Anuj Bidve, 23, was shot at point-blank range as he walked with friends near their hotel in Salford, Greater Manchester, in the early hours of Boxing Day.
Police investigating the shooting have arrested a 16-year-old boy, two boys aged 17 and a 19-year-old man on suspicion of murder.
Norman Owen, leader of Salford's Liberal Democrats, said: "I am really concerned. If you look at the history over the last three or four years we have had some severe shootings in this city.
"We are being dragged down by this issue and the police have got to get into gun crime," said Mr Owen.
Asked if he thought this shooting had brought shame on the city, he responded: "It does. This city is trying to drive itself forward in many ways."
But he said Salford needed to "rid itself" of the gun "culture" and work with police, parents and schools to address it.
Police said a 16-year-old boy and 17-year-old boy were arrested yesterday on suspicion of murder.
A third teenager, also 17, handed himself into a police station at about 10.30pm last night and was arrested on suspicion of murder.
In the early hours of this morning a fourth man, 19, was also arrested on suspicion of murder.
All four males are being questioned and remain in police custody.
Chief Superintendent Kevin Mulligan, divisional commander for Salford, said today: "This is a fast moving investigation and we now have four people in custody.
"Those people will be questioned throughout the day by detectives and it is important we do not speculate on the possible motives for this murder.
"However, I would still appeal to anyone who has information about the murder to come forward.
"While we have made arrests, this is just the beginning of our investigation and we need people to come forward with anything they know."
He added: "I want to take this opportunity to praise the response of the Ordsall community who are clearly outraged by the murder of this young man and have come forward with information that is greatly assisting the investigation.
"We absolutely need this support to continue."
A large police presence, including armed officers, are patrolling in the Ordsall area and that will continue over the next few days, police said.
Tributes have been left on Facebook, including one page set up by a fellow Lancaster student which plans a "peace march" in the young man's memory.
According to the page, set up by Sonakshi Saran, Mr Bidve "was killed for not answering a simple question - 'What's the time?"'
Greater Manchester Police have not disclosed what was said during the short conversation between the killer and his victim.
Speaking from Mr Bidve's home town of Pune in Maharashtra yesterday, Rakesh Sonawane, 30, who is married to the student's sister Nehal, said the devastated family had "lost faith in everything".
Mr Bidve arrived in the UK in September after completing an electronics degree at Pune University.
"Anuj had been very happy. It was his dream to go to the UK but unfortunately his dream could not last longer than three months," Mr Sonawane said.
He added that his brother-in-law had planned to return to India "to serve his nation" following his postgraduate studies in Lancaster.
Mr Bidve was described as "clever and sporty", and a fan of football, supporting Manchester United.
Mr Sonawane added that the family have received lots of messages of support, especially from the Indian student community in Lancaster where Mr Bidve was studying for a micro-electronics postgraduate qualification.
They have spoken to some of those who were with Mr Bidve when he was killed and said the eight other students have now returned to the university.
Mr Bidve was in a group of nine male and female Indian students visiting Manchester for a short break over the Christmas holidays.
Police said the students, who had not been drinking, were walking from their hotel towards the city centre and became aware of two men on the other side of the street.
The gunman, a white male in his 20s who was wearing a grey top, walked across the road and engaged the victim in a short conversation before producing the gun and shooting the student at close range to the side of the head.
The killer then ran back across the street before the pair fled on foot towards Asgard Drive and the Ordsall housing estate.
Armed response units were sent to the scene at around 1.35am.
Mr Bidve collapsed to the ground and died in hospital a short time later.
The student's father, Subhash, a branch manager with an energy company, has called on the UK Government to speed up the process of returning his son's body to his home country.
In a message on his son's Facebook wall, he said: "Please help in getting Anuj Bidve's dead body back to India. He was shot in Manchester today morning. We appeal to all his friends in UK and India to help spread the word. Please help us get Anuj back. Please share this message."
Mr Sonawane said he had tried to contact the Indian High Commission in London but it has been shut for Christmas.
A spokesman for the commission said today that officers had visited Manchester for talks with the police and the body would be released for repatriation when the "investigation requirements" were completed.
Last night Mr Bidve's family paid tribute to a "loving son".
In a statement issued through GMP, they said: "Anuj was a loving son, a super caring brother and first and forever a friend for many.
"He valued his relationships and put them over anything else. He was the first person you could call when in trouble.
"Anuj loved to travel and loved his food. He loved to cook.
"He could easily bring a smile on anyone's face with his innocent jokes.
"He loved playing football. He enjoyed his time and life and is an inspiration for us to live life to its fullest.
"His passing will not only leave a void in our lives, but in the hearts of all those who knew him."
Professor Bob McKinlay, deputy vice chancellor of Lancaster University, said it will offer Mr Bidve's family help with the costs of repatriation of the student's body.
Anyone with information is asked to call the incident room on 0161 856 5448 or Crimestoppers on 0800 555 111
PA
