Having done the whole SRA to fastq conversion a few times myself, I know that the fastq files can be REALLY big. If all you are interested in is the read number, then you can use shell 'pipes' to avoid creating the intermediate fastq by using -Z or --stdout option for fastq-dump, e.g.
$ fastq-dump --stdout *.sra | wc -l
Then divide the output number by 4 as suggested above.
Note: This process combines all split files into one 'stream' so it probably would not work if you split the fastq.
If its paired end then it gives two fastq files (left and right),
if its single end then it doesn't split it and gives single file as output, so --split-3 is harmless for single end and helpful for paired ends
Neverthless, I agree that if user just want to know the number of reads then piping wc -l is good option but if user want to analyze the data the he/she has to save the output for further analysis
Usually you can get a good idea from the number of spots (Illumina sequencing) shown right on the download page. These are the number of distinct flowcell grid locations, so should correspond to reads. In my experience, they are exactly the same.
For the example you gave, it says 1,754,082 spots. I downloaded the data and confirmed that there are in fact that many reads.
So, use spots to give you a (close) ballpark estimate, but for an exact count, you probably want to download the file.
You can check the number of reads from the command line by running a grep pattern match against the fasta style header. Navigate to the directory your data file is in, and type:
grep ">" | wc -l
this command will find all the instances of ">", the greater-than symbol, within the file you designated, and then pipe that output to a line count. Output on your screen will be the number of reads in the file, because each read is going to begin with a ">" symbol.
Then you see the first 10 reads and you see how many times you would see 10 reads (in this case: 175409). So this number by 10 (as ten reads per page) -> 1754090 and you have roughly your number of reads. The most error is that you are off by 9 :).
I did not check it , but I don't see why it would not work. Plus it is certainly faster then downloading it first. Btw. I always use the sratoolkit method called: fastq-dump
Its easy and fast and results in Fastq files or even paired files for paired end data.