1]-(double)count[0])*(double)tsum*(double)t;
}
}
fprintf(Outfp," %e\n",sum17);
continue;
}
if (out==36) {
sum4=0.0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
if (delta==0)
sum4=sum4+((double)count[1]-(double)count[0])*numdiv(i,5);
else
sum4=sum4+((double)count[0]-(double)count[1]+1.0/6.0)*numdiv(i,5);
}
fprintf(Outfp," %e\n",sum4);
continue;
}
if (out==37) {
sum4=0.0;
for (i=1; i<=N; i++) {
mertens9(N/i, count, S, flag);
if (delta==0)
sum4=sum4+((double)count[1]-(double)count[0])*dirchar(i,prime);
else
sum4=sum4+((double)count[0]-(double)count[1]+1.0/6.0)*dirchar(i,prime);
}
fprintf(Outfp," %e\n",sum4);
continue;
}
}
if ((out==17)&&(hflag==1)) {
mean=0.0;
msum2=0.0;
for (i=0; i<201; i++) {
fprintf(Outfp," %d\n",histo[i]);
mean=mean+((double)i-100.0)*histo[i];
//    printf(" %d %d %d %e\n",(int)(i-100),histo[i],(int)(i-100)*histo[i],mean);
msum2=msum2+((double)i-100.0)*histo[i]*((double)i-100.0);
}
msum=mean;
mean=mean/(double)(MAXN-1);
std=(double)(MAXN-1)*msum2-msum*msum;
std=sqrt(std/((double)(MAXN-1)*(double)(MAXN-2)));
printf("mean=%e, std=%e \n",mean,std);
}
zskip:
free(S);
fclose(Outfp);
return;
}
```